tag:blogger.com,1999:blog-2879676981915371822024-03-13T00:21:58.668-07:00Zeta 137zibhttp://www.blogger.com/profile/17401589710495280112noreply@blogger.comBlogger18125tag:blogger.com,1999:blog-287967698191537182.post-65145922941047564402021-05-23T12:32:00.004-07:002021-05-23T12:49:08.021-07:0017. Benford law<p class="MsoNormal" style="line-height: normal; margin-bottom: 0in; margin-left: 3.0in; margin-right: 0in; margin-top: 0in; margin: 0in 0in 0in 3in; text-align: justify;"><span face=""Arial",sans-serif" lang="IT" style="font-size: 12pt;">Benford's law (binary
numbering): <o:p></o:p></span></p>
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0in; margin-left: 3.0in; margin-right: 0in; margin-top: 0in; margin: 0in 0in 0in 3in; text-align: justify;"><span face=""Arial",sans-serif" lang="IT" style="font-size: 12pt;">for any number greater
than zero, <o:p></o:p></span></p>
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0in; margin-left: 3.0in; margin-right: 0in; margin-top: 0in; margin: 0in 0in 0in 3in; text-align: justify;"><span face=""Arial",sans-serif" style="font-size: 12pt; mso-ansi-language: EN-US;">"1"
appears as the first digit in<o:p></o:p></span></p>
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0in; margin-left: 3.0in; margin-right: 0in; margin-top: 0in; margin: 0in 0in 0in 3in; text-align: justify;"><span face=""Arial",sans-serif" style="font-size: 12pt; mso-ansi-language: EN-US;"><span style="mso-spacerun: yes;"> </span>100% of cases.</span></p>
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0in; text-align: justify;"><span face=""Arial",sans-serif" lang="IT" style="font-size: 14pt;">The
problem was this: consider the first digit in the decimal expansion of 2<sup>n</sup>
for n ≥ 0: 1, 2, 4, 8, 1, 3, 6, 1, 2, 5, 1, 2, 4, ... <o:p></o:p></span></p>
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0in; text-align: justify;"><span face=""Arial",sans-serif" style="font-size: 14pt; mso-ansi-language: EN-US;"><i>Does the digit 7 appear in this sequence?</i></span></p><p class="MsoNormal" style="line-height: normal; margin-bottom: 0in; text-align: justify;"><span face="Arial, sans-serif" style="font-size: 14pt;"><i>Does 7 or 8 appear more frequently? </i></span></p><p class="MsoNormal" style="line-height: normal; margin-bottom: 0in; text-align: justify;"><span face=""Arial",sans-serif" lang="IT" style="font-size: 14pt;"><i>How much more
frequently? </i><o:p></o:p></span></p>
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0in; text-align: justify;"><span face="Arial, sans-serif" style="font-size: 14pt;">At the time I did not know which the correct way was
to solve the problem, but I approached it like this: on a logarithmic scale the
products are transformed into sums, for example multiplying by 2 is equivalent
to adding log (2) and we obtain in sequence: 1, 2, 4, 8, 16, 32, 64, 128, ...</span></p>
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0in; text-align: justify;"><span face=""Arial",sans-serif" style="font-size: 14pt; mso-ansi-language: EN-US;">The first digits of the sequence are: 1, 2, 4, 8, 1,
3, 6, 1, 2, 5, 1, 2, 4, 8, 1, 3, 6, 1, 2, 5, 1, 2, 4, 8, 1, 3, 6, 1, 2, 5, 1,
2, 4, 8, 1, 3, 6, 1, 2, 5, 1, 2, 4, 8, 1, 3, <b>7</b>, 1, 2, 5, 1, 2, 4, 9, 1,
3, <b>7</b>, 1, 2, 5, 1, 2, 4, 9, 1, 3, <b>7</b>, 1, 2, 5, 1, 2, 4, 9, 1, 3, <b>7</b>,
1, 3, 6, 1, 2, 4, 9, 1, 3, <b>7</b>, 1, 3, 6, 1, 2, 4, 9, 1, 3, ...</span></p>
<p align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 0in; text-align: center;"><span face=""Arial",sans-serif" style="font-size: 14pt; mso-ansi-language: EN-US;"><i>Sloane's On-Line Encyclopedia of Integer</i> <i>Sequences</i>
A008952</span></p><p align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 0in; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiR63mZjBqkuUku_iRpoNNw6NpJrQpxuvIQuivqGvX3pLw-Rx4cvj1_keBDWdOcs_PgpLRCM-wrFNplCi_2U1l-yhdqcpz04I3ekv_gzljpExlhnR1DaKWc-5gJFG1j5MYL-7CqiEs7Mb8/s1810/2021-03-05+23_59_39-PowerPoint+Slide+Show++-++New+Microsoft+PowerPoint+Presentation.pptx.png" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="200" data-original-width="1810" height="70" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiR63mZjBqkuUku_iRpoNNw6NpJrQpxuvIQuivqGvX3pLw-Rx4cvj1_keBDWdOcs_PgpLRCM-wrFNplCi_2U1l-yhdqcpz04I3ekv_gzljpExlhnR1DaKWc-5gJFG1j5MYL-7CqiEs7Mb8/w640-h70/2021-03-05+23_59_39-PowerPoint+Slide+Show++-++New+Microsoft+PowerPoint+Presentation.pptx.png" width="640" /></a></p>
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0in; text-align: justify;"><span face="Arial, sans-serif" style="font-size: 14pt;">By definition, position </span><b style="font-family: Arial, sans-serif; font-size: 14pt;">1</b><span face="Arial, sans-serif" style="font-size: 14pt;"> is set to log</span><sub style="font-family: Arial, sans-serif;">10</sub><span face="Arial, sans-serif" style="font-size: 14pt;">
1 = 0, while position </span><b style="font-family: Arial, sans-serif; font-size: 14pt;">10</b><span face="Arial, sans-serif" style="font-size: 14pt;"> to 1 (log</span><sub style="font-family: Arial, sans-serif;">10</sub><span face="Arial, sans-serif" style="font-size: 14pt;"> </span><b style="font-family: Arial, sans-serif; font-size: 14pt;">10</b><span face="Arial, sans-serif" style="font-size: 14pt;"> = 1) and </span><b style="font-family: Arial, sans-serif; font-size: 14pt;">100</b><span face="Arial, sans-serif" style="font-size: 14pt;">
to 2 (log</span><sub style="font-family: Arial, sans-serif;">10</sub><span face="Arial, sans-serif" style="font-size: 14pt;"> </span><b style="font-family: Arial, sans-serif; font-size: 14pt;">100</b><span face="Arial, sans-serif" style="font-size: 14pt;"> = 2), the intermediate positions are at:</span></p>
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0in; text-align: justify;"><span face=""Arial",sans-serif" style="font-size: 14pt; mso-ansi-language: EN-US;">log<sub>10</sub> <b>2</b> = 0.3010, log<sub>10</sub> <b>3</b>
= 0.4771, log<sub>10</sub> <b>4</b> = 0.6020,…, log<sub>10</sub> <b>9</b> =
0.9542</span></p>
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0in; text-align: justify;"><span face=""Arial",sans-serif" style="font-size: 14pt; mso-ansi-language: EN-US;">Each interval I (<b>m</b>) of numbers starting with
the digit <b>m</b> is between log (<b>m</b>) and log (<b>m + 1</b>), so the
various intervals hold:</span></p>
<p align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 0in; text-align: center;"><span face=""Arial",sans-serif" lang="IT" style="font-size: 14pt;">log
(<b>m + 1</b>) - log (<b>m</b>) = log (<b>(m + 1) / m</b>) = log (<b>1 + 1/m</b>)<o:p></o:p></span></p>
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0in; text-align: justify;"><span face="Arial, sans-serif" style="font-size: 14pt;">Returning to the problem, the digit </span><b style="font-family: Arial, sans-serif; font-size: 14pt;">7</b><span face="Arial, sans-serif" style="font-size: 14pt;"> appears
log (1 + 1/7) = 0.05799 = 5.8% and to see it appear we have to wait 2</span><sup style="font-family: Arial, sans-serif;">46</sup><span face="Arial, sans-serif" style="font-size: 14pt;">
= 70368744177664.</span></p>
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0in; text-align: justify;"><span face=""Arial",sans-serif" style="font-size: 14pt; mso-ansi-language: EN-US;">The second question can be answered that it appears
more 7, and <o:p></o:p></span></p>
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0in; text-align: justify;"><span face=""Arial",sans-serif" style="font-size: 14pt; mso-ansi-language: EN-US;">for n > 209, the frequency f (7) of the digit 7 is
greater than that of 8:</span></p>
<p align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 0in; text-align: center;"><span face=""Arial",sans-serif" style="font-size: 14pt; mso-ansi-language: EN-US;">f (7) / f (8) tends to log<sub>10</sub> (1 + 1/7) /
log<sub>10</sub> (1 + 1/8) = 1.133706496<o:p></o:p></span></p>
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0in; text-align: justify;"><span face="Arial, sans-serif" lang="IT" style="font-size: 14pt;">Note:
</span><span lang="IT" style="font-family: "Comic Sans MS"; font-size: 14pt;">I(<b>1</b>) = I(<b>2</b>) + I(<b>3</b>) = I(<b>4</b>)
+ I(<b>5</b>) + I(<b>6</b>) + I(<b>7</b>)</span></p>
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0in; text-align: justify;"><span face=""Arial",sans-serif" style="font-size: 14pt; mso-ansi-language: EN-US;">Benford's law arises from the observation that in many
collections of numbers, eg. mathematical tables, real-life data, or
combinations thereof in various units of measurement, the initial significant
digits are not evenly distributed, as you might expect, but are larger for the
smaller digits. <o:p></o:p></span></p>
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0in; text-align: justify;"><span face=""Arial",sans-serif" style="font-size: 14pt; mso-ansi-language: EN-US;">It claims that the significant figures in many data
sets follow a logarithmic distribution. In its most common formulation, it is
also known as the "law of the first digit" and is named</span><span face=""Arial",sans-serif" lang="IT" style="font-size: 14pt;"> after the
American physicist <b>Frank Benford</b> (1883-1948) who formulated it in 1938,
although it had already been highlighted by the American astronomer <b>Simon
Newcomb</b> (1835 -1909) in 1881. <o:p></o:p></span></p>
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0in; text-align: justify;"><span face=""Arial",sans-serif" style="font-size: 14pt; mso-ansi-language: EN-US;">Benford observed that the logarithmic tables, used at
the time for calculations, had very crumpled first pages and therefore deduced
that the numbers beginning with 1 occurred more often than those beginning for
the other digits. <o:p></o:p></span></p>
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0in; text-align: justify;"><span face=""Arial",sans-serif" style="font-size: 14pt; mso-ansi-language: EN-US;">This distribution has a characteristic property known
as "scale invariance" and is often used to discover
"forged" data. <o:p></o:p></span></p>
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0in; text-align: justify;"><span face=""Arial",sans-serif" style="font-size: 14pt; mso-ansi-language: EN-US;">If you were to falsify numbers, make sure that the
number 1 appears in about 30% of the cases, 2 in 17% and so on. <o:p></o:p></span></p>
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0in; text-align: justify;"><span face=""Arial",sans-serif" style="font-size: 14pt; mso-ansi-language: EN-US;">For a given number of digits, it is possible to
calculate the probability of encountering a number starting with the digit
string n of that length. Below what is reported in Wikipedia under
"<i>Benford's law</i>". <o:p></o:p></span></p><p class="MsoNormal" style="line-height: normal; margin-bottom: 0in; text-align: justify;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgSKG1bs9qsnP2s7kZKXJFK148Q7kTK0_AK2CZCe2HTBDXH-iDupQrLFxnVcdGtT06izyPP7acOl2pB8nfyr4p7Gc8O6H6tjCiydqDkyBCFbsDKJeVfoUZe6-J57fpEZrVnyYngtHJ4OVw/s1057/2021-03-18+00_17_48-Benford%2527s+law+-+Wikipedia+and+4+more+pages+-+Work+-+Microsoft%25E2%2580%258B+Edge.png" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em; text-align: center;"><img border="0" data-original-height="721" data-original-width="1057" height="436" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgSKG1bs9qsnP2s7kZKXJFK148Q7kTK0_AK2CZCe2HTBDXH-iDupQrLFxnVcdGtT06izyPP7acOl2pB8nfyr4p7Gc8O6H6tjCiydqDkyBCFbsDKJeVfoUZe6-J57fpEZrVnyYngtHJ4OVw/w640-h436/2021-03-18+00_17_48-Benford%2527s+law+-+Wikipedia+and+4+more+pages+-+Work+-+Microsoft%25E2%2580%258B+Edge.png" width="640" /></a></p>
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0in; text-align: justify;"><span face=""Arial",sans-serif" style="font-size: 14pt; mso-ansi-language: EN-US;"><br /></span></p><p class="MsoNormal" style="line-height: normal; margin-bottom: 0in; text-align: justify;"><span face=""Arial",sans-serif" style="font-size: 14pt; mso-ansi-language: EN-US;"><br /></span></p><p class="MsoNormal" style="line-height: normal; margin-bottom: 0in; text-align: justify;"><span face=""Arial",sans-serif" style="font-size: 14pt; mso-ansi-language: EN-US;">The distribution of the n-th digit, as <b>n</b> increases,
quickly approaches a uniform distribution with 10% for each of the ten digits,
as shown below:</span></p><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh_a6PUoCLruzglgkQzozrKcP0o91nsBSWyirJy_Mwn-mOraIAdmOBZ5ENK9cCxFvfL8xtbleBnbWLFuefNVRuvVCdBRgzjsPa4pjDYsON_sgvTN5x7JAsxXlV1wnARnN8xWCuauXRRg3U/s1311/2021-03-18+00_14_34-Benford_Primi_Binari.xlsx+-+Excel.png" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="394" data-original-width="1311" height="192" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh_a6PUoCLruzglgkQzozrKcP0o91nsBSWyirJy_Mwn-mOraIAdmOBZ5ENK9cCxFvfL8xtbleBnbWLFuefNVRuvVCdBRgzjsPa4pjDYsON_sgvTN5x7JAsxXlV1wnARnN8xWCuauXRRg3U/w640-h192/2021-03-18+00_14_34-Benford_Primi_Binari.xlsx+-+Excel.png" width="640" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjWSEwyJ4tjsysQJ-4mAJytF5IOwYqpL3JFdCl3RWL5iSHjVvAECo2MgQNrOm2_AxEBqhVA2zBUp3xWxR9OE2PX9pGnd98OkSzbNCgjSroXG_N2CuT3R0mxaHe5nUmeG2Uc1GLUYmS5LEE/s818/2021-03-17+22_56_17-Benford_Primi_Binari.xlsx+-+Excel.png" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="585" data-original-width="818" height="458" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjWSEwyJ4tjsysQJ-4mAJytF5IOwYqpL3JFdCl3RWL5iSHjVvAECo2MgQNrOm2_AxEBqhVA2zBUp3xWxR9OE2PX9pGnd98OkSzbNCgjSroXG_N2CuT3R0mxaHe5nUmeG2Uc1GLUYmS5LEE/w640-h458/2021-03-17+22_56_17-Benford_Primi_Binari.xlsx+-+Excel.png" width="640" /></a></div></div><br /><p class="MsoNormal" style="line-height: normal; margin-bottom: 0in; text-align: justify;"><br /></p><p class="MsoNormal" style="line-height: normal; margin-bottom: 0in; text-align: justify;"><br /></p><p class="MsoNormal" style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px; line-height: normal; margin-bottom: 0in;"><a href="http://www.benfordonline.net/" style="color: #33aaff; font-size: 14pt;">Benford Online Bibliography</a></p><p class="MsoNormal" style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px; line-height: normal; margin-bottom: 0in;"><span lang="IT"><span lang="EN-US" style="font-size: 14pt;"><a href="https://mathworld.wolfram.com/BenfordsLaw.html" style="color: #2288bb; text-decoration-line: none;">https://mathworld.wolfram.com/BenfordsLaw.html</a></span></span><span style="font-size: 14pt;"><o:p></o:p></span></p><p class="MsoNormal" style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px; line-height: normal; margin-bottom: 0in;"><span lang="IT" style="font-size: medium;"><a href="https://www.mathpages.com/home/kmath302/kmath302.htm" style="color: #2288bb; text-decoration-line: none;">Benford's Law (mathpages.com)</a></span></p><p class="MsoNormal" style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px; line-height: normal; margin-bottom: 0in;"><span lang="IT"><a href="http://oeis.org/wiki/Index_to_OEIS:_Section_Be#Benford" style="color: #33aaff;"><span lang="EN-US" style="font-size: 14pt;">Index to OEIS: Section Be - OeisWiki</span></a></span><span style="font-size: 14pt;"><o:p></o:p></span></p><p class="MsoNormal" style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px; line-height: normal; margin-bottom: 0in;"><span lang="IT"><a href="https://oeis.org/A008952" style="color: #2288bb; text-decoration-line: none;"><span lang="EN-US" style="font-size: 14pt;">https://oeis.org/A008952</span></a></span></p>zibhttp://www.blogger.com/profile/17401589710495280112noreply@blogger.com0tag:blogger.com,1999:blog-287967698191537182.post-62282165792534605152021-03-08T14:08:00.002-08:002021-03-08T14:12:44.210-08:0016. Binary code<p><span face="Arial, sans-serif" style="font-size: 12pt; text-align: justify;">If you divide a square of unit area into 2 equal
parts, each will have a value of 0.5.</span></p>
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0in; text-align: justify;"><span face=""Arial",sans-serif" style="font-size: 12pt; mso-ansi-language: EN-US;">By dividing one of these into 2 equal parts and continuing
to dissect in the same way, the values will be obtained: 0.5; 0.25; 0.125; 0.0625;
0.03125; ... <o:p></o:p></span></p>
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0in; text-align: justify;"><span face=""Arial",sans-serif" style="font-size: 12pt; mso-ansi-language: EN-US;">In other words, a geometric series is obtained whose
sum converges to 1: <o:p></o:p></span></p><p class="MsoNormal" style="line-height: normal; margin-bottom: 0in; text-align: justify;"><span face=""Arial",sans-serif" style="font-size: 12pt; mso-ansi-language: EN-US;"><br /></span></p>
<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjCd5oTN0k2TjuHAPV2ggH8Tfn5RqbCzxcMIFQ-z7Ys9WV6IUfqN_YApHw_I-rCpy47IlxQI-2or20qDml7xJ_Yy9AT0iTD70apjVONv6p6M88nt3Q1mMOmTfEIM-mG4qkMflv0I3kgJBA/s687/2021-03-07+00_49_56-Benford.pptx+-+PowerPoint.png" style="font-family: Arial, sans-serif; font-size: 12pt; margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" data-original-height="687" data-original-width="687" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjCd5oTN0k2TjuHAPV2ggH8Tfn5RqbCzxcMIFQ-z7Ys9WV6IUfqN_YApHw_I-rCpy47IlxQI-2or20qDml7xJ_Yy9AT0iTD70apjVONv6p6M88nt3Q1mMOmTfEIM-mG4qkMflv0I3kgJBA/s320/2021-03-07+00_49_56-Benford.pptx+-+PowerPoint.png" /></a></div><p class="MsoNormal" style="line-height: normal; margin-bottom: 0in; text-align: justify;"><span face="Arial, sans-serif" style="font-size: 12pt;">A scale with available weights: 0.5 lb; 0.25 lb; etc.,
could measure any object less than 1 pound. What we are doing is using a
numbering system based on 2.</span></p>
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0in; text-align: justify;"><span face=""Arial",sans-serif" style="font-size: 12pt; mso-ansi-language: EN-US;">Probably everyone has tried their hand at binary
numbers, but few people will have used this base to deal with decimals, here we
propose an example:</span><span face="Arial, sans-serif" style="background-color: whitesmoke; font-size: 12pt;"> </span></p>
<p align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 0in; text-align: center;"><span face=""Arial",sans-serif" style="font-size: 14pt; mso-ansi-language: EN-US;">3,14159265358979323846… = 11,00100100001111110110101</span><span face=""Arial",sans-serif" style="font-size: 12pt; mso-ansi-language: EN-US;">...</span></p>
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0in; text-align: justify;"><span face=""Arial",sans-serif" style="font-size: 12pt; mso-ansi-language: EN-US;">The whole part will probably be clear to everyone,
while the other part will require a minimum of reasoning. <o:p></o:p></span></p>
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0in; text-align: justify;"><span face=""Arial",sans-serif" style="font-size: 12pt; mso-ansi-language: EN-US;">These tables can perhaps help:</span></p><p class="MsoNormal" style="line-height: normal; margin-bottom: 0in; text-align: justify;"><span face=""Arial",sans-serif" style="font-size: 12pt; mso-ansi-language: EN-US;"><br /></span></p>
<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi5IgvUtNJiVbqmDy3ZC_cEixvZP3krdTLuDh1tazCvo5TFPQ_dr940Ns4qAi26nT8uPTRdDdrXbg1NnLrBeg6gKAAG-VxzX7_Jvw01UCqHlQqZWsKnSL_eE03ixZ02jQQdtpQc2ZHHL3s/s933/2021-03-08+22_41_57-Benford_Primi_Binari.xlsx+-+Excel.png" style="clear: left; float: left; font-family: Arial, sans-serif; font-size: 16px; margin-bottom: 1em; margin-right: 1em; text-align: center;"><img border="0" data-original-height="605" data-original-width="933" height="416" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi5IgvUtNJiVbqmDy3ZC_cEixvZP3krdTLuDh1tazCvo5TFPQ_dr940Ns4qAi26nT8uPTRdDdrXbg1NnLrBeg6gKAAG-VxzX7_Jvw01UCqHlQqZWsKnSL_eE03ixZ02jQQdtpQc2ZHHL3s/w640-h416/2021-03-08+22_41_57-Benford_Primi_Binari.xlsx+-+Excel.png" width="640" /></a></div><p class="MsoNormal" style="line-height: normal; margin-bottom: 0in; text-align: justify;"><span face=""Arial",sans-serif" style="font-size: 12pt; mso-ansi-language: EN-US;"> </span></p>
<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi6E1zJK1QfjZ-QZUJiQx_kAqDo9rj4k-dOzjXIJ3QXJpYOSd5xljGoaeexJ0enf-Vkmvm98YrDYTltWlBYURUSFB9XVqShqLvkqYQguYeN6RisRH9fHPydarM-JwpGRqpF94od8cQ_dnA/s755/2021-03-07+14_53_15-Benford_Primi_Binari.xlsx+-+Excel.png" style="font-family: Arial, sans-serif; font-size: 16px; margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" data-original-height="755" data-original-width="661" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi6E1zJK1QfjZ-QZUJiQx_kAqDo9rj4k-dOzjXIJ3QXJpYOSd5xljGoaeexJ0enf-Vkmvm98YrDYTltWlBYURUSFB9XVqShqLvkqYQguYeN6RisRH9fHPydarM-JwpGRqpF94od8cQ_dnA/w350-h400/2021-03-07+14_53_15-Benford_Primi_Binari.xlsx+-+Excel.png" width="350" /></a></div><p class="MsoNormal" style="line-height: normal; margin-bottom: 0in; text-align: justify;"><span face="Arial, sans-serif" style="font-size: 12pt;">Here are some examples of pi in various numbering
systems.</span></p>
<span face=""Arial",sans-serif" style="font-size: 12pt; line-height: 115%; mso-ansi-language: EN-US; mso-bidi-language: AR-SA; mso-fareast-font-family: Calibri; mso-fareast-language: EN-US;">As an exercise, you can see how simple the
transition from base 2 to base 4 and from base 4 to base 16 is.</span><div><span face=""Arial",sans-serif" style="font-size: 12pt; line-height: 115%; mso-ansi-language: EN-US; mso-bidi-language: AR-SA; mso-fareast-font-family: Calibri; mso-fareast-language: EN-US;"><br /></span></div><div><span face=""Arial",sans-serif" style="font-size: 12pt; line-height: 115%; mso-ansi-language: EN-US; mso-bidi-language: AR-SA; mso-fareast-font-family: Calibri; mso-fareast-language: EN-US;"><p class="MsoNormal" style="line-height: normal; margin-bottom: 0in;"><span face=""Arial",sans-serif" lang="IT" style="mso-bidi-font-size: 10.0pt;">base-2<span style="mso-spacerun: yes;"> </span>(binary code) :<span style="mso-spacerun: yes;"> </span></span><span face=""Arial",sans-serif" lang="IT" style="font-size: 12pt; mso-bidi-font-size: 11.0pt;">11.0010010000111111011010101000100010000101101000110000100011010011000100110001100110001010001011100000
</span><span face=""Arial",sans-serif" lang="IT" style="mso-bidi-font-size: 10.0pt;">…<o:p></o:p></span></p>
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0in;"><span style="font-size: 12pt;">base-4 :
3.0210033312222020201122030020310301030121202202320003130013031010221000210320020202212133030131000020
…</span></p>
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0in;"><span style="font-size: 12pt;">base-16 </span><span style="font-size: 12pt;"> </span><span style="font-size: 12pt;">(exadecimal code): 3.243F6A8885A308D313198A2E03707344A4093822299F31D0082EFA98EC4E6C89452821E638D01377BE5466CF34E90C6CC0AC
…</span></p>
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0in;"><span face=""Arial",sans-serif" lang="IT" style="mso-bidi-font-size: 10.0pt;"><o:p> </o:p></span></p>
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0in;"><span face=""Arial",sans-serif" lang="IT" style="mso-bidi-font-size: 10.0pt;">base-3 : 10.0102110122220102110021111102212222201112012121212001211001001012220222120120121112101210112002201202
...<o:p></o:p></span></p>
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0in;"><span style="font-size: 12pt;">base-5 :
3.0323221430334324112412240414023142111430203100220034441322110104033213440043244401441042334133011323
...</span></p>
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0in;"><span style="font-size: 12pt;">base-6 :
3.0503300514151241052344140531253211023012144420041152525533142033313113553513123345533410015154344401
...</span></p>
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0in;"><span style="font-size: 12pt;">base-7 : 3.0663651432036134110263402244652226643520650240155443215426431025161154565220002622436103301443233631
...</span></p>
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0in;"><span style="font-size: 12pt;">base-8 :
3.1103755242102643021514230630505600670163211220111602105147630720020273724616611633104505120207461615
...</span></p><p class="MsoNormal" style="line-height: normal; margin-bottom: 0in;"><span style="font-size: 12pt;">base-9 :
3.12418812407442788645177761731035828516545353462652301126321450283864034354163303086781327871588
...</span></p>
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0in;"><span face=""Arial",sans-serif" style="mso-ansi-language: EN-US; mso-bidi-font-size: 10.0pt;"><o:p> </o:p></span></p>
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0in;"><span lang="IT"><a href="https://en.wikipedia.org/wiki/Binary_number"><span lang="EN-US" style="font-family: "Comic Sans MS"; mso-ansi-language: EN-US; mso-bidi-font-family: Arial;">Binary number - Wikipedia</span></a></span><span style="font-family: "Comic Sans MS"; mso-ansi-language: EN-US; mso-bidi-font-family: Arial;"><o:p></o:p></span></p>
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0in; text-align: justify;"><span lang="IT"><a href="http://zibalsc.blogspot.com/search/label/pi"><span style="font-family: "Comic Sans MS"; mso-bidi-font-family: Arial;">Zibaldone
Scientifico: pi (zibalsc.blogspot.com)</span></a></span><span class="MsoHyperlink"><span lang="IT" style="font-family: "Comic Sans MS"; mso-bidi-font-family: Arial;"><o:p></o:p></span></span></p>
<p class="MsoNormal" style="line-height: normal; margin-bottom: 0in; text-align: justify;"><span class="MsoHyperlink"><span lang="IT" style="font-family: "Comic Sans MS"; mso-bidi-font-family: Arial;"><a href="https://robertlovespi.net/2014/06/09/the-beginning-of-the-number-pi-in-binary-through-hexadecimal-etc/">https://robertlovespi.net/2014/06/09/the-beginning-of-the-number-pi-in-binary-through-hexadecimal-etc/</a></span></span><span face=""Arial",sans-serif" lang="IT" style="font-size: 12pt;"><o:p></o:p></span></p><br /></span></div>zibhttp://www.blogger.com/profile/17401589710495280112noreply@blogger.com0tag:blogger.com,1999:blog-287967698191537182.post-14631321461844447342019-07-26T13:49:00.000-07:002019-07-26T13:49:03.377-07:0015. Oloid<div class="separator" style="clear: both; text-align: left;">
<span lang="EN" style="color: #222222; font-family: Arial, sans-serif; font-size: 12pt; text-align: justify;">Discovered by <b><a href="https://en.wikipedia.org/wiki/Paul_Schatz" target="_blank">Paul Schatz</a></b>
in 1929, the <b>Oloid</b> is defined as the
convex hull of 2 circumferences of </span><span lang="EN" style="color: #222222; font-family: "Comic Sans MS"; font-size: 14pt; text-align: justify;">radius R</span><span lang="EN" style="color: #222222; font-family: "Comic Sans MS"; font-size: 12pt; text-align: justify;"> </span><span lang="EN" style="color: #222222; font-family: Arial, sans-serif; font-size: 12pt; text-align: justify;">equal
to each other, arranged on two orthogonal planes and such that each of the 2
steps to the center of the other.</span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgFBSqFjDhKU6kv0MzBFDO9mkLPZ1KX9uloCRPZv4V4yQwGd_zeGJwwXxdMWrewFirgpDo6x5SCz9XCZx5KuU6-cKfWeo5LcQOoiwhXHnMxB0efzx1FNnJA0CiNyVt431Eor5gf9b7jwrk/s1600/2019-07-24+23_58_05-Start.jpg" imageanchor="1" style="clear: left; float: left; font-family: "Times New Roman"; font-size: medium; margin-bottom: 1em; margin-right: 1em; text-align: center;"><img border="0" data-original-height="599" data-original-width="932" height="204" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgFBSqFjDhKU6kv0MzBFDO9mkLPZ1KX9uloCRPZv4V4yQwGd_zeGJwwXxdMWrewFirgpDo6x5SCz9XCZx5KuU6-cKfWeo5LcQOoiwhXHnMxB0efzx1FNnJA0CiNyVt431Eor5gf9b7jwrk/s320/2019-07-24+23_58_05-Start.jpg" width="320" /></a><span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12.0pt; mso-ansi-language: EN-US;"></span></div>
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<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12.0pt; mso-ansi-language: EN-US;"><br /></span></div>
This <b>grooved surface</b>
has many interesting properties (for example, all of its generating lines have
the same length).<o:p></o:p><br />
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<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12.0pt; mso-ansi-language: EN-US;">The <b>Oloid</b> has a shape
particularly suitable for the mixing of fluids.<o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: Arial, sans-serif; font-size: 12pt;">The Oloid surface </span><b><span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 14.0pt; mso-ansi-language: EN-US; mso-bidi-font-size: 12.0pt;">S</span></b><span lang="EN-US" style="font-family: Arial, sans-serif; font-size: 12pt;"> is equal to the area of the sphere of </span><span lang="EN" style="color: #222222; font-family: "Comic Sans MS"; font-size: 14pt;">radius <b>R</b></span><span lang="EN-US" style="font-family: Arial, sans-serif; font-size: 12pt;">;</span></div>
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<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12.0pt; mso-ansi-language: EN-US;">while by means of
numerical calculation, it can estimate the volume </span><b style="mso-bidi-font-weight: normal;"><span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 14.0pt; mso-ansi-language: EN-US; mso-bidi-font-size: 12.0pt;">V</span></b><span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12.0pt; mso-ansi-language: EN-US;">:<o:p></o:p></span></div>
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<span style="font-family: arial, sans-serif; font-size: 12pt;">The Oloid is the only three-dimensional shape that can rotate on </span><i style="font-family: Arial, sans-serif; font-size: 12pt;">its entire surface</i><span style="font-family: arial, sans-serif; font-size: 12pt;">.</span></div>
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<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12.0pt; mso-ansi-language: EN-US;">When is made to roll on
a flat horizontal surface, the Oloid moves uniformly because the distance from
its center of mass at the surface is almost constant.<o:p></o:p></span></div>
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<span style="font-family: "arial" , sans-serif; font-size: 12pt;"></span></div>
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<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12pt;">Prof. Dr. Peter Berger: </span><span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12pt;"><a href="http://www.prof-dr-berger.de/_figurint-tork.htm" target="_blank"><span style="vertical-align: inherit;">http://www.prof-dr-berger.de/_figurint-tork.htm</span></a></span></div>
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<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0pt; text-align: center;">
<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 14pt;"> </span><span style="font-family: arial, sans-serif; font-size: 12pt;"><span style="vertical-align: inherit;"><span style="vertical-align: inherit;">The dell'Oloide equation corresponds to an </span></span><b><span style="vertical-align: inherit;"><span style="vertical-align: inherit;">algebraic surface of order 8</span></span></b><span style="vertical-align: inherit;"><span style="vertical-align: inherit;"> :</span></span></span></div>
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<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12.0pt; mso-ansi-language: EN-US;"></span></div>
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<span style="font-family: "arial" , sans-serif; font-size: 12pt;"><a href="http://dictionnaire.sensagent.leparisien.fr/Oloide/es-es/" target="_blank">http://dictionnaire.sensagent.leparisien.fr/Oloide/es-es/</a></span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj2gFSo9I29MtZBw9cv-HXSYdtpi3qKWxPGs2r5iRsx-o98C4NCXz2bGxKWWNjw0MHIt5WxjCx7tnofjrJnu063Rw35FOqeff25zRUBV0YZq2Z8fX_78WHJN3hmYeh3m-Xqmv5HmBuqH3M/s1600/2019-07-24+22_52_47-Start.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="148" data-original-width="863" height="109" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj2gFSo9I29MtZBw9cv-HXSYdtpi3qKWxPGs2r5iRsx-o98C4NCXz2bGxKWWNjw0MHIt5WxjCx7tnofjrJnu063Rw35FOqeff25zRUBV0YZq2Z8fX_78WHJN3hmYeh3m-Xqmv5HmBuqH3M/s640/2019-07-24+22_52_47-Start.jpg" width="640" /></a></div>
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<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12.0pt; mso-ansi-language: EN-US;"><o:p> </o:p></span><a href="https://demonstrations.wolfram.com/MovementOfAnOloid/" style="font-family: arial, sans-serif; font-size: 12pt; text-align: center;" target="_blank"><span style="vertical-align: inherit;">https://demonstrations.wolfram.com/MovementOfAnOloid/</span></a></div>
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<a href="https://en.wikipedia.org/wiki/Oloid"><span style="font-family: "arial" , sans-serif; font-size: 12pt;">https://en.wikipedia.org/wiki/Oloid</span></a><span style="font-family: "arial" , sans-serif; font-size: 12pt;"><o:p></o:p></span></div>
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<a href="https://www.youtube.com/watch?v=GM3_JuFgJ2E"><span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 14pt;">https://www.youtube.com/watch?v=GM3_JuFgJ2E</span></a><span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 14pt;"><o:p></o:p></span></div>
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<a href="http://www.dimnp.unipi.it/gabiccini-m/robotica.html" target="_blank"><span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 14pt;">http://www.dimnp.unipi.it/gabiccini-m/robotica.html</span></a></div>
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<a href="http://www.ucke.de/christian/physik/ftp/lectures/udine2003.pdf" target="_blank"><span style="font-family: "arial" , sans-serif; font-size: 14pt;">http://www.ucke.de/christian/physik/ftp/lectures/udine2003.pdf</span></a><span style="font-family: "arial" , sans-serif; font-size: 14pt;"><o:p></o:p></span></div>
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<br /></div>
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<span style="background: yellow; font-family: "Arial",sans-serif; font-size: 16pt; mso-bidi-font-size: 12.0pt; mso-highlight: yellow;"><a href="http://www.heldermann-verlag.de/jgg/jgg01_05/jgg0113.pdf"><span style="vertical-align: inherit;"><span class="goog-text-highlight" style="vertical-align: inherit;">http://www.heldermann-verlag.de/jgg/jgg01_05/jgg0113.pdf</span></span></a></span><span style="font-family: "Arial",sans-serif; font-size: 16pt; mso-bidi-font-size: 12.0pt;"><o:p></o:p></span></div>
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<a href="http://www.mathcurve.com/surfaces.gb/orthobicycle/orthobicycle.shtml" target="_blank"><span style="font-family: "arial" , sans-serif; font-size: 14pt;">http://www.mathcurve.com/surfaces.gb/orthobicycle/orthobicycle.shtml</span></a></div>
</div>
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<a href="http://www.mathcurve.com/surfaces/orthobicycle/orthobicycle.shtml" target="_blank"><span style="font-family: "arial" , sans-serif; font-size: 14pt;">http://www.mathcurve.com/surfaces/orthobicycle/orthobicycle.shtml</span></a><span style="font-family: "arial" , sans-serif; font-size: 14pt;"><o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12.0pt; mso-ansi-language: EN-US;"><o:p><br /></o:p></span></div>
<br />zibhttp://www.blogger.com/profile/17401589710495280112noreply@blogger.com0tag:blogger.com,1999:blog-287967698191537182.post-83273538982812801772019-06-16T14:04:00.003-07:002019-06-16T14:10:45.949-07:0014. Geodesics on a Polyhedron<span style="font-family: arial, sans-serif; font-size: 12pt; text-align: justify;">In mathematics, a </span><b style="font-family: arial, sans-serif; font-size: 12pt; text-align: justify;">geodesic</b><span style="font-family: arial, sans-serif; font-size: 12pt; text-align: justify;"> is the </span><b style="font-family: arial, sans-serif; font-size: 12pt; text-align: justify;">shortest curve</b><span style="font-family: arial, sans-serif; font-size: 12pt; text-align: justify;"> that connects two points. The geodesics in the plane
are straight lines, on a sphere they are the arcs of maximum circle. At each
point the main normal to it coincides with the normal to the surface at that
point; that is, the osculating plane line is normal to the surface at that
point.</span><br />
<br />
<div style="line-height: normal; margin: 0in 0in 0pt; text-align: justify;">
<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12pt;">What are the shortest
paths on polyhedron?</span></div>
<br />
<div style="line-height: normal; margin: 0in 0in 0pt; text-align: justify;">
<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12pt;">A polyhedron is a solid
bounded by a finite number of polygonal plane faces and, if 2 points lie on the
same face, the shortest path is the straight line segment. If instead the 2
points lie on adjacent faces, then the shortest path belongs to a straight line
on the developed surface (open on the plane).</span></div>
<span style="font-family: arial, sans-serif; font-size: 12pt; text-align: justify;">Prolonging this
segment, beyond its extremes, we can join all the pairs of points on the
surface of the polyhedron.</span><br />
<br />
<div style="line-height: normal; margin: 0in 0in 0pt; text-align: justify;">
<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12pt;">A cubic-edged planet
with an edge equal to the <b style="mso-bidi-font-weight: normal;">diameter of
the Earth</b> (12,746 km) would have a diagonal of 22,077 km. The difference
between the highest and the deepest point would be 4,665 km (4,665,000 meters).
On Earth between <b style="mso-bidi-font-weight: normal;">Mount Everest</b> (8,844
m) and the <b style="mso-bidi-font-weight: normal;">Mariana Trench</b> (-10,911
m) there is a difference in height of 19,755 m. Assuming 10,000 m deep oceans,
the 8 vertices would be mountains 500 times the height of Mount Everest. To go
from one of the 6 faces to the other you should overcome passes with an
altitude equal to 300 times Mount Everest. The oceans could be at most 6.</span><br />
<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12pt;"><br /></span>
<span style="font-family: "arial";"><br /></span>
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<div style="border-image: none;">
<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12pt;">On the <b style="mso-bidi-font-weight: normal;">Carlos
Furuti</b> website, you can find many <b style="mso-bidi-font-weight: normal;">Cartographic
Projections</b>:</span></div>
</div>
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<div style="border-image: none;">
<a href="http://www.progonos.com/furuti/MapProj/Normal/ProjPoly/Foldout/Cube/cube.html"><span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12pt;"><span style="color: blue;">http://www.progonos.com/furuti/MapProj/Normal/ProjPoly/Foldout/Cube/cube.html</span></span></a></div>
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<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12pt;">and you can even print a <i style="mso-bidi-font-style: normal;">cubic Earth</i>.</span></div>
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<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12pt;">The great British mathematician <b style="mso-bidi-font-weight: normal;">Henry Ernest Dudeney</b> (1857 - 1930) invented hundreds of
mathematical games <span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12pt;">a couple will be reported dealing with geodesics on
polyhedron.</span></span></div>
<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12pt;"><span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12pt;"></span></span><br />
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</span></span></div>
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<b style="mso-bidi-font-weight: normal;"><span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12pt;">THE
JOURNEY OF THE MOSCOW</span></b></div>
</div>
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<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12pt;">A fly, starting from point <b style="mso-bidi-font-weight: normal;">A</b>, can travel the 4 sides of the base of a cubic block in 4
minutes. How long will it take to get from <b style="mso-bidi-font-weight: normal;">A</b>
to the opposite corner <b style="mso-bidi-font-weight: normal;">B</b>?</span></div>
</div>
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<div style="border-image: none;">
<span lang="EN-US" style="font-family: "consolas"; font-size: 14pt;">The
fly would select the path shown by the line in the illustration, which will
require 2.236 minutes. It will not go in the direction indicated by the dotted
line that might seem the suggested one. This path is longer and takes more time
(2.414 minutes).</span><br />
<span lang="EN-US" style="font-family: "consolas"; font-size: 14pt;"><br /></span></div>
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<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12pt;"><br /></span></div>
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<b style="mso-bidi-font-weight: normal;"><span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12pt;"><br /></span></b>
<b style="mso-bidi-font-weight: normal;"><span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12pt;"><br /></span></b>
<b style="mso-bidi-font-weight: normal;"><span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12pt;">THE
SPIDER AND THE FLY</span></b></div>
<br />
<div style="line-height: normal; margin: 0in 0in 0pt; text-align: justify;">
<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12pt;">A rectangular room has
the dimensions shown in the figure (in English measurements). A spider,
indicated with the yellow star, is located in the center of one of the two
bottom walls one foot from the ceiling. A fly, indicated with the dark star, is
instead found on the opposite wall, one foot from the floor. What is the
shortest distance the spider has to travel to reach the fly?</span></div>
<br />
<div style="line-height: normal; margin: 0in 0in 0pt; text-align: justify;">
<span lang="EN-US" style="font-family: "consolas"; font-size: 14pt;">The easiest way to solve the problem is to develop the
room on the floor. In this way the shortest path is the hypotenuse of a right
triangle, equal to 40 feet.</span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhiuRwY2LqjgfLYlNwHvVF9hV3YknWI4LtKE3AgMVc9vD59lAi8N5Lx14HDmVX8gj0R0Bg7YEB9EYrWtKbNjtofuH4z0WJFTeTGjYNYpd4B6KP4bI4uLB5k4LVQ4fq_ue3I-gTMhBOwFcI/s1600/2018-02-10+00_18_55-PowerPoint+Slide+Show+-+%255BFormule_e_Illusioni_1.pptx%255D.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="801" data-original-width="1164" height="440" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhiuRwY2LqjgfLYlNwHvVF9hV3YknWI4LtKE3AgMVc9vD59lAi8N5Lx14HDmVX8gj0R0Bg7YEB9EYrWtKbNjtofuH4z0WJFTeTGjYNYpd4B6KP4bI4uLB5k4LVQ4fq_ue3I-gTMhBOwFcI/s640/2018-02-10+00_18_55-PowerPoint+Slide+Show+-+%255BFormule_e_Illusioni_1.pptx%255D.jpg" width="640" /></a></div>
<span style="font-family: "arial";"><br /></span>
<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12pt;"><br /><br />
</span><br />
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<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12pt;"><br /></span>
<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12pt;">The paradox, in this
case, consists in the fact that the horizontal path might seem shorter, but it
is not difficult to verify that it is equal to 42 feet (2 more).</span></div>
<br />
<br />
<div style="line-height: normal; margin: 0in 0in 0pt;">
<a href="http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Dudeney.html"><span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12pt;"><span style="color: blue;">http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Dudeney.html</span></span></a></div>
<a href="http://mathworld.wolfram.com/SpiderandFlyProblem.html"><span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12pt;"><span style="color: blue;">http://mathworld.wolfram.com/SpiderandFlyProblem.html</span></span></a><br />
<a href="https://pbbmath.weebly.com/blog/the-logic-puzzles-of-henry-dudeney"><span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12pt;"><span style="color: blue;">https://pbbmath.weebly.com/blog/the-logic-puzzles-of-henry-dudeney</span></span></a><br />
<a href="http://www.ilovephilosophy.com/viewtopic.php?f=4&t=181293"><span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12pt;"><span style="color: blue;">http://www.ilovephilosophy.com/viewtopic.php?f=4&t=181293</span></span></a><br />
<a href="https://math.stackexchange.com/questions/292495/gentle-introduction-to-quasi-geodesics"><span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12pt;"><span style="color: blue;">https://math.stackexchange.com/questions/292495/gentle-introduction-to-quasi-geodesics</span></span></a><br />
<a href="http://www.progonos.com/furuti/MapProj/Normal/ProjPoly/projPoly3.html"><span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12pt;"><span style="color: blue;">http://www.progonos.com/furuti/MapProj/Normal/ProjPoly/projPoly3.html</span></span></a><br />
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</div>
zibhttp://www.blogger.com/profile/17401589710495280112noreply@blogger.com0tag:blogger.com,1999:blog-287967698191537182.post-28594448448615621222019-06-12T13:11:00.001-07:002021-02-07T07:12:50.413-08:0013. Squaring the Circle in n-Dimensions<br />
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0in; text-align: justify;">
<span face=""arial" , sans-serif" lang="EN-US" style="font-size: 12pt;">In the previous post we
saw how, starting from the <b style="mso-bidi-font-weight: normal;">Wallis sieve</b>,
we can arrive at a 3D-<i style="mso-bidi-font-style: normal;">cubic</i> <b style="mso-bidi-font-weight: normal;">fractal</b>-like and by assembling 8 of
these cubes we have the same volume as a sphere of unitary radius. Below we
will explain in an analytical way how to build a sieve in 2D, 3D, etc.<o:p></o:p></span></div>
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<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0in; text-align: justify;">
<span face=""arial" , sans-serif" lang="EN-US" style="font-size: 12pt;">For 2D we have a square
divided into <b style="mso-bidi-font-weight: normal;">9 equal parts</b>:<o:p></o:p></span></div>
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<br /></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEheVdiW0fnFdAR0Oc2MF6t4_58ofoJslr9F7WwJu-Sb6nRQ7ZmMqOJ-6tbAIxjonKrqIwVoxjYJl-rDQ04JmO29DQusvMSnyFYAE9H5aQM8rMUQDInkpMiWtzSx0Q0LkjQFlDF3YswK4OE/s1600/2019-06-11+00_53_29-Start.jpg" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em; text-align: center;"><img border="0" data-original-height="275" data-original-width="841" height="208" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEheVdiW0fnFdAR0Oc2MF6t4_58ofoJslr9F7WwJu-Sb6nRQ7ZmMqOJ-6tbAIxjonKrqIwVoxjYJl-rDQ04JmO29DQusvMSnyFYAE9H5aQM8rMUQDInkpMiWtzSx0Q0LkjQFlDF3YswK4OE/s640/2019-06-11+00_53_29-Start.jpg" width="640" /></a></div>
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<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0in; text-align: justify;">
<span face=""arial" , sans-serif" lang="EN-US" style="font-size: 12pt;">The number of elements
(or <i style="mso-bidi-font-style: normal;">cardinality</i>) of the Cartesian
product of <b style="mso-bidi-font-weight: normal;">2</b> sets is the product of
the number of elements of the <b style="mso-bidi-font-weight: normal;">2</b>
sets. <i style="mso-bidi-font-style: normal;">Generalizing</i>, the number of
elements of the Cartesian product of n sets is the product of the number of
elements of each set.<o:p></o:p></span></div>
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<br /></div>
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<span face=""arial" , sans-serif" lang="EN-US" style="font-size: 12pt;">We can give a <b style="mso-bidi-font-weight: normal;">tabular representation</b>, which consists
in writing all the possible pairs enclosed in braces:<o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0in;">
<br /></div>
<div align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 0in; text-align: center;">
<span face=""arial" , sans-serif" lang="EN-US" style="font-size: 12pt;">{(1, 1); (1, 2); (1,
3); (2, 1); (<span style="background: yellow; mso-highlight: yellow;">2, 2</span>);
(2, 3); (3, 1); (3, 2); (3, 3)}<o:p></o:p></span></div>
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<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0in; text-align: justify;">
<span face=""arial" , sans-serif" lang="EN-US" style="font-size: 12pt;">or by means of a <b style="mso-bidi-font-weight: normal;">double entry table</b>, obtained with the
elements of the first set placed in the column and those of the second in the
first row; the boxes contain the various pairs that are obtained:<o:p></o:p></span></div>
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<br /></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgfLk-5FFbsBPbA1nQDD9pOMI7U_5Pg4zL7uccLWFPOFg8vId2_FAcBJx6SUDqqh8UVU1UJsfM0sRuQnLGpIcWmBvfvMPAYMpKOqhq7ux490uMm2VRAf0jZJCi0zC79pPiXnGSNGL0RbZU/s1600/2019-06-10+22_08_26-Search.jpg" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" data-original-height="368" data-original-width="384" height="381" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgfLk-5FFbsBPbA1nQDD9pOMI7U_5Pg4zL7uccLWFPOFg8vId2_FAcBJx6SUDqqh8UVU1UJsfM0sRuQnLGpIcWmBvfvMPAYMpKOqhq7ux490uMm2VRAf0jZJCi0zC79pPiXnGSNGL0RbZU/s400/2019-06-10+22_08_26-Search.jpg" width="400" /></a></div>
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</div>
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<br /></div>
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<span face=""arial" , sans-serif" lang="EN-US" style="font-size: 12pt;">As can be seen, the
only square removed is (<span style="background: yellow; mso-highlight: yellow;">2,
2</span>).<o:p></o:p></span></div>
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<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0in; text-align: justify;">
<span face=""arial" , sans-serif" lang="EN-US" style="font-size: 12pt;">In the case of 3D <b style="mso-bidi-font-weight: normal;">fractal</b>-like we can do a similar
reasoning and the <b style="mso-bidi-font-weight: normal;">7</b> cubes removed
will be:<o:p></o:p></span></div>
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<br /></div>
<div align="center" class="MsoNormal" style="line-height: normal; margin-bottom: 0in; text-align: center;">
<span face=""arial" , sans-serif" lang="EN-US" style="font-size: 12pt;">(1, <b style="mso-bidi-font-weight: normal;"><span style="background: yellow; mso-highlight: yellow;">2, 2</span></b>); (3, <b style="mso-bidi-font-weight: normal;"><span style="background: yellow; mso-highlight: yellow;">2, 2</span></b>); (<b style="mso-bidi-font-weight: normal;"><span style="background: yellow; mso-highlight: yellow;">2</span></b>, 1, <b style="mso-bidi-font-weight: normal;"><span style="background: yellow; mso-highlight: yellow;">2</span></b>); (<b style="mso-bidi-font-weight: normal;"><span style="background: yellow; mso-highlight: yellow;">2, 2, 2</span></b>); (<b style="mso-bidi-font-weight: normal;"><span style="background: yellow; mso-highlight: yellow;">2</span></b>, 3, <b style="mso-bidi-font-weight: normal;"><span style="background: yellow; mso-highlight: yellow;">2</span></b>); (<b style="mso-bidi-font-weight: normal;"><span style="background: yellow; mso-highlight: yellow;">2, 2</span></b>, 1); <span style="background: yellow; mso-highlight: yellow;">(<b style="mso-bidi-font-weight: normal;">2, 2</b></span>, 3)<o:p></o:p></span></div>
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<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0in; text-align: justify;">
<span face=""arial" , sans-serif" lang="EN-US" style="font-size: 12pt;">The following table
summarizes (for each dimension) how many groups of numbers <b style="mso-bidi-font-weight: normal;">2</b> we can find; eg in <b style="mso-bidi-font-weight: normal;">3D</b>
we have <b style="mso-bidi-font-weight: normal;">8 triples</b> with zero <b style="mso-bidi-font-weight: normal;">2</b>, <b style="mso-bidi-font-weight: normal;">12</b>
with only <b style="mso-bidi-font-weight: normal;">2</b>, <b style="mso-bidi-font-weight: normal;">6</b> with <b style="mso-bidi-font-weight: normal;">2</b> elements equal
to <b style="mso-bidi-font-weight: normal;">2</b> and <b style="mso-bidi-font-weight: normal;">1</b> with all the numbers equal to <b style="mso-bidi-font-weight: normal;">2</b> (which corresponds to the central cube):<o:p></o:p></span></div>
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<br /></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgkmmTmRmpk1AYrHUM1pC1m6z0SRYqno6VWnhfzDGwNPY11OpFX_042FtK-8XmzvomzMPSEXHEvyF5ZRaewMm2YP4Z4pGsK8F71RuFebgWlrVDURToACiB1EpsRoKsbw3J2XkaAAIjlG-c/s1600/2019-06-12+21_31_02-Start.jpg" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em; text-align: center;"><img border="0" data-original-height="315" data-original-width="914" height="220" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgkmmTmRmpk1AYrHUM1pC1m6z0SRYqno6VWnhfzDGwNPY11OpFX_042FtK-8XmzvomzMPSEXHEvyF5ZRaewMm2YP4Z4pGsK8F71RuFebgWlrVDURToACiB1EpsRoKsbw3J2XkaAAIjlG-c/s640/2019-06-12+21_31_02-Start.jpg" width="640" /></a></div>
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<br /></div>
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<span face=""arial" , sans-serif" lang="EN-US" style="font-size: 12pt;">the combinations with
at least <b style="mso-bidi-font-weight: normal;">2</b> elements equal to <b style="mso-bidi-font-weight: normal;">2</b> are <span style="color: red;">6 + 1</span>
= <b style="mso-bidi-font-weight: normal;">7</b>, on a total of <b style="mso-bidi-font-weight: normal;">27</b> cubes (shown in the last column) and
for a cube of side <b style="mso-bidi-font-weight: normal;"><span style="background: yellow; mso-highlight: yellow;">3</span></b>.<o:p></o:p></span></div>
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<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0in; text-align: justify;">
<span face=""arial" , sans-serif" lang="EN-US" style="font-size: 12pt;">We can do the same
reasoning for a cube of side <b style="mso-bidi-font-weight: normal;"><span style="background: yellow; mso-highlight: yellow;">5</span></b>:<o:p></o:p></span></div>
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<br /></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiUfEWpoRCt2N-7DoFX4ewMaF_2KSgApX047mibE_QwCgN7xv0C_hc02WPfkyZkhnAAhg0o87nf6Zu5kaNTQ1unfRnTbdh5i0YmjQ768kkpihxj_dv2UzL991T9prDCz_vdYri2ECUeUeU/s1600/2019-06-12+21_32_02-Start.jpg" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em; text-align: center;"><img border="0" data-original-height="315" data-original-width="914" height="220" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiUfEWpoRCt2N-7DoFX4ewMaF_2KSgApX047mibE_QwCgN7xv0C_hc02WPfkyZkhnAAhg0o87nf6Zu5kaNTQ1unfRnTbdh5i0YmjQ768kkpihxj_dv2UzL991T9prDCz_vdYri2ECUeUeU/s640/2019-06-12+21_32_02-Start.jpg" width="640" /></a></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0in; text-align: justify;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0in; text-align: justify;">
<span face=""arial" , sans-serif" lang="EN-US" style="font-size: 12pt;">last
column shows the number of cubes remaining.<o:p></o:p></span></div>
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<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0in; text-align: justify;">
<i style="mso-bidi-font-style: normal;"><span face=""arial" , sans-serif" lang="EN-US" style="font-size: 12pt;">Note</span></i><span face=""arial" , sans-serif" lang="EN-US" style="font-size: 12pt;">: the coefficients that appear in
the table can be easily calculated using the <b style="mso-bidi-font-weight: normal;"><a href="https://en.wikipedia.org/wiki/Binomial_theorem" target="_blank">formula of the Newton binomial</a></b> - in this case (4 + 1) n -<o:p></o:p></span></div>
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<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0in; text-align: justify;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0in; text-align: justify;">
<span face=""arial" , sans-serif" lang="EN-US" style="font-size: 12pt;">Now, in the previous
post we saw that in <b style="mso-bidi-font-weight: normal;">2D</b> for an
initial value of 4, multiplied by 8/9, 24/25, 48/49, 80/81, ..., (n<sup>2</sup>
- 1) / n<sup>2</sup><o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0in; text-align: justify;">
<span face=""arial" , sans-serif" lang="EN-US" style="font-size: 12pt;">we have the formula
derived from the <b style="mso-bidi-font-weight: normal;"><a href="https://en.wikipedia.org/wiki/Wallis_product" target="_blank">Wallis product</a></b>:<o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0in; text-align: justify;">
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgxjLNPMAZbLAMSl5n0JEhx9B1nAe5yXF_h-91ehxX_kslQ2giVD2DDTsA7sZJRt7rgQ5lu10YIIaf9Sw98ehdfWYfMjZHmhGmvXyG-mBD40u3eSd-juen54bkiB-F5DmF2DzqGiE2OxZ8/s1600/2019-06-03+01_09_02-Start.jpg" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" data-original-height="148" data-original-width="559" height="84" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgxjLNPMAZbLAMSl5n0JEhx9B1nAe5yXF_h-91ehxX_kslQ2giVD2DDTsA7sZJRt7rgQ5lu10YIIaf9Sw98ehdfWYfMjZHmhGmvXyG-mBD40u3eSd-juen54bkiB-F5DmF2DzqGiE2OxZ8/s320/2019-06-03+01_09_02-Start.jpg" width="320" /></a></div>
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</div>
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<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0in; text-align: justify;">
<span face=""arial" , sans-serif" lang="EN-US" style="font-size: 12pt;">which can be summarized
as follows:<o:p></o:p></span></div>
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<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjCKxGflVITGTV-cmWDilHnXnbnExQa8u-ZfQQjSX0RL0YeUh5zUnPN95lIU385wXqX9Kmjv9UGdWePFhtiMj53E4Q9aNumhrOAmWQDYlRCxixe-r3-_YmSFpDBf8hUWs8zWMpD8EKMK2U/s1600/2019-06-10+23_37_29-Start.jpg" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" data-original-height="117" data-original-width="549" height="85" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjCKxGflVITGTV-cmWDilHnXnbnExQa8u-ZfQQjSX0RL0YeUh5zUnPN95lIU385wXqX9Kmjv9UGdWePFhtiMj53E4Q9aNumhrOAmWQDYlRCxixe-r3-_YmSFpDBf8hUWs8zWMpD8EKMK2U/s400/2019-06-10+23_37_29-Start.jpg" width="400" /></a></div>
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</div>
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<br /></div>
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<span face=""arial" , sans-serif" lang="EN-US" style="font-size: 12pt;">While in <b style="mso-bidi-font-weight: normal;">3D</b> we have this result:<o:p></o:p></span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgLkHWKNAtTMekdCxgc4KilWo4dMNf_ZypXVQ0AdZUy9U8kVA-WSUMVkPZ45aClN1ZVELkyfrk4mC09va0FQLbGPjkqInc15KHlWppviuecleGDtIuSwnLc0R_35eXCx2Pu23hjx8ADJ-M/s1600/2019-06-10+23_37_58-Start.jpg" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" data-original-height="155" data-original-width="549" height="112" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgLkHWKNAtTMekdCxgc4KilWo4dMNf_ZypXVQ0AdZUy9U8kVA-WSUMVkPZ45aClN1ZVELkyfrk4mC09va0FQLbGPjkqInc15KHlWppviuecleGDtIuSwnLc0R_35eXCx2Pu23hjx8ADJ-M/s400/2019-06-10+23_37_58-Start.jpg" width="400" /></a></div>
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</div>
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<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0in; text-align: justify;">
<span face=""arial" , sans-serif" lang="EN-US" style="font-size: 12pt;">where the numbers in
gray represent the various denominators.<o:p></o:p></span></div>
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<br /></div>
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<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0in; text-align: justify;">
<span face=""arial" , sans-serif" lang="EN-US" style="font-size: 12pt;">I omit the passages and
report what I calculated with Wolfram|Alpha:<o:p></o:p></span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjKEacJtUSQzICuV0UhMQ4m9z1Y3c1FV3lYWjwFCsP5gupoLsLF60nMsULPyMiDVPDsoO_0hQY7fL3DGQNirWQXKeQWxiGlXOOtaa4I-rNGLHRDvojInYxOe535eVATlNxL__z8audh0gg/s1600/2019-06-10+23_59_11-Start.jpg" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em; text-align: center;"><img border="0" data-original-height="568" data-original-width="1197" height="302" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjKEacJtUSQzICuV0UhMQ4m9z1Y3c1FV3lYWjwFCsP5gupoLsLF60nMsULPyMiDVPDsoO_0hQY7fL3DGQNirWQXKeQWxiGlXOOtaa4I-rNGLHRDvojInYxOe535eVATlNxL__z8audh0gg/s640/2019-06-10+23_59_11-Start.jpg" width="640" /></a></div>
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<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0in; text-align: justify;">
<span face=""arial" , sans-serif" lang="EN-US" style="font-size: 12pt;">which can be
summarized:<o:p></o:p></span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgPOvRywOCUinHpylETNqVJLeAlN37PaBGbyxPji5V_B8f_s04BpjTiiaz_edoJPs_HdGt5gX0IDLJ684uzzKpEuh2Q-axFf7CFmqPfObgV7AfZxi27Jubluw9NKA0yAtT6WrW2W5nIHwo/s1600/2019-06-10+23_48_48-Start.jpg" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em; text-align: center;"><img border="0" data-original-height="614" data-original-width="843" height="466" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgPOvRywOCUinHpylETNqVJLeAlN37PaBGbyxPji5V_B8f_s04BpjTiiaz_edoJPs_HdGt5gX0IDLJ684uzzKpEuh2Q-axFf7CFmqPfObgV7AfZxi27Jubluw9NKA0yAtT6WrW2W5nIHwo/s640/2019-06-10+23_48_48-Start.jpg" width="640" /></a></div>
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<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0in; text-align: justify;">
<span face=""arial" , sans-serif" lang="EN-US" style="font-size: 12pt;">In the formulas the <b style="mso-bidi-font-weight: normal;">Euler Gamma function</b> appears which
extends the factorial concept (even to complex numbers). At the end of the post
I report the chart and some notable formulas concerning it.<o:p></o:p></span></div>
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<br /></div>
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<span face=""arial" , sans-serif" lang="EN-US" style="font-size: 12pt;">Here I am only
interested in noting that the right part of the formulas containing the index </span><b style="mso-bidi-font-weight: normal;"><span face=""arial" , sans-serif" lang="EN-US" style="font-size: 14pt;">n</span></b><span face=""arial" , sans-serif" lang="EN-US" style="font-size: 12pt;"> <i style="mso-bidi-font-style: normal;">tends to</i> <b style="mso-bidi-font-weight: normal;">1</b> when <i style="mso-bidi-font-style: normal;">tending</i> </span><b style="mso-bidi-font-weight: normal;"><span face=""arial" , sans-serif" lang="EN-US" style="font-size: 14pt;">n</span></b><span face=""arial" , sans-serif" lang="EN-US" style="font-size: 12pt;"> to <b style="mso-bidi-font-weight: normal;">infinity</b>.<o:p></o:p></span></div>
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<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: 0in; text-align: justify;">
<span face=""arial" , sans-serif" lang="EN-US" style="font-size: 12pt;">So what remains of the
3 formulas is <b style="mso-bidi-font-weight: normal;">pi</b> and the number is
in the denominator. Remembering that each formula must be multiplied by the
number of "quadrants", the first (<b style="mso-bidi-font-weight: normal;">2D</b>) 2<sup>2</sup> = 4, the second (<b style="mso-bidi-font-weight: normal;">3D</b>) for 2<sup>3</sup> = 8 and the third (<b style="mso-bidi-font-weight: normal;">4D</b>) for 16.<o:p></o:p></span></div>
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<br /></div>
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<span face=""arial" , sans-serif" lang="EN-US" style="font-size: 12pt;">Thus obtaining the
"<b style="mso-bidi-font-weight: normal;">Volumes</b>" <b style="mso-bidi-font-weight: normal;">of the n-dimensional spheres</b>:<o:p></o:p></span></div>
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<span face=""arial" , sans-serif" lang="EN-US" style="font-size: 12pt;">This is not yet a real
demonstration, but it contains all the elements and could also be used as an
alternative method to calculate the volume of a hyper-sphere.<o:p></o:p></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: 0in; text-align: justify;"><span face=""arial" , sans-serif" lang="EN-US" style="font-size: 12pt;"><a href="goog_485374476"><br /></a></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: 0in; text-align: justify;"><span face=""arial" , sans-serif" lang="EN-US" style="font-size: medium;"><a href="https://community.wolfram.com/groups/-/m/t/822984">https://community.wolfram.com/groups/-/m/t/822984</a></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: 0in; text-align: justify;"><br /></div>
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<span face=""arial" , sans-serif" style="color: #222222; font-size: 12pt; line-height: 115%;"><a href="https://www.wolframalpha.com/input/?i=product+(2k%2F(2k%2B1))%5E4+((2k%2B5)%2F(2k%2B1)),+k%3D1+to+infinity" target="_blank"><span lang="EN-US" style="mso-ansi-language: EN-US;">https://www.wolframalpha.com/input/?i=product+(2k%2F(2k%2B1))%5E4+((2k%2B5)%2F(2k%2B1)),+k%3D1+to+infinity</span></a></span><span face=""arial" , sans-serif" lang="EN-US" style="color: #222222; font-size: 12pt; line-height: 115%;"><o:p></o:p></span></div>
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<span face=""arial" , sans-serif" style="color: #222222; font-size: 12pt; line-height: 115%;"><a href="https://math.stackexchange.com/questions/1783732/is-a-hypersphere-of-non-integer-dimension-a-fractal" target="_blank"><span lang="EN-US" style="mso-ansi-language: EN-US;">https://math.stackexchange.com/questions/1783732/is-a-<span style="background: yellow; mso-highlight: yellow;">hypersphere</span>-of-non-integer-dimension-a-fractal</span></a></span><span face=""arial" , sans-serif" lang="EN-US" style="color: #222222; font-size: 12pt; line-height: 115%;"><o:p></o:p></span></div>
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<span face=""arial" , sans-serif" style="color: blue; font-size: 12pt; line-height: 115%;"><a href="https://math.stackexchange.com/search?q=fractal+sphere+gamma" target="_blank"><span lang="EN-US" style="mso-ansi-language: EN-US;">https://math.stackexchange.com/search?q=<span style="background: yellow; mso-highlight: yellow;">fractal</span>+sphere+gamma</span></a><o:p></o:p></span></div>
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<span face=""arial" , sans-serif" style="color: #222222; font-size: 12pt; line-height: 115%;"><a href="http://zeta137.blogspot.com/2019/06/12-wallis-and-squaring-of-circle.html?_sm_au_=iQW2QM3DR7VWvD1j" target="_blank">http://zeta137.blogspot.com/2019/06/12-<span style="background: yellow; mso-highlight: yellow;">wallis</span>-and-squaring-of-circle.html?_sm_au_=iQW2QM3DR7VWvD1j</a><o:p></o:p></span></div>
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<span face=""arial" , sans-serif" style="color: #222222; font-size: 12pt; line-height: 115%;"><a href="http://zeta137.blogspot.com/2015/06/6-wallis-and-pippenger-infinite-products.html?_sm_au_=iQW2QM3DR7VWvD1j" target="_blank">http://zeta137.blogspot.com/2015/06/6-wallis-and-<span style="background: yellow; mso-highlight: yellow;">pippenger</span>-infinite-products.html?_sm_au_=iQW2QM3DR7VWvD1j</a><o:p></o:p></span></div>
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<br />zibhttp://www.blogger.com/profile/17401589710495280112noreply@blogger.com0tag:blogger.com,1999:blog-287967698191537182.post-58064625476798105502019-06-11T14:48:00.003-07:002019-06-12T13:27:15.136-07:0012. Wallis and Squaring of the Circle<br />
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<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt; line-height: 115%;">The <b style="mso-bidi-font-weight: normal;">squaring of the circle</b>, together with
the problem of <b style="mso-bidi-font-weight: normal;">angle trisection</b> and
that of <b style="mso-bidi-font-weight: normal;">cube duplication</b>, is a
classical problem in Greek mathematics, whose purpose is to construct a square
that has the same area as a given circle, with exclusive use of straightedge</span><span style="font-family: "arial" , sans-serif; font-size: 12pt;"> and
compass.</span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiBm5xnDYtpiPLxitYm_9nCzS2mW2ecRSmSuTa8JXsmSnxJs3GfP2RN4Gp7m1FOD0pjomyCDav7oAeCZ_vf3WK6uCzVZQm5zEHzWFNC9XPFPBlLdfYBsFQ4UooqRbutv3Rb5uI47qDnslY/s1600/2019-06-02+22_05_31-Start.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" data-original-height="423" data-original-width="437" height="309" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiBm5xnDYtpiPLxitYm_9nCzS2mW2ecRSmSuTa8JXsmSnxJs3GfP2RN4Gp7m1FOD0pjomyCDav7oAeCZ_vf3WK6uCzVZQm5zEHzWFNC9XPFPBlLdfYBsFQ4UooqRbutv3Rb5uI47qDnslY/s320/2019-06-02+22_05_31-Start.jpg" width="320" /></a></div>
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<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt; line-height: 115%;">In 1882, <b style="mso-bidi-font-weight: normal;">Ferdinand Lindemann</b>, of the University
of Munich, showed that <b>pi</b> is transcendent, thus forever drawing a line on the
problem of squaring the circle; from the proof provided by Lindemann, it turns
out that it is impossible to construct, only with straightedge and compass, a square
of area equal to that of a given circle, a problem that has tormented entire
generations of mathematicians since before <b style="mso-bidi-font-weight: normal;">Euclid</b>.
Lindemann showed that pi is not an <i style="mso-bidi-font-style: normal;">algebraic
number</i>.<o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt; line-height: 115%;">Any geometry
problem that can be solved only with the ruler and the compass, when placed
under its equivalent algebraic form, leads to one or more algebraic equations
with integer rational coefficients that can be solved by successive extractions
of square roots. Since π does not satisfy any of these equations, we cannot
arrive at the quadrature of the circle with the instruments in question.<o:p></o:p></span></div>
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<i style="mso-bidi-font-style: normal;"><span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 14.0pt; line-height: 115%;">Note</span></i><span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt; line-height: 115%;">: with row and compass it is obviously possible to
"construct" the positive integers as whole distances from the given
origin. Every rational number is constructible, and this thanks to Thales'
beautiful theorem on similar triangles; furthermore it can be shown that the
square root of any constructible number is itself constructible.<o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt; line-height: 115%;"><br /></span></div>
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<span lang="EN-US" style="font-family: "consolas"; font-size: 14.0pt; line-height: 115%;">A
<b style="mso-bidi-font-weight: normal;">fractal</b> is a geometric object that
is repeated in its form in the same way on different scales, and by enlarging
any part of it you get a figure similar to the original</span><span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt; line-height: 115%;">.<o:p></o:p></span></div>
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<br /></div>
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<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt; line-height: 115%;">In the previous
post we saw the <b style="mso-bidi-font-weight: normal;">Sierpinski carpet</b>, a
fractal obtained from a square, described by the Polish mathematician <b style="mso-bidi-font-weight: normal;">Wacław Sierpiński</b> in 1916. At each step
the squares that make up the figure are divided into 9 smaller squares and the
central square is removed. In this way, for each step the area is reduced by
8/9, so the fractal dimension of the carpet is <b style="mso-bidi-font-weight: normal;">log 8 / log 3</b>, equal to 1.892789 ...<o:p></o:p></span></div>
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<br /></div>
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<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;">Now let's see another object that "might"
look similar to the previous one.<o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;">The <b style="mso-bidi-font-weight: normal;"><span style="background: yellow; mso-highlight: yellow;">sieve of Wallis</span></b> is
thus constructed:<o:p></o:p></span></div>
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<span style="font-family: "arial" , sans-serif; font-size: 12pt;"><br /></span></div>
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<span style="font-family: "arial" , sans-serif; font-size: 12pt;">it starts with a 2 x 2 square and is divided into 4
squares;</span></div>
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<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;">divide each new sub-panel by 9 and remove the central
square (8/9);<o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;">each new sub-panel is divided by 25 and the central
square (24/25) is removed;<o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;">each new sub-panel is divided by 49 and the central
square (48/49) is removed;<o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;">each new sub-panel is divided by 81 and the central
square (80/81) is removed<o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;">and so on.<o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;"><br /></span></div>
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<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;">Here's how it looks after a few steps:<o:p></o:p></span></div>
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<br /></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiJmYQPePw6UHQwkBH0bnWRr4ROnxUjohYtRarK9-3wpP2dUwIycvRVvDDz5T5cyjb3GPm2Iueup3GFus27YyqV4QRFMI3Jhxic3AoWio0oiURa-11CjM8UyhJWbKOjaEgVwAkKxlxewLw/s1600/2019-06-04+08_23_50-Start.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em; text-align: center;"><img border="0" data-original-height="619" data-original-width="620" height="638" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiJmYQPePw6UHQwkBH0bnWRr4ROnxUjohYtRarK9-3wpP2dUwIycvRVvDDz5T5cyjb3GPm2Iueup3GFus27YyqV4QRFMI3Jhxic3AoWio0oiURa-11CjM8UyhJWbKOjaEgVwAkKxlxewLw/s640/2019-06-04+08_23_50-Start.jpg" width="640" /></a></div>
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<br /></div>
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<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;"><br /></span></div>
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<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;"><br /></span></div>
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<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;">I said "<i style="mso-bidi-font-style: normal;">could</i>" because the <b style="mso-bidi-font-weight: normal;">Wallis sieve</b> is a <b style="mso-bidi-font-weight: normal;">fractal</b>-like, at every step the rule changes and therefore
self-similarity is not maintained. However, while in the case of the carpet the
area (Lebesgue measure) is zero, for the sieve the area has a well-known value.<o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;">The initial value is by
definition 4, then in the order it is multiplied by 8/9, 24/25, 48/49, 80/81,
..., (n<sup>2</sup> - 1) / n<sup>2</sup><o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;"><br /></span></div>
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<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;">This is the famous <b style="mso-bidi-font-weight: normal;">Wallis product</b>:<o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;"><br /></span></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhhLdiKB7cmv5CVLskTvKeWk2xeleRByx0_MhL6lZg_bIjUSLFwOprdXH6x54bcLTtF1TqpufJtajQa5ruMo6QdfdNShxjOXmvWOYx1zGoAnI1dTT9hW06Ur4o9Js-276HAm7izFs7qNnk/s1600/2019-06-03+01_08_31-Start.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" data-original-height="161" data-original-width="551" height="93" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhhLdiKB7cmv5CVLskTvKeWk2xeleRByx0_MhL6lZg_bIjUSLFwOprdXH6x54bcLTtF1TqpufJtajQa5ruMo6QdfdNShxjOXmvWOYx1zGoAnI1dTT9hW06Ur4o9Js-276HAm7izFs7qNnk/s320/2019-06-03+01_08_31-Start.jpg" width="320" /></a></div>
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<br /></div>
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<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;">which can also be rewritten in this way:<o:p></o:p></span></div>
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<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjSd-kkF80n06z0YmolC-lt0UdcsrE6tDyerJd17FYOZPfPcyPlBMKvyDe98tHAJmICkstIOFstQQe90j87mNzrVtnp4xhiSfNPeGmoHIhMxJ0sKgwO9IN8Iumkw2KNfQvDn3C7q5ntjBY/s1600/2019-06-03+01_09_02-Start.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" data-original-height="148" data-original-width="559" height="84" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjSd-kkF80n06z0YmolC-lt0UdcsrE6tDyerJd17FYOZPfPcyPlBMKvyDe98tHAJmICkstIOFstQQe90j87mNzrVtnp4xhiSfNPeGmoHIhMxJ0sKgwO9IN8Iumkw2KNfQvDn3C7q5ntjBY/s320/2019-06-03+01_09_02-Start.jpg" width="320" /></a></div>
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<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;">The wonder lies in the
fact that the areas of the <b style="mso-bidi-font-weight: normal;">Wallis sieve</b>
and the <b style="mso-bidi-font-weight: normal;">Circle</b> are the same.<o:p></o:p></span></div>
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<br /></div>
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<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;">Let's see 2 more views:<o:p></o:p></span></div>
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<br /></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi9wqoy51d3gI2P4Luu4iYkW2VUsJXyKBQQ9AZK-QDh5RL2lJ9yvm1b89TEh7FXqRfJo99zNQiUhGEPhHHmFwVW1p_OafvvWe5s2_Mq3TUJAw3i7lohMQEz-1N-sk2tgoXLAkUFxpEfFT0/s1600/2019-06-03+00_32_50-Start.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em; text-align: center;"><img border="0" data-original-height="466" data-original-width="950" height="312" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi9wqoy51d3gI2P4Luu4iYkW2VUsJXyKBQQ9AZK-QDh5RL2lJ9yvm1b89TEh7FXqRfJo99zNQiUhGEPhHHmFwVW1p_OafvvWe5s2_Mq3TUJAw3i7lohMQEz-1N-sk2tgoXLAkUFxpEfFT0/s640/2019-06-03+00_32_50-Start.jpg" width="640" /></a></div>
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<br /></div>
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<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;">in the first <i style="mso-bidi-font-style: normal;">figure</i> we start with only 1 square and
relate to the relative inscribed circle, while the same object is then
decomposed and recomposed to better highlight the equivalence of <i style="mso-bidi-font-style: normal;">circle</i> and <i style="mso-bidi-font-style: normal;">sieve</i>.<o:p></o:p></span></div>
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<br /></div>
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<span lang="EN-US" style="background: yellow; font-family: "arial" , sans-serif; font-size: 14.0pt;">With straightedge and compass you can therefore build a "square"
equivalent to the circle, just have patience and the necessary time ... if it
is not exactly a square, it somehow looks like it</span><span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;">.<o:p></o:p></span></div>
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<br /></div>
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<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;">Another way could be
the following: take a square of side 2, remove a concentric square of area 4/3,
add one of area 4/5 and continue like this to infinity ... you will have thus
obtained the series of <b style="mso-bidi-font-weight: normal;">Gregory – Leibniz</b>
multiplied by 4, that is </span><b style="mso-bidi-font-weight: normal;"><span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 14.0pt;">pi</span></b><span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;">.<o:p></o:p></span></div>
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<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEidvko2uUv98vchXmcOiG_P2caavezykXchzeuYRYdil7P8k1tldonEpEuzNDFWQSnPq3uYY0s5-grlW4m6nwRUBqQKptjM4ia0r3wSI-OAFTe1rc1EeV0-nWifeeL_d3grtQCDJeBlNkI/s1600/2019-06-03+23_35_17-Start.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" data-original-height="103" data-original-width="358" height="92" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEidvko2uUv98vchXmcOiG_P2caavezykXchzeuYRYdil7P8k1tldonEpEuzNDFWQSnPq3uYY0s5-grlW4m6nwRUBqQKptjM4ia0r3wSI-OAFTe1rc1EeV0-nWifeeL_d3grtQCDJeBlNkI/s320/2019-06-03+23_35_17-Start.jpg" width="320" /></a></div>
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgMVE8oRnzANqjbt3bVqpUykfBrp6u5EfjtMrtwhJDzSUjL6FA_q7TATSuoqkDlDebxPNcpHvWUgJ3EF-I2SfVyMPBcUJ7s7luZeTZ32OIifk-j7RnibDQxM8u9ehsvSN2iyc69TtN9BbY/s1600/2019-06-04+23_04_27-Start.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto; text-align: center;"><img border="0" data-original-height="612" data-original-width="606" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgMVE8oRnzANqjbt3bVqpUykfBrp6u5EfjtMrtwhJDzSUjL6FA_q7TATSuoqkDlDebxPNcpHvWUgJ3EF-I2SfVyMPBcUJ7s7luZeTZ32OIifk-j7RnibDQxM8u9ehsvSN2iyc69TtN9BbY/s400/2019-06-04+23_04_27-Start.jpg" width="396" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><span style="font-size: large;">Size of yellow area is equal to <b>pi</b></span></td></tr>
</tbody></table>
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</div>
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<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;">Let us ask ourselves
this question:<o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "consolas"; font-size: 14.0pt;">is there a <b style="mso-bidi-font-weight: normal;">Menger
sponge</b> equivalent to the <b style="mso-bidi-font-weight: normal;">Wallis sieve</b></span><span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;">?<o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;">It starts from a cube
of side 1 and divides the side into 3 parts (27 cubes as for the Rubik's cube),
the central cube and the 6 central cubes are removed with each face (so 20
cubes remain), in the next step divide the side of each new cube by 5 removing
the central cube and the cubes that extend from the center, then repeat the
same procedure with 7, 9, 11 and the subsequent odd numbers.<o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "consolas"; font-size: 14.0pt;"><i>Wonder of wonders</i></span><span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;">,
<b style="mso-bidi-font-weight: normal;">8 of these cubes have the same volume as
a sphere of unit radius</b></span> </div>
<div align="center" class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in; text-align: center;">
<i style="mso-bidi-font-style: normal;"><span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;">or equivalent<o:p></o:p></span></i></div>
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<br /></div>
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<b style="mso-bidi-font-weight: normal;"><span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 14.0pt;">cube of side 1 has the same volume as sphere of
unitary diameter</span></b><b style="mso-bidi-font-weight: normal;"><span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;"><o:p></o:p></span></b></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhwBsnkqWTmXems46jCcBEv2ZCI7UTeJOsHegY9a7KuMeajFTU9TN6HQv9xqh3JXFOSChIiEeLe67bC85o_kYfNYSvEmrYmZew2o7YoHgHdujc78KqUmufCSBpYoYBGO44mA3mKB6KtTQI/s1600/2019-06-03+00_32_08-Start.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="460" data-original-width="1021" height="284" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhwBsnkqWTmXems46jCcBEv2ZCI7UTeJOsHegY9a7KuMeajFTU9TN6HQv9xqh3JXFOSChIiEeLe67bC85o_kYfNYSvEmrYmZew2o7YoHgHdujc78KqUmufCSBpYoYBGO44mA3mKB6KtTQI/s640/2019-06-03+00_32_08-Start.jpg" width="640" /></a></div>
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<span style="color: #222222; font-family: "arial" , sans-serif; font-size: 10.0pt; line-height: 115%;"><br /></span>
<span style="color: #222222; font-family: "arial" , sans-serif; font-size: 10.0pt; line-height: 115%;"><span lang="EN-US" style="font-size: 11.0pt; line-height: 115%; mso-ansi-language: EN-US;"><a href="https://blogs.scientificamerican.com/roots-of-unity/a-few-of-my-favorite-spaces-the-wallis-sieve/?redirect=1" target="_blank">https://blogs.scientificamerican.com/roots-of-unity/a-few-of-my-favorite-spaces-the-wallis-sieve/?redirect=1</a></span></span><span lang="EN-US" style="color: #222222; font-family: "arial" , sans-serif; font-size: 10.0pt; line-height: 115%;"><u1:p></u1:p><o:p></o:p></span></div>
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<span style="color: #222222; font-family: "arial" , sans-serif; font-size: 10.0pt; line-height: 115%;"><a href="https://demonstrations.wolfram.com/WallisSievePiApproximation/" target="_blank"><span style="background: yellow; font-size: 14.0pt; line-height: 115%;">https://demonstrations.wolfram.com/WallisSievePiApproximation/</span></a><o:p></o:p></span></div>
<u1:p></u1:p>
<br />
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<span style="color: #222222; font-family: "arial" , sans-serif; font-size: 10.0pt; line-height: 115%;"><a href="https://community.wolfram.com/groups/-/m/t/822984" target="_blank"><span style="font-size: 11.0pt; line-height: 115%;">https://community.wolfram.com/groups/-/m/t/822984</span></a><o:p></o:p></span></div>
<u1:p></u1:p>
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<span style="color: #222222; font-family: "arial" , sans-serif; font-size: 10.0pt; line-height: 115%;"><a href="https://mathsedideas.blogspot.com/2018/02/on-pi-day.html?_sm_au_=iQWkRvfFNHZnNJj6" target="_blank"><span style="color: red; line-height: 115%;">https://mathsedideas.blogspot.com/2018/02/on-pi-day.html?_sm_au_=iQWkRvfFNHZnNJj6</span></a><o:p></o:p></span></div>
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<br />zibhttp://www.blogger.com/profile/17401589710495280112noreply@blogger.com0tag:blogger.com,1999:blog-287967698191537182.post-5258100334821870312019-05-22T13:16:00.000-07:002019-06-13T08:27:50.022-07:0011. Section of a Menger sponge<br />
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<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;">By sectioning with a
plane perpendicular to a major diagonal passing through the center of gravity
of the cube, a regular hexagon can be obtained:<o:p></o:p></span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjA_b4-_kzIslCb5pKhVTpd43LyyNibfIiU91DVv4x91OI-4sEz0wdsV0tTimYvYzPdchX0CiGIDF8scx6W0whkVF64pHwG4r5L6mAwvWntN1XcwNHRlH37Jjltd8dZApnrFhfLzGVSEuY/s1600/SP32-20110402-225610.gif" imageanchor="1" style="font-family: "Times New Roman"; font-size: medium; margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" data-original-height="332" data-original-width="374" height="355" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjA_b4-_kzIslCb5pKhVTpd43LyyNibfIiU91DVv4x91OI-4sEz0wdsV0tTimYvYzPdchX0CiGIDF8scx6W0whkVF64pHwG4r5L6mAwvWntN1XcwNHRlH37Jjltd8dZApnrFhfLzGVSEuY/s400/SP32-20110402-225610.gif" width="400" /></a></div>
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<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;">It is not intuitive,
but it is easy to understand if you think that the number of sides of a hexagon
is 6 and the number of faces of a cube is also 6.<o:p></o:p></span></div>
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<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhfhrYaonw6qkySWN18Z9Wz1ZG-ICm5YK1v3tZFc8ZkOdE6nwiH3buI3vYasZUvRvK4JmwWyCZXly3a0ppTob8CyRi5-roImBsADiO7MnvHAMg4iXPB6SBnfEymhhG5Jw5rAydqt-ZTB_w/s1600/2019-06-13+17_23_12-Start.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img alt="" border="0" data-original-height="457" data-original-width="837" height="174" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhfhrYaonw6qkySWN18Z9Wz1ZG-ICm5YK1v3tZFc8ZkOdE6nwiH3buI3vYasZUvRvK4JmwWyCZXly3a0ppTob8CyRi5-roImBsADiO7MnvHAMg4iXPB6SBnfEymhhG5Jw5rAydqt-ZTB_w/s320/2019-06-13+17_23_12-Start.jpg" title="" width="320" /></a></td></tr>
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<span style="color: #002052; font-family: Arial, sans-serif;"><span style="font-size: 10pt;"> </span><a href="https://youtu.be/5vTS--uuErA"><span style="background-color: yellow; font-size: large;">https://youtu.be/5vTS--uuErA</span></a><span style="font-size: 10pt;"><o:p></o:p></span></span></div>
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<span style="font-family: arial, sans-serif; font-size: 12pt;">Below we will consider
some fractal objects and the section of a </span><i style="font-family: arial, sans-serif; font-size: 12pt;">fractal
object</i><span style="font-family: arial, sans-serif; font-size: 12pt;"> </span><i style="font-family: arial, sans-serif; font-size: 12pt;">corresponding to the cube</i><span style="font-family: arial, sans-serif; font-size: 12pt;">.</span></div>
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<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;">A <b style="mso-bidi-font-weight: normal;">Menger sponge</b>
is defined as follows:<o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;">1. we start from a cube,<o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;">2. divide the cube into 27 cubes (as in the Rubik
cube),<o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;">3. the central cube and the 6 central cubes are
removed from each face (20 cubes remain),<o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;">4. repeat steps 1 - 3 on each new cube.</span></div>
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<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;">At each iteration the
number of holes increases, as shown in the figure:<o:p></o:p></span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgJ6WHWGeyyqIvcR3bVCW4EAw4FSrrnVH5_pGneV3SVHwmDiS2U7awLKMNBZwlBQgRkLdgChlJXIAyMdpQz0tUSUnjQcVFap1K0dgVYzXZaMOX19JSUeX9R2DhUfPG-btK4hDG7RLHK2Uw/s1600/Menger_sponge_%2528Level_0-3%2529.jpg" imageanchor="1" style="clear: left; float: left; font-family: "Times New Roman"; font-size: medium; margin-bottom: 1em; margin-right: 1em; text-align: center;"><img border="0" data-original-height="228" data-original-width="800" height="182" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgJ6WHWGeyyqIvcR3bVCW4EAw4FSrrnVH5_pGneV3SVHwmDiS2U7awLKMNBZwlBQgRkLdgChlJXIAyMdpQz0tUSUnjQcVFap1K0dgVYzXZaMOX19JSUeX9R2DhUfPG-btK4hDG7RLHK2Uw/s640/Menger_sponge_%2528Level_0-3%2529.jpg" width="640" /></a></div>
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<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgc_Q9NC61x9x7NsW9QIFXPJA3lkUiMZqjpYuj7PF5PyYk82vFMgejfL4W4gP75KNbJCDiNww5GsMYc7t3Yq56v3j05FT-zZn-PdCnfS5_u9wmpdhPkyQCT3i6S7vfkL78g52KLZrz3ZcI/s1600/2019-05-06+16_54_59-Start.jpg" imageanchor="1" style="font-family: "times new roman"; font-size: medium; margin-left: auto; margin-right: auto; text-align: center;"><img border="0" data-original-height="779" data-original-width="876" height="355" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgc_Q9NC61x9x7NsW9QIFXPJA3lkUiMZqjpYuj7PF5PyYk82vFMgejfL4W4gP75KNbJCDiNww5GsMYc7t3Yq56v3j05FT-zZn-PdCnfS5_u9wmpdhPkyQCT3i6S7vfkL78g52KLZrz3ZcI/s400/2019-05-06+16_54_59-Start.jpg" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><span style="font-size: large;"><a href="https://en.wikipedia.org/wiki/Menger_sponge">https://en.wikipedia.org/wiki/Menger_sponge</a></span></td></tr>
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<b style="mso-bidi-font-weight: normal;"><span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;"><br /></span></b></div>
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<b style="mso-bidi-font-weight: normal;"><span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;">Menger sponge</span></b><span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> is the space that is
obtained as the limit of these operations.<o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;">In its construction of
1926, <b style="mso-bidi-font-weight: normal;">Karl Menger</b> showed that the <b style="mso-bidi-font-weight: normal;">Hausdorff dimension</b> of the sponge is
log 20 / log 3 approximately 2,726833 ...<o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;"><br /></span></div>
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<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;">Well, if you take a
sponge from Menger and dissect it like the cube seen above, here is what you
will get:</span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhXNwkPetMeIK-YIb6wa3p8XjvoJxw6zb2HNf_KZCkKj8Kb5jcnQSRV_9DVCp2r5wVZ_MTR27AFbnAeoMAhfbzJmOqkpikytCraulHS3r3KZI7Iov3kJuUbAMpoYnNDUu-rWJF92nxADFk/s1600/2019-05-08+00_33_53-Start.jpg" imageanchor="1" style="clear: right; float: right; font-family: "Times New Roman"; font-size: medium; margin-bottom: 1em; margin-left: 1em; text-align: center;"><img border="0" data-original-height="508" data-original-width="446" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhXNwkPetMeIK-YIb6wa3p8XjvoJxw6zb2HNf_KZCkKj8Kb5jcnQSRV_9DVCp2r5wVZ_MTR27AFbnAeoMAhfbzJmOqkpikytCraulHS3r3KZI7Iov3kJuUbAMpoYnNDUu-rWJF92nxADFk/s320/2019-05-08+00_33_53-Start.jpg" width="280" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgj2H8wzPpABiYIJt3gnumss0wIpuxktmhIpnt0wcAhOHJdR2pnSsE87NBVFjSjULMFO0VU3Lw4hK6w6szi2Es1hgkiC9XgZ3jrLqDtuHWys7viEoXM9wD9wpiHpfFX0tHZEEAs-PvMNmI/s1600/2019-05-08+00_36_55-Start.jpg" imageanchor="1" style="clear: left; display: inline !important; margin-bottom: 1em; margin-right: 1em; text-align: center;"><img border="0" data-original-height="649" data-original-width="639" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgj2H8wzPpABiYIJt3gnumss0wIpuxktmhIpnt0wcAhOHJdR2pnSsE87NBVFjSjULMFO0VU3Lw4hK6w6szi2Es1hgkiC9XgZ3jrLqDtuHWys7viEoXM9wD9wpiHpfFX0tHZEEAs-PvMNmI/s320/2019-05-08+00_36_55-Start.jpg" width="312" /></a></div>
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<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;"></span></div>
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<span style="color: #222222; font-family: "arial" , sans-serif; font-size: 14pt;">These figures are taken from the site of</span><span style="color: #222222; font-family: "arial" , sans-serif; font-size: 14pt;"> </span><b style="color: #222222; font-family: Arial, sans-serif; font-size: 14pt;"><span style="background: yellow;">George W. Hart</span></b></div>
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<span lang="EN-US" style="color: #222222; font-family: "arial" , sans-serif;"> </span><span style="color: #222222; font-family: "arial" , sans-serif;"><a href="http://www.georgehart.com/index.html">http://www.georgehart.com/index.html</a></span></div>
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<span style="color: #222222; font-family: "arial" , sans-serif;"><a href="http://www.georgehart.com/rp/half-menger-sponge.html">http://www.georgehart.com/rp/half-menger-sponge.html</a></span><span style="color: #222222; font-family: "arial" , sans-serif; font-size: 10.0pt; line-height: 115%;"><o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;"><br /></span></div>
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<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;">In the <b style="mso-bidi-font-weight: normal;"><a href="http://blog.zacharyabel.com/2012/02/seeing-stars/" target="_blank">Seeing Stars</a></b> article it is shown that this section has <b>Hausdorff dimension</b>:<o:p></o:p></span><br />
<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;"><br /></span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiYn4zBDzECln3Pj5Y5JnZznl10q9fk5PpuUpAvX28lYezqTTARw7iQAgjC_Oh6vklNRN8VoiXzNYLMIzpnewnZqT1UBXbUfzv6bx0e-n2l2qD7KUQduMTMfAFQARDNfpS_yipyyLEXaEo/s1600/2019-05-07+23_56_27-Start.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" data-original-height="457" data-original-width="549" height="332" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiYn4zBDzECln3Pj5Y5JnZznl10q9fk5PpuUpAvX28lYezqTTARw7iQAgjC_Oh6vklNRN8VoiXzNYLMIzpnewnZqT1UBXbUfzv6bx0e-n2l2qD7KUQduMTMfAFQARDNfpS_yipyyLEXaEo/s400/2019-05-07+23_56_27-Start.jpg" width="400" /></a></div>
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<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;">The <b style="background-color: yellow;">Cantor set</b>, introduced by the German
mathematician <b style="mso-bidi-font-weight: normal;"><a href="https://en.wikipedia.org/wiki/Georg_Cantor" target="_blank">Georg Cantor</a></b>, is a
subset of the interval [0, 1] of real numbers, defined recursively, removing at
each step an open central segment from each interval.<o:p></o:p></span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEglgWzSJ-oGSdGc4K0Djc335o5GpT5ZZDdptQnA6Ryqkv74XE-8iBK28ceP1MWcmuUsDlkTDPwLKEG5ZRiqjfoSlDRF6AZKWCAk-EzVVIMwdPCKNDT5JkQGNTGPfYN2ucYVifSgtE_o6_c/s1600/2019-05-08+00_04_11-Start.jpg" imageanchor="1" style="clear: left; float: left; font-family: "Times New Roman"; font-size: medium; margin-bottom: 1em; margin-right: 1em; text-align: center;"><img border="0" data-original-height="320" data-original-width="1577" height="129" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEglgWzSJ-oGSdGc4K0Djc335o5GpT5ZZDdptQnA6Ryqkv74XE-8iBK28ceP1MWcmuUsDlkTDPwLKEG5ZRiqjfoSlDRF6AZKWCAk-EzVVIMwdPCKNDT5JkQGNTGPfYN2ucYVifSgtE_o6_c/s640/2019-05-08+00_04_11-Start.jpg" width="640" /></a><span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;"></span></div>
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<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;">The <b>Cantor set</b> consists
of all the points of the initial interval [0, 1] that are never removed by this
recursive procedure: in other words, the set that remains after iterating this
procedure countless times, called suggestively dust of Cantor. This set is a
fractal with Hausdorff dimension ln 2 / ln 3, equal to 0.6309 ...<o:p></o:p></span></div>
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<b style="mso-bidi-font-weight: normal;"><span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;">Sierpinski carpet</span></b><span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> is also a fractal
similar to the Cantor collection obtained from a square, by the
Polish mathematician <b style="mso-bidi-font-weight: normal;">Wacław Sierpiński</b>
in 1916.<o:p></o:p></span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEheWg38PPc0d00bLmZaKv1xORe5WlCIAbY55laiTe-28kV49fWRaXyoqvzeqHAyY1TmpMwWRQLeLH1bn-1zBXOI15o2_wHLJf6zD3y5m0jvM4NAx69fa3GdiMnvkGWr9_KW-bQsP-kWLS0/s1600/2019-05-08+00_04_49-Start.jpg" imageanchor="1" style="clear: left; float: left; font-family: "Times New Roman"; font-size: medium; margin-bottom: 1em; margin-right: 1em; text-align: center;"><img border="0" data-original-height="760" data-original-width="763" height="636" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEheWg38PPc0d00bLmZaKv1xORe5WlCIAbY55laiTe-28kV49fWRaXyoqvzeqHAyY1TmpMwWRQLeLH1bn-1zBXOI15o2_wHLJf6zD3y5m0jvM4NAx69fa3GdiMnvkGWr9_KW-bQsP-kWLS0/s640/2019-05-08+00_04_49-Start.jpg" width="640" /></a><span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;"></span></div>
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<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;">Its fractal dimension
is ln 8 / ln 3, equal to = 1.8927 ... (</span><span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 14.0pt;">it is interesting to note that it is equal to <b style="mso-bidi-font-weight: normal;">3 times</b> the Hausdorff dimension of the <b style="mso-bidi-font-weight: normal;">Cantor set</b></span><span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;">).
This value is slightly higher than the one shown above.<o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 14.0pt;">The three-dimensional version of the <b>carpet</b> is the <b style="mso-bidi-font-weight: normal;">Menger sponge</b></span><span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;">.<o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;">The figure shows an
extract from the </span><b style="mso-bidi-font-weight: normal;"><span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 14.0pt;">list of fractals</span></b><span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;">
by <b style="mso-bidi-font-weight: normal;"><span style="color: red;">Hausdorff
dimension</span></b> reported in <b style="mso-bidi-font-weight: normal;">Wikipedia</b>:<o:p></o:p></span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhj-7JKK_MLGGsBUQ4DKFG_HxP0ZD2H_6MKg-9I0gpMh9cWiwkdj7QGG57sNWeN21fzhHeHD_wcc8oB8M5PSK4-mbZ5w0Wow5t9RYjcwyYG35pgYcmZwO4LYYL4n4wovaitIiYhA3G4csQ/s1600/2019-05-07+14_49_04-Start.jpg" imageanchor="1" style="clear: left; float: left; font-family: "Times New Roman"; font-size: medium; margin-bottom: 1em; margin-right: 1em; text-align: center;"><img border="0" data-original-height="845" data-original-width="1154" height="468" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhj-7JKK_MLGGsBUQ4DKFG_HxP0ZD2H_6MKg-9I0gpMh9cWiwkdj7QGG57sNWeN21fzhHeHD_wcc8oB8M5PSK4-mbZ5w0Wow5t9RYjcwyYG35pgYcmZwO4LYYL4n4wovaitIiYhA3G4csQ/s640/2019-05-07+14_49_04-Start.jpg" width="640" /></a><span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;"></span></div>
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<span style="color: #222222; font-family: "arial" , sans-serif; font-size: 12.0pt; line-height: 115%;"><a href="https://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension" target="_blank">https://en.wikipedia.org/wiki/<b style="mso-bidi-font-weight: normal;"><span style="font-size: 14.0pt; line-height: 115%; mso-bidi-font-size: 10.0pt;">List_of_fractals_by_Hausdorff_dimension</span></b></a><o:p></o:p></span></div>
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<span style="color: #222222; font-family: "arial" , sans-serif;"><a href="http://blog.zacharyabel.com/tag/menger-sponge/" target="_blank">http://blog.zacharyabel.com/tag/menger-sponge/</a></span><span style="color: #222222; font-family: "arial" , sans-serif; font-size: 10.0pt; line-height: 115%;"><o:p></o:p></span></div>
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<span style="color: #222222; font-family: "arial" , sans-serif;"><a href="http://mathworld.wolfram.com/MengerSponge.html">http://mathworld.wolfram.com/MengerSponge.html</a></span><span style="color: #222222; font-family: "arial" , sans-serif; font-size: 10.0pt; line-height: 115%;"><br />
<br />
</span><span style="background: yellow; color: #222222; font-family: "arial" , sans-serif; font-size: 18.0pt; line-height: 115%;"><a href="http://blog.zacharyabel.com/2012/02/seeing-stars/" target="_blank">http://blog.zacharyabel.com/2012/02/seeing-stars/</a></span><span style="color: #222222; font-family: "arial" , sans-serif; font-size: 10.0pt; line-height: 115%;"><o:p></o:p></span></div>
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<span style="color: #222222; font-family: "arial" , sans-serif;"><a href="https://www.johndcook.com/blog/2019/05/05/area-volume-menger-sponge/?utm_source=feedburner&utm_medium=email&utm_campaign=Feed%3A+TheEndeavour+%28The+Endeavour%29" target="_blank">https://www.johndcook.com/blog/2019/05/05/area-volume-menger-sponge/?utm_source=feedburner&utm_medium=email&utm_campaign=Feed%3A+TheEndeavour+%28The+Endeavour%29</a></span><span style="color: #222222; font-family: "arial" , sans-serif; font-size: 10.0pt; line-height: 115%;"><o:p></o:p></span></div>
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<span style="font-size: large;"><span style="color: #222222; font-family: "arial" , sans-serif;"><a href="http://www.georgehart.com/rp/half-menger-sponge.html" target="_blank">http://www.georgehart.com/rp/half-menger-sponge.html</a></span><span style="color: #222222; font-family: "arial" , sans-serif; line-height: 115%;"><br />
</span><span style="color: #222222; font-family: "arial" , sans-serif; line-height: 115%;"><a href="http://www-history.mcs.st-and.ac.uk/HistTopics/fractals.html" target="_blank">http://www-history.mcs.st-and.ac.uk/HistTopics/fractals.html</a></span><span style="color: #222222; font-family: "arial" , sans-serif; line-height: 115%;"><o:p></o:p></span></span></div>
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<br />zibhttp://www.blogger.com/profile/17401589710495280112noreply@blogger.com0tag:blogger.com,1999:blog-287967698191537182.post-46845499778740309472019-05-21T14:00:00.000-07:002019-05-21T14:11:12.422-07:0010. Black Holes<div style="text-align: left;">
<span style="font-family: "arial" , sans-serif; font-size: 12pt; text-align: justify;">A few weeks ago it was
announced that the </span><b style="font-family: arial, sans-serif; font-size: 12pt; text-align: justify;">Event Horizon
Telescope</b><span style="font-family: "arial" , sans-serif; font-size: 12pt; text-align: justify;">, a group of eight radio telescopes (on a planetary scale),
provided the first direct visual evidence ever obtained of a </span><b style="font-family: arial, sans-serif; font-size: 12pt; text-align: justify;"><a href="https://en.wikipedia.org/wiki/Black_hole" target="_blank">Black Hole</a></b><span style="font-family: "arial" , sans-serif; font-size: 12pt; text-align: justify;"> positioned in the heart of </span><b style="font-family: arial, sans-serif; font-size: 12pt; text-align: justify;">Messier 87</b><span style="font-family: "arial" , sans-serif; font-size: 12pt; text-align: justify;">, a huge galaxy located in
the nearby Virgo cluster. In a series of six articles in "</span><i style="font-family: arial, sans-serif; font-size: 12pt; text-align: justify;">The Astrophysical Journal Letters</i><span style="font-family: "arial" , sans-serif; font-size: 12pt; text-align: justify;">",
the image reveals a supermassive Black Hole with a mass equal to 6.5 billion
times that of the Sun and that is 55 million light years from the Earth.</span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj_1E3q-SuhqQLV0kKBAFt5aqJTvSjB9KpmyFo-zCC8sENqDpceRBZDpAecdqXvVq7Jk9lx2Mc2r6SLxowVw76fz66oCpuZ2grly7_hVVHuMC94sxW8puDCttr1QGsr4CgwRmDar099l08/s1600/buco_nero.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1200" data-original-width="1600" height="300" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj_1E3q-SuhqQLV0kKBAFt5aqJTvSjB9KpmyFo-zCC8sENqDpceRBZDpAecdqXvVq7Jk9lx2Mc2r6SLxowVw76fz66oCpuZ2grly7_hVVHuMC94sxW8puDCttr1QGsr4CgwRmDar099l08/s400/buco_nero.jpg" width="400" /></a></div>
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<u><span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;">When we talk
about a Black Hole of this mass what are we talking about</span></u><span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;">?<o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;">For example the mass of
the planet Earth is 5.97 × 10<sup>24</sup> kg (diameter of 12,750 km), while
that of the Sun is 1.99 × 10<sup>30</sup> kg (diameter 1.39 × 10<sup>6</sup>
km); the relationship between the values of the 2 masses is easy to remember: </span><b style="mso-bidi-font-weight: normal;"><span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 14.0pt;">333,333</span></b><span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;">. <o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;">The mass of <b style="mso-bidi-font-weight: normal;">Sgr A</b>* (Black Hole at the center of our
galaxy) is 4.31 × 10<sup>6</sup> M<sub>o</sub> and finally that of the BH of <b style="mso-bidi-font-weight: normal;">M87</b> is equal to 6.5 × 10<sup>9</sup> M<sub>o</sub>
(1.29 × 10<sup>40</sup> kg ).<o:p></o:p></span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgQhDWE797V9lW-3Qnirv147ZmVOpU6TPMvVc4VIp5Cv2iQ3nAoHn_Q7Ds4b5vy03yYeRvIdsomfzJa53AN_xJ0Aum6I6jqvq9HJygc9DsR1WU3dmTt3z6sKcfkyHH3IZx0HQ_2-ekq34c/s1600/2019-05-16+22_49_11-Start.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="491" data-original-width="850" height="368" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgQhDWE797V9lW-3Qnirv147ZmVOpU6TPMvVc4VIp5Cv2iQ3nAoHn_Q7Ds4b5vy03yYeRvIdsomfzJa53AN_xJ0Aum6I6jqvq9HJygc9DsR1WU3dmTt3z6sKcfkyHH3IZx0HQ_2-ekq34c/s640/2019-05-16+22_49_11-Start.jpg" width="640" /></a></div>
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<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;">From the table, it can
be seen that the BH diameter is directly proportional to its mass, and that if
the Sun became a BH it would have a radius of 2.95 km; multiplying this last
value by the number of solar masses (M<sub>o</sub>) we obtain the radius of the
BH.<o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;">This is because the <b style="mso-bidi-font-weight: normal;">Schwarzschild Radius</b> is directly
proportional to its mass:<o:p></o:p></span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgCJSe75pV0f4gnuiepiHojc3bIGGAP_ICYSbKOq0QPsNeUqATY3ZSe62PPoeJiFQMHiqn_koGlJgenus6mpJPTeFCHwqD0hijH0hsgRlkafKfzktiSI5pv-VnCeO1YW7_BKTV7vkXA7mY/s1600/2019-05-19+18_41_41-Start.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="204" data-original-width="441" height="91" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgCJSe75pV0f4gnuiepiHojc3bIGGAP_ICYSbKOq0QPsNeUqATY3ZSe62PPoeJiFQMHiqn_koGlJgenus6mpJPTeFCHwqD0hijH0hsgRlkafKfzktiSI5pv-VnCeO1YW7_BKTV7vkXA7mY/s200/2019-05-19+18_41_41-Start.jpg" width="200" /></a></div>
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<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;">The Black Hole of M87
which has a mass 6.5 billion solar masses therefore has a <b style="mso-bidi-font-weight: normal;">R<sub>S</sub></b> of about 20 billion km (127 <b style="mso-bidi-font-weight: normal;">A</b>stronomical <b style="mso-bidi-font-weight: normal;">U</b>nits). <o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;">Pluto aphelion is
located at about 49.3 AU (from the Sun), so the Solar System could be
conveniently <b style="mso-bidi-font-weight: normal;"><span style="color: red;">contained
in the Black Hole of M87</span></b>.<o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;">The <i>Schwarzschild
Radius</i> <b style="mso-bidi-font-weight: normal;">R<sub>S</sub></b> is the distance
at which the <b style="mso-bidi-font-weight: normal;">escape velocity</b> is <b style="mso-bidi-font-weight: normal;"><i style="mso-bidi-font-style: normal;">c</i></b>
(speed of light).<o:p></o:p></span></div>
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<b style="mso-bidi-font-weight: normal;"><span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;">Titius-Bode </span></b><span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;">law, an empirical
formula that describes with good approximation the values of the semi-axes
greater than the orbits of the planets of the solar system (expressed in <b style="mso-bidi-font-weight: normal;">A</b>stronomical <b style="mso-bidi-font-weight: normal;">U</b>nits) and is expressed with the simple formula:<o:p></o:p></span><br />
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<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;"><span style="mso-spacerun: yes;">
</span><span style="mso-spacerun: yes;"> </span></span><span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 16.0pt;">(3</span><b style="mso-bidi-font-weight: normal;"><i style="mso-bidi-font-style: normal;"><span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 18.0pt;">n</span></i></b><span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 16.0pt;"> + 4) / 10</span><span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;"><o:p></o:p></span><br />
<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 16.0pt;"><br /></span></div>
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<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;">where </span><b style="mso-bidi-font-weight: normal;"><i style="mso-bidi-font-style: normal;"><span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 14.0pt;">n</span></i></b><span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> takes the
values 0, 1, 2, 4, 8, 16, ...<o:p></o:p></span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiFOOyp3h8wyTqmPnp-9UZIyWauj71LEm2BP0YEHQyfWZKrWlcJB1ywUepbSpElRRLnvfwr8-f_eOksa-xjOc3A_nv4L8ViRmfkVmyauFzmtpJhtM7K6QSrlApmvgzIc6nPkTLitmufrew/s1600/2019-05-17+23_28_26-Start.jpg" imageanchor="1" style="font-family: arial, sans-serif; font-size: 16px; margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" data-original-height="445" data-original-width="514" height="346" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiFOOyp3h8wyTqmPnp-9UZIyWauj71LEm2BP0YEHQyfWZKrWlcJB1ywUepbSpElRRLnvfwr8-f_eOksa-xjOc3A_nv4L8ViRmfkVmyauFzmtpJhtM7K6QSrlApmvgzIc6nPkTLitmufrew/s400/2019-05-17+23_28_26-Start.jpg" width="400" /></a></div>
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<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;">This simple geometric
progression has a nice graphic representation using </span><span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 14.0pt;">logarithmic scale</span><span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;">:<o:p></o:p></span><br />
<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;"><br /></span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiXZlvHfJVKm-62bFmGZXsoc9LzIovWQDo-dU29lQVSl8C9Xu4eUxrvdk49DZsj5b9SMAcTRtcSz6tuTTj6Y2-lE8dFl85P2gfx-i1C1RYA2y5PKUI2I9b0CPiAstXLZRoQKNboikJQL64/s1600/2019-04-24+22_48_35-Anni_Giorni_Secondi.xlsx+-+Excel.jpg" imageanchor="1" style="clear: left; float: left; font-family: arial, sans-serif; font-size: 16px; margin-bottom: 1em; margin-right: 1em; text-align: center;"><img border="0" data-original-height="453" data-original-width="1468" height="197" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiXZlvHfJVKm-62bFmGZXsoc9LzIovWQDo-dU29lQVSl8C9Xu4eUxrvdk49DZsj5b9SMAcTRtcSz6tuTTj6Y2-lE8dFl85P2gfx-i1C1RYA2y5PKUI2I9b0CPiAstXLZRoQKNboikJQL64/s640/2019-04-24+22_48_35-Anni_Giorni_Secondi.xlsx+-+Excel.jpg" width="640" /></a></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in; text-align: justify;">
<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;"><br /></span>
<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;"><br /></span>
<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;">where a <b style="mso-bidi-font-weight: normal;">1</b> AU we find <i style="mso-bidi-font-style: normal;">by definition</i> the <b style="mso-bidi-font-weight: normal;">Earth</b>
and about <b style="mso-bidi-font-weight: normal;">10</b> AU <b style="mso-bidi-font-weight: normal;">Saturn</b>. As mentioned above, at <b style="mso-bidi-font-weight: normal;">127</b>
we can position the <b style="mso-bidi-font-weight: normal;">event horizon of the
Black Hole of M87</b>.<o:p></o:p></span><br />
<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;"><br /></span></div>
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<br /></div>
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<u><span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 14.0pt;">But what is the <b style="mso-bidi-font-weight: normal;">average density</b>
of a Black Hole</span></u><span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;">?<o:p></o:p></span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEijJjQJBSAJWBo1Mi-q8of0zC5olTyM-jT8rr9Ka0gaS_yRNmU849_mRaZdNDdgTDftXcvjUbw-6yVogLGgCNEyFQJQifDj9XqQ2S818iYvDsnARxcga-I5QZFq-sJYctjXLe9yFP1KcbM/s1600/2019-05-19+18_47_54-Start.jpg" imageanchor="1" style="clear: left; float: left; font-family: Arial, sans-serif; margin-bottom: 1em; margin-right: 1em; text-align: center;"><img border="0" data-original-height="137" data-original-width="739" height="73" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEijJjQJBSAJWBo1Mi-q8of0zC5olTyM-jT8rr9Ka0gaS_yRNmU849_mRaZdNDdgTDftXcvjUbw-6yVogLGgCNEyFQJQifDj9XqQ2S818iYvDsnARxcga-I5QZFq-sJYctjXLe9yFP1KcbM/s400/2019-05-19+18_47_54-Start.jpg" width="400" /></a><span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;"></span><br />
<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;"><span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;"><br /></span></span>
<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;">For Black Holes of
"small" dimensions, the average density within the <b style="mso-bidi-font-weight: normal;">event horizon</b> is incredibly high (see
the first table) and at the borders of the Black Hole there are tidal forces
greater than a trillion times the gravitational force . However (quite
surprisingly) the average density decreases dramatically for the <b style="mso-bidi-font-weight: normal;">massive</b> Black Holes. A BH of <b style="mso-bidi-font-weight: normal;">387 million solar masses</b> would have the
average <b style="mso-bidi-font-weight: normal;">water density</b> and would be
comparable to a giant water balloon extending from the Sun to almost Jupiter. A
BH of <b style="mso-bidi-font-weight: normal;">11 billion solar masses</b> would
have the average <b style="mso-bidi-font-weight: normal;">air density</b> and
would be analogous to a giant balloon 2.5 times larger than Pluto's orbit. The
average mass density in space itself, however small, may eventually become a
low-density BH. If the average density of the Universe corresponds to the
critical density of only 5.67 hydrogen atoms per cubic meter, a Hole would form
Low density black of about 13.8 billion light years, corresponding to the Big
Bang model of the Universe. A BH can use rotation and/or electric charge to
avoid collapse. Gravitational forces become negligible for large low density
Black Holes.<o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;">So you can live in a
large low density BH without even realizing it.<o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;"><br /></span></div>
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<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;">I stop here and let the
reader fantasize ….<o:p></o:p></span></div>
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<br /></div>
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<b style="mso-bidi-font-weight: normal;"><span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;">AU
Astronomical Unit</span></b><span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> (Earth - Sun
distance) : 149,597,870 km = 8.5 minutes light<o:p></o:p></span></div>
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<b style="mso-bidi-font-weight: normal;"><span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;">Light
year</span></b><span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;"> (distance traveled by light in a year) : 9,460 billion
km = 63,300 AU<o:p></o:p></span></div>
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<b style="mso-bidi-font-weight: normal;"><span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;">Parsec </span></b><span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;">: 3.262 light years = 30,860 billion km = 206,000 AU<o:p></o:p></span></div>
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<b style="mso-bidi-font-weight: normal;"><span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;">Speed
of light </span></b><span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;">: 299,792 km/sec<o:p></o:p></span><br />
<span lang="EN-US" style="font-family: "arial" , sans-serif; font-size: 12.0pt;"><br /></span>
<span lang="EN-US" style="font-family: "arial" , sans-serif;"><a href="https://commons.wikimedia.org/w/index.php?curid=74584660">https://commons.wikimedia.org/w/index.php?curid=74584660</a></span><br />
<span lang="EN-US" style="font-family: "arial" , sans-serif;"><br /></span>
<span lang="EN-US" style="font-family: "arial" , sans-serif;"><br /></span></div>
<br />zibhttp://www.blogger.com/profile/17401589710495280112noreply@blogger.com0tag:blogger.com,1999:blog-287967698191537182.post-56121466641531533522015-07-11T14:17:00.001-07:002015-07-12T13:29:12.907-07:009. Avatar Flower<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"><strong><a href="https://en.wikipedia.org/wiki/Dini%27s_surface" target="_blank">Dini’s surface</a></strong> has the interesting feature of having
constant negative curvature that can be created by twisting a <b style="mso-bidi-font-weight: normal;"><u><span style="color: red;"><a href="https://en.wikipedia.org/wiki/Pseudosphere" target="_blank" title="Pseudosphere"><span style="color: red;">pseudosphere</span></a></span></u></b>:<o:p></o:p></span><br />
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"><o:p> </o:p></span><br />
<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgMwwFInCUj2aCsgV-SwBLBfvjOOKPuWT9daMsvLmck0RMOQAw_MWv1T-UVP2g2FOszJhfszyFtaZnE5GJyt4h0Q9Yy4kxbab9QkqIlLnFF-dcg_tT2lMPzus3sFUenfnAYHCtpZD562tM/s1600/SP32-20150711-225652.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="356" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgMwwFInCUj2aCsgV-SwBLBfvjOOKPuWT9daMsvLmck0RMOQAw_MWv1T-UVP2g2FOszJhfszyFtaZnE5GJyt4h0Q9Yy4kxbab9QkqIlLnFF-dcg_tT2lMPzus3sFUenfnAYHCtpZD562tM/s640/SP32-20150711-225652.gif" width="640" /></a></div>
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<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"><o:p> </o:p></span></div>
<br />
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<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;">It is named after <b style="mso-bidi-font-weight: normal;">Ulisse
Dini</b> and described by the following parametric equations:<o:p></o:p></span></div>
<br />
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<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"><o:p> </o:p></span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjPBLdaJE5its479mkEuoKtYKEsZFWE_GhA53L63cV4YMZVuzH_KIclxXEwOOw76j4zAXsttuaiZX1vY_APRNJO3l-cqeVujzclPb6DYotHwAquXaDFX3XgrsW-VodzlOrgEEPrvUWf2q0/s1600/SP32-20150711-225743.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="108" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjPBLdaJE5its479mkEuoKtYKEsZFWE_GhA53L63cV4YMZVuzH_KIclxXEwOOw76j4zAXsttuaiZX1vY_APRNJO3l-cqeVujzclPb6DYotHwAquXaDFX3XgrsW-VodzlOrgEEPrvUWf2q0/s400/SP32-20150711-225743.gif" width="400" /></a></div>
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<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"><o:p> </o:p></span></div>
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<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;">The flowers that appear in the movie
"Avatar" seem to have the same shape:<o:p></o:p></span></div>
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"><o:p> </o:p></span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiXiPutGa5sw9Ognjd6ZXyi9-YBBVnHFWVP0xwJhunLW2gjsrZZpuaoWjl10n5TiW3TFaQ2l4T4F5T7SLOftjBfunSeWS70bt8qYU_Q0llVlhKyzW-ZtnGoOqZCS-S-N111O_jgweMj9jE/s1600/SP32-20150709-093427.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="312" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiXiPutGa5sw9Ognjd6ZXyi9-YBBVnHFWVP0xwJhunLW2gjsrZZpuaoWjl10n5TiW3TFaQ2l4T4F5T7SLOftjBfunSeWS70bt8qYU_Q0llVlhKyzW-ZtnGoOqZCS-S-N111O_jgweMj9jE/s400/SP32-20150709-093427.gif" width="400" /></a></div>
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zibhttp://www.blogger.com/profile/17401589710495280112noreply@blogger.com0tag:blogger.com,1999:blog-287967698191537182.post-10980351413044325382015-06-13T14:26:00.000-07:002015-06-13T14:54:54.206-07:008. Hyperspheres<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;">As reported in wikipedia: <span style="mso-spacerun: yes;"> </span><a href="http://en.wikipedia.org/wiki/Hypersphere"><span style="color: blue;">http://en.wikipedia.org/wiki/Hypersphere</span></a></span>, <span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;">a hypersphere is the set of points at a constant
distance from a given point called its center.<o:p></o:p></span><br />
<br />
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<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;">The "volume" of a hypersphere is given by:<o:p></o:p></span></div>
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"></span><br />
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"><div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 0pt;">
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgRDCHh74UbHqgLdEwj8icM0nALFqJWsMKsHmJjCx8yShi6kwyaC5KDPATSfBh_Uzqz46tfkOYE50yNo3d_4u7gClkxVRL8V9gDvkAjj0kMmBuGcdPKJTAin_I2MGp4X_fNohYxIVieq7Q/s1600/SP32-20150613-231227.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="158" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgRDCHh74UbHqgLdEwj8icM0nALFqJWsMKsHmJjCx8yShi6kwyaC5KDPATSfBh_Uzqz46tfkOYE50yNo3d_4u7gClkxVRL8V9gDvkAjj0kMmBuGcdPKJTAin_I2MGp4X_fNohYxIVieq7Q/s400/SP32-20150613-231227.gif" width="400" /></a></div>
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<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"></span> </div>
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<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;">where Γ denotes the gamma function. <o:p></o:p></span></div>
<br />
<br />
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<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;">The "surface area" is instead given by:<o:p></o:p></span></div>
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"><span style="mso-spacerun: yes;"> </span><o:p></o:p></span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgtzsBv5XlXMLgJ4WGUlAyuc9nhFeUeLiJPYvXM2ZftkBGVV-qTJZY3C3nLJ41JVaoQQvLRq5EMG1ml8xDPuZE_ISZuOVwcJAM6bFpvhPf92YjOE8SvDzKTWGgIfh0I09kNF78iYedJoG4/s1600/SP32-20150613-231254.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="163" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgtzsBv5XlXMLgJ4WGUlAyuc9nhFeUeLiJPYvXM2ZftkBGVV-qTJZY3C3nLJ41JVaoQQvLRq5EMG1ml8xDPuZE_ISZuOVwcJAM6bFpvhPf92YjOE8SvDzKTWGgIfh0I09kNF78iYedJoG4/s400/SP32-20150613-231254.gif" width="400" /></a></div>
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<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;">In three dimensions, the formula for the calculation
of volume is: <o:p></o:p></span></div>
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"><o:p> </o:p></span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhptDry4hGPwJY0uu44g6c5i9-10xJu0jXCrG2uMTvGlBjtH313yHGCiS-ibdVi4NFV4yLD0a7ZIQZHBDazV7GptuQBQZhCj99pBxrjQBzUb9QBxIzEaS4lUiIb4zwUxEjtZU5RTZzuaYw/s1600/SP32-20150613-231320.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="127" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhptDry4hGPwJY0uu44g6c5i9-10xJu0jXCrG2uMTvGlBjtH313yHGCiS-ibdVi4NFV4yLD0a7ZIQZHBDazV7GptuQBQZhCj99pBxrjQBzUb9QBxIzEaS4lUiIb4zwUxEjtZU5RTZzuaYw/s400/SP32-20150613-231320.gif" width="400" /></a></div>
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<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"></span><br />
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;">As for the surface is: <o:p></o:p></span></div>
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"><o:p></o:p></span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjpjHL4hpWWVa6sjR-7y0RhkCVxfdb-WFQyluqV4cZkry-zyyMotgcb7uDXiU5zqAvVaWGd4zeA8gGeHHbKkctRdvJVfNuQHqJXaXXW1QplIUfjaV8xxwcaVbiWH7xubJNKrUWDuVpSCMI/s1600/SP32-20150613-231344.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjpjHL4hpWWVa6sjR-7y0RhkCVxfdb-WFQyluqV4cZkry-zyyMotgcb7uDXiU5zqAvVaWGd4zeA8gGeHHbKkctRdvJVfNuQHqJXaXXW1QplIUfjaV8xxwcaVbiWH7xubJNKrUWDuVpSCMI/s1600/SP32-20150613-231344.gif" /></a></div>
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<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;">For the definition of <b style="mso-bidi-font-weight: normal;">derivative</b>, as the limit of the quotient, we have: </span></div>
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<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"><span style="color: red; font-size: large;">t</span></span><span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"><span style="font-size: large;"><span style="color: red;">he</span> <b style="mso-bidi-font-weight: normal;"><span style="color: red;">surface</span></b><span style="color: red;"> is the <b style="mso-bidi-font-weight: normal;">derivative</b>
of the <b style="mso-bidi-font-weight: normal;">volume</b></span></span>. <o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;">And this </span><span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 14pt; mso-ansi-language: EN-US; mso-bidi-font-size: 12.0pt;">in each</span><span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"> dimension:<span style="mso-spacerun: yes;"> </span><span style="mso-spacerun: yes;"> </span></span><i style="mso-bidi-font-style: normal;"><span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: small; mso-ansi-language: EN-US; mso-bidi-font-size: 12.0pt;">D</span></i><span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 16pt; mso-ansi-language: EN-US; mso-bidi-font-size: 12.0pt;"><span style="font-size: large;"> (V<sub>n</sub>) = S<sub>n-1</sub></span> </span><span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"><o:p></o:p></span></div>
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"><o:p> </o:p></span><br />
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<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;">Their relationship in every dimension is remarkably
simple: <o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"><span style="mso-spacerun: yes;"> </span><span style="mso-spacerun: yes;"> </span></span><span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 18pt; mso-ansi-language: EN-US; mso-bidi-font-size: 12.0pt;">V<sub>n</sub> / S<sub>n-1</sub> = R / n </span><span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"><span style="mso-spacerun: yes;"> </span><span style="mso-spacerun: yes;"> </span></span><span lang="EN-US" style="font-family: "Arial",sans-serif; mso-ansi-language: EN-US; mso-bidi-font-size: 12.0pt;">e.g. <span style="mso-spacerun: yes;"> </span>in</span><span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"> <span style="mso-spacerun: yes;"> </span></span><span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 14pt; mso-ansi-language: EN-US; mso-bidi-font-size: 12.0pt;">3 </span><span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"><span style="mso-spacerun: yes;"> </span></span><span lang="EN-US" style="font-family: "Arial",sans-serif; mso-ansi-language: EN-US; mso-bidi-font-size: 12.0pt;">dimensions we
have: </span><span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"><span style="mso-spacerun: yes;"> </span></span><span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 16pt; mso-ansi-language: EN-US; mso-bidi-font-size: 12.0pt;">V<sub>3</sub> / S<sub>2</sub> = R / 3</span><span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"><o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"><o:p> </o:p></span></div>
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<a href="http://en.wikipedia.org/wiki/N-sphere" target="_blank"><b style="mso-bidi-font-weight: normal;"><span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"><span style="color: blue; font-size: large;">http://en.wikipedia.org/wiki/N-sphere</span></span></b></a><b style="mso-bidi-font-weight: normal;"><span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"><o:p></o:p></span></b></div>
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<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"></span><br /></div>
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<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;">For the calculation of hypervolumes and hypersurfaces,
formulas contain: <o:p></o:p></span></div>
<strong><span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;">in 2 or 3 dim. appears </span><span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 16pt; mso-ansi-language: EN-US; mso-bidi-font-size: 12.0pt;">π</span><span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;">, while in 4 or 5 dim. </span><span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 16pt; mso-ansi-language: EN-US; mso-bidi-font-size: 12.0pt;">π<sup>2</sup></span><span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;">,<span style="mso-spacerun: yes;"> </span>6 or 7 dim. </span><span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 16pt; mso-ansi-language: EN-US; mso-bidi-font-size: 12.0pt;">π<sup>3</sup></span></strong><span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"><strong> <span style="mso-spacerun: yes;"> </span>and so on</strong>. <o:p></o:p></span><br />
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"><o:p> </o:p></span><br />
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<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;">As an example for even number of dimensions, we have: <o:p></o:p></span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhDZm0YuKM637WvEI0XV-_39KqEr7hkKljza1UVBhNqLZxWtE5Osi1Acp7aErcqo_F1AQjVdJ_o2G4zef6hljBbkzjQzuOG-qjLN8uHxDVyhiKHQQNdL-Boo0gyj4oiYYrJGu6uK-bkVOQ/s1600/SP32-20150613-231405.gif" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="100" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhDZm0YuKM637WvEI0XV-_39KqEr7hkKljza1UVBhNqLZxWtE5Osi1Acp7aErcqo_F1AQjVdJ_o2G4zef6hljBbkzjQzuOG-qjLN8uHxDVyhiKHQQNdL-Boo0gyj4oiYYrJGu6uK-bkVOQ/s640/SP32-20150613-231405.gif" width="640" /></a></div>
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<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"><o:p> </o:p></span></div>
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<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;">while for odd number of dimensions: <o:p></o:p></span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjtc8DhDnlFJEZiOhZ-h1EzqTMruVTUGrshGfoyiFBRJ6jny48J1cMmyCWDD5D6Kkfoddf8e1-LuOFaaBueNM_hyqYRjW2J0IofhUowX9GeApu_mzn_bI2ZyqkrordTGSuotd7EK0XvM-g/s1600/SP32-20150613-231434.gif" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="89" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjtc8DhDnlFJEZiOhZ-h1EzqTMruVTUGrshGfoyiFBRJ6jny48J1cMmyCWDD5D6Kkfoddf8e1-LuOFaaBueNM_hyqYRjW2J0IofhUowX9GeApu_mzn_bI2ZyqkrordTGSuotd7EK0XvM-g/s640/SP32-20150613-231434.gif" width="640" /></a></div>
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<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"><o:p> </o:p></span></div>
zibhttp://www.blogger.com/profile/17401589710495280112noreply@blogger.com0tag:blogger.com,1999:blog-287967698191537182.post-43244069636532126672015-06-04T14:44:00.001-07:002015-06-04T14:44:56.610-07:007. M.C.Escher - Study for Stars<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhtEc2BIPN0E2duujGNlbZBqYfT9-lQX-8TvdThkwbP5CxhsQ_NTHGaF1uMrkLHyo21FeOA2weLFgKibGMHXUiGb9NIk1l4vlCEaNnGFXgFuo_pPFJ_y2-JLC1z8Fg2gYx4dRYJhJ9Fn10/s1600/Escher.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="482" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhtEc2BIPN0E2duujGNlbZBqYfT9-lQX-8TvdThkwbP5CxhsQ_NTHGaF1uMrkLHyo21FeOA2weLFgKibGMHXUiGb9NIk1l4vlCEaNnGFXgFuo_pPFJ_y2-JLC1z8Fg2gYx4dRYJhJ9Fn10/s640/Escher.jpg" width="640" /></a></div>
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<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"><o:p> </o:p></span></div>
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<b style="mso-bidi-font-weight: normal;"><span lang="EN-US" style="color: red; font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"><a href="http://www.youtube.com/watch?v=9WHdyG9mJaI&context=C3ceadd3ADOEgsToPDskLlwtChO3oZKUxKT4LOX5P_" target="_blank"><span style="color: blue;">http://www.youtube.com/watch?v=9WHdyG9mJaI&context=C3ceadd3ADOEgsToPDskLlwtChO3oZKUxKT4LOX5P_</span></a><o:p></o:p></span></b></div>
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<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"><a href="http://www.mcescher.com/" target="_blank"><span style="color: blue;">http://www.mcescher.com/</span></a><o:p></o:p></span></div>
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"><a href="http://britton.disted.camosun.bc.ca/escher/jbescher.htm"><span style="color: blue;">http://britton.disted.camosun.bc.ca/escher/jbescher.htm</span></a><o:p></o:p></span><br />
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"><a href="http://www.mathacademy.com/pr/minitext/escher/index.asp" target="_blank"><span style="color: blue;">http://www.mathacademy.com/pr/minitext/escher/index.asp</span></a><o:p></o:p></span><br />
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"><a href="http://aixa.ugr.es/escher/table.html" target="_blank"><span style="color: blue;">http://aixa.ugr.es/escher/table.html</span></a><o:p></o:p></span><br />
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"><a href="http://zibalsc.blogspot.com/2012/01/95-galleria-di-stampe.html"><span style="color: blue;">http://zibalsc.blogspot.com/2012/01/95-galleria-di-stampe.html</span></a><o:p></o:p></span><br />
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;">.<o:p></o:p></span><br />
zibhttp://www.blogger.com/profile/17401589710495280112noreply@blogger.com0tag:blogger.com,1999:blog-287967698191537182.post-37338085230303310912015-06-01T13:43:00.000-07:002015-06-01T13:50:51.168-07:006. Wallis and Pippenger Infinite Products<div style="text-align: justify;">
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;">The Wallis product is a formula found in 1655 by <strong><a href="http://en.wikipedia.org/wiki/John_Wallis" target="_blank">John Wallis</a></strong>, </span><span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;">which calculates the </span><span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;">simple infinite product:<o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"><o:p> </o:p></span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEicOajHY0-hiBd9nLyVtKYOaBpKJRJQbNdDo4YuAXghNlBKmYKJrCrkdodEPaBFqw-XY3PqomKwODAwgG8wa9_5Y9tH6hRTSGSelEkN_7ilsxgviFaYcSF8aOIG7X1aiVaSiY8yM5cjHXM/s1600/SP32-20150120-003442.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="84" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEicOajHY0-hiBd9nLyVtKYOaBpKJRJQbNdDo4YuAXghNlBKmYKJrCrkdodEPaBFqw-XY3PqomKwODAwgG8wa9_5Y9tH6hRTSGSelEkN_7ilsxgviFaYcSF8aOIG7X1aiVaSiY8yM5cjHXM/s320/SP32-20150120-003442.gif" width="320" /></a></div>
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<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"><strong><a href="http://en.wikipedia.org/wiki/Nick_Pippenger" target="_blank">Nick Pippenger</a></strong> obtained an infinite product for <span style="mso-spacerun: yes;"> </span></span><i style="mso-bidi-font-style: normal;"><span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 18pt; mso-ansi-language: EN-US; mso-bidi-font-size: 12.0pt;">e</span></i><span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;">:<o:p></o:p></span></div>
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"><o:p></o:p></span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiZp-wIJ70FGPTqV7oqC12I2nswm-CW-U8PfF0e7GJYdJwpxRRhX9mEbyP-6hyphenhyphenFqXfS940uo0WTJzHc4mRESWzfq4A_VtgDqJJsddJ0phD03XHA7aDbhXXr9mh4wungOZ2G9OeVD7oUCEQ/s1600/SP32-20150120-003459.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="70" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiZp-wIJ70FGPTqV7oqC12I2nswm-CW-U8PfF0e7GJYdJwpxRRhX9mEbyP-6hyphenhyphenFqXfS940uo0WTJzHc4mRESWzfq4A_VtgDqJJsddJ0phD03XHA7aDbhXXr9mh4wungOZ2G9OeVD7oUCEQ/s400/SP32-20150120-003459.gif" width="400" /></a></div>
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<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"><a href="http://en.wikipedia.org/wiki/Wallis_product" target="_blank"><span style="color: blue;">http://en.wikipedia.org/wiki/</span><span style="color: red;"><strong>Wallis_product</strong></span></a><o:p></o:p></span></div>
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"><a href="http://en.wikipedia.org/wiki/Infinite_product#Product_representations_of_functions" target="_blank"><span style="color: blue;">http://en.wikipedia.org/wiki/Infinite_product#Product_representations_of_functions</span></a><o:p></o:p></span><br />
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<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"><a href="http://mathworld.wolfram.com/WallisFormula.html" target="_blank"><span style="color: blue;">http://mathworld.wolfram.com/<span style="color: red;"><strong>WallisFormula</strong></span>.html</span></a><o:p></o:p></span></div>
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"><a href="http://mathworld.wolfram.com/PippengerProduct.html" target="_blank"><span style="color: blue;">http://mathworld.wolfram.com/<span style="color: red;"><strong>PippengerProduct</strong></span>.html</span></a><o:p></o:p></span><br />
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<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"><a href="http://en.wikipedia.org/wiki/Leibniz_formula_for_%CF%80" target="_blank"><span style="color: blue;">http://en.wikipedia.org/wiki/Leibniz_formula_for_%CF%80</span></a><o:p></o:p></span></div>
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"><a href="http://en.wikipedia.org/wiki/Vi%C3%A8te%27s_formula" target="_blank"><span style="color: blue;">http://en.wikipedia.org/wiki/Vi%C3%A8te%27s_formula</span></a><o:p></o:p></span><br />
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<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"><o:p> </o:p></span></div>
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<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;">From the Wallis formula you can also get: <o:p></o:p></span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjOpwLJKFMvzt5UtqaclZ7I43WzdHQ2jYltOEkaJPFMYMRSJtA9OdVjcJfErX-fQh-lTbKKtzYZzl3TVzc_dR6bxWlh7-H2OFZ5clPUzA2jUT2Cwr68kw4CZ7-v09b9h13rkHVfHBM3ZQY/s1600/SP32-20150120-003536.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="75" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjOpwLJKFMvzt5UtqaclZ7I43WzdHQ2jYltOEkaJPFMYMRSJtA9OdVjcJfErX-fQh-lTbKKtzYZzl3TVzc_dR6bxWlh7-H2OFZ5clPUzA2jUT2Cwr68kw4CZ7-v09b9h13rkHVfHBM3ZQY/s400/SP32-20150120-003536.gif" width="400" /></a></div>
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<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"></span><br /></div>
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<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;">In Wolfram MathWorld site: <o:p></o:p></span></div>
<br />
<div align="center" class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 0pt; text-align: center;">
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"><a href="http://mathworld.wolfram.com/InfiniteProduct.html" target="_blank"><span style="color: blue;">http://mathworld.wolfram.com/InfiniteProduct.html</span></a><o:p></o:p></span></div>
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"><o:p> </o:p></span><br />
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; line-height: 115%; mso-ansi-language: EN-US; mso-bidi-language: AR-SA; mso-fareast-font-family: Calibri; mso-fareast-language: EN-US;">you can find an extensive list of infinite
products. In particular, the formula obtained above is a simple case (21) of a more
general set (D.W.Cantrell, 2006) </span></div>
zibhttp://www.blogger.com/profile/17401589710495280112noreply@blogger.com0tag:blogger.com,1999:blog-287967698191537182.post-41331172283651596782015-05-31T14:16:00.001-07:002015-07-11T14:22:23.778-07:005. Tesseract<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;">In <strong><a href="http://en.wikipedia.org/wiki/Geometry" target="_blank">geometry</a></strong>, the <strong><a href="http://en.wikipedia.org/wiki/Tesseract" target="_blank">tesseract</a></strong> is the four-dimensional
analog of the cube.<o:p></o:p></span><br />
<br />
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<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"><strong><a href="http://en.wikipedia.org/wiki/Hypercube" target="_blank">Hypercube</a></strong> is composed of: <o:p></o:p></span></div>
<br />
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<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 16pt; mso-ansi-language: EN-US; mso-bidi-font-size: 12.0pt;">8 Cells (</span><span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 14pt; mso-ansi-language: EN-US; mso-bidi-font-size: 12.0pt;">or Cubes</span><span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 16pt; mso-ansi-language: EN-US; mso-bidi-font-size: 12.0pt;">),<span style="mso-spacerun: yes;"> </span>24
Faces,<span style="mso-spacerun: yes;"> </span>32 Edges <span style="mso-spacerun: yes;"> </span>and <span style="mso-spacerun: yes;"> </span>16 Vertices<o:p></o:p></span></div>
<br />
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;">and it meets the extension of Euler's formula:<span style="mso-spacerun: yes;"> </span></span><br />
<br />
<div style="text-align: center;">
<b style="mso-bidi-font-weight: normal;"><span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 18pt; mso-ansi-language: EN-US; mso-bidi-font-size: 12.0pt;">Faces + Vertices = Cells + Edges</span></b><span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"><span style="mso-spacerun: yes;"> </span><o:p></o:p></span></div>
<br />
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<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"></span><br /></div>
<div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 0pt;">
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;">There may be different types of representations; such
as this where all edges have the same length:<o:p></o:p></span></div>
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"><o:p> </o:p></span><br />
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<br />
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"><o:p> </o:p></span><br />
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<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;">Or a central projection:</span></div>
<div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 0pt;">
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"><o:p></o:p></span> </div>
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<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"><o:p> </o:p></span></div>
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<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"></span><br /></div>
<div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 0pt; text-align: justify;">
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;">In four dimensions, the
hypercube is also called tesseract (from the greek τέσσερις ακτίνες or
"four beams"). <o:p></o:p></span></div>
<div style="text-align: justify;">
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"><o:p> </o:p></span></div>
<div style="text-align: justify;">
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;">A famous example of a
tesseract is the <b style="mso-bidi-font-weight: normal;"><span style="background: yellow; mso-highlight: yellow;">Arch of La Défense</span></b>, a monument located
in the modern district of La Défense in Paris. The official name in French is <b style="mso-bidi-font-weight: normal;"><a href="http://en.wikipedia.org/wiki/Grande_Arche" target="_blank">Grande Arche de la Fraternité</a></b>.<o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"><o:p> </o:p></span></div>
<div class="separator" style="clear: both; text-align: center;">
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<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"><o:p> </o:p></span></div>
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<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"><a href="http://www.mathematische-basteleien.de/hypercube.htm" target="_blank"><span style="color: blue;">http://www.mathematische-basteleien.de/hypercube.htm</span></a><o:p></o:p></span></div>
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"><a href="http://zibalsc.blogspot.fr/2010/12/2-formula-di-eulero-per-i-poliedri.html" target="_blank"><span style="color: blue;">http://zibalsc.blogspot.fr/2010/12/2-formula-di-eulero-per-i-poliedri.html</span></a><o:p></o:p></span><br />
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<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"><a href="http://en.wikipedia.org/wiki/Tesseract" target="_blank"><span style="color: blue;">http://en.wikipedia.org/wiki/Tesseract</span></a><o:p></o:p></span></div>
<br />
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<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"><o:p> </o:p></span></div>
zibhttp://www.blogger.com/profile/17401589710495280112noreply@blogger.com0tag:blogger.com,1999:blog-287967698191537182.post-70252913264548648822015-05-20T13:51:00.002-07:002015-05-20T14:31:59.694-07:004. Vincent – Starry Nights<em></em><br />
<div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 0pt 144pt;">
<i style="mso-bidi-font-style: normal;"><span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US; mso-bidi-font-size: 11.0pt;">And if the cloud bursts, thunder in your ear</span></i></div>
<div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 0pt 144pt;">
<i style="mso-bidi-font-style: normal;"><span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US; mso-bidi-font-size: 11.0pt;">You shout and no one seems to hear</span></i></div>
<div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 0pt 144pt;">
<i style="mso-bidi-font-style: normal;"><span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US; mso-bidi-font-size: 11.0pt;">And if the band you're in starts playing different tunes</span></i></div>
<div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 0pt 144pt;">
<i style="mso-bidi-font-style: normal;"><span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US; mso-bidi-font-size: 11.0pt;">I'll see you on the dark side of the moon.</span></i></div>
<div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 0pt 144pt;">
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US; mso-bidi-font-size: 11.0pt;"><span style="mso-spacerun: yes;"> </span><span style="mso-tab-count: 2;"> </span><span style="mso-spacerun: yes;"> </span></span><span lang="EN-US" style="font-family: "Arial",sans-serif; mso-ansi-language: EN-US;"><span style="font-size: x-small;">Roger Waters - <i style="mso-bidi-font-style: normal;">Brain Damage</i></span><o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US; mso-bidi-font-size: 11.0pt;"><o:p></o:p></span> </div>
<div class="separator" style="clear: both; text-align: center;">
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<br />
<div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 0pt;">
<span lang="EN-US" style="font-family: "Comic Sans MS"; font-size: 14pt; mso-ansi-language: EN-US; mso-bidi-font-family: Arial; mso-bidi-font-size: 11.0pt;">Starry,
starry nights<o:p></o:p></span></div>
<span lang="EN-US" style="font-family: "Comic Sans MS"; font-size: 14pt; mso-ansi-language: EN-US; mso-bidi-font-family: Arial; mso-bidi-font-size: 11.0pt;">Paint
your palette blue and grey<o:p></o:p></span><br />
<span lang="EN-US" style="font-family: "Comic Sans MS"; font-size: 14pt; mso-ansi-language: EN-US; mso-bidi-font-family: Arial; mso-bidi-font-size: 11.0pt;">Look
out on a summer's day <o:p></o:p></span><br />
<span lang="EN-US" style="font-family: "Comic Sans MS"; font-size: 14pt; mso-ansi-language: EN-US; mso-bidi-font-family: Arial; mso-bidi-font-size: 11.0pt;">With
eyes that know the darkness in my soul</span><span lang="EN-US" style="font-family: "Comic Sans MS"; font-size: 12pt; mso-ansi-language: EN-US; mso-bidi-font-family: Arial; mso-bidi-font-size: 11.0pt;"><o:p></o:p></span><br />
<span lang="EN-US" style="font-family: "Comic Sans MS"; mso-ansi-language: EN-US; mso-bidi-font-family: Arial;"><o:p> </o:p></span><br />
<div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 0pt;">
<span lang="EN-US" style="font-family: "Comic Sans MS"; mso-ansi-language: EN-US; mso-bidi-font-family: Arial;">Shadows on the hills<o:p></o:p></span></div>
<span lang="EN-US" style="font-family: "Comic Sans MS"; mso-ansi-language: EN-US; mso-bidi-font-family: Arial;">Sketch the trees and the daffodils<o:p></o:p></span><br />
<span lang="EN-US" style="font-family: "Comic Sans MS"; mso-ansi-language: EN-US; mso-bidi-font-family: Arial;">Catch the breeze and the winter chills<o:p></o:p></span><br />
<span lang="EN-US" style="font-family: "Comic Sans MS"; mso-ansi-language: EN-US; mso-bidi-font-family: Arial;">In colors on the snowy linen land<o:p></o:p></span><br />
<br />
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<span lang="EN-US" style="font-family: "Comic Sans MS"; mso-ansi-language: EN-US; mso-bidi-font-family: Arial;">Now I understand<o:p></o:p></span></div>
<span lang="EN-US" style="font-family: "Comic Sans MS"; mso-ansi-language: EN-US; mso-bidi-font-family: Arial;">What you tried to say to me<o:p></o:p></span><br />
<span lang="EN-US" style="font-family: "Comic Sans MS"; mso-ansi-language: EN-US; mso-bidi-font-family: Arial;">And how you suffered for your sanity<o:p></o:p></span><br />
<span lang="EN-US" style="font-family: "Comic Sans MS"; mso-ansi-language: EN-US; mso-bidi-font-family: Arial;">And how you tried to set them free<o:p></o:p></span><br />
<span lang="EN-US" style="font-family: "Comic Sans MS"; mso-ansi-language: EN-US; mso-bidi-font-family: Arial;">They would not listen, they did not know how,<o:p></o:p></span><br />
<span lang="EN-US" style="font-family: "Comic Sans MS"; mso-ansi-language: EN-US; mso-bidi-font-family: Arial;">Perhaps they'll listen now<o:p></o:p></span><br />
<br />
<div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 0pt;">
<span lang="EN-US" style="font-family: "Comic Sans MS"; mso-ansi-language: EN-US; mso-bidi-font-family: Arial;">Starry, starry night<o:p></o:p></span></div>
<span lang="EN-US" style="font-family: "Comic Sans MS"; mso-ansi-language: EN-US; mso-bidi-font-family: Arial;">Flaming flowers that brightly blaze<o:p></o:p></span><br />
<span lang="EN-US" style="font-family: "Comic Sans MS"; mso-ansi-language: EN-US; mso-bidi-font-family: Arial;">Swirling clouds in violet haze<o:p></o:p></span><br />
<span lang="EN-US" style="font-family: "Comic Sans MS"; mso-ansi-language: EN-US; mso-bidi-font-family: Arial;">Reflect in Vincent's eyes of china blue<o:p></o:p></span><br />
<br />
<div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 0pt;">
<span lang="EN-US" style="font-family: "Comic Sans MS"; mso-ansi-language: EN-US; mso-bidi-font-family: Arial;">Colors changing hue<o:p></o:p></span></div>
<span lang="EN-US" style="font-family: "Comic Sans MS"; mso-ansi-language: EN-US; mso-bidi-font-family: Arial;">Morning fields of amber grain<o:p></o:p></span><br />
<span lang="EN-US" style="font-family: "Comic Sans MS"; mso-ansi-language: EN-US; mso-bidi-font-family: Arial;">Weathered faces lined in pain<o:p></o:p></span><br />
<span lang="EN-US" style="font-family: "Comic Sans MS"; mso-ansi-language: EN-US; mso-bidi-font-family: Arial;">Are soothed beneath the artist's loving hand<o:p></o:p></span><br />
<br />
<div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 0pt;">
<span lang="EN-US" style="font-family: "Comic Sans MS"; mso-ansi-language: EN-US; mso-bidi-font-family: Arial;">Now I understand<o:p></o:p></span></div>
<span lang="EN-US" style="font-family: "Comic Sans MS"; mso-ansi-language: EN-US; mso-bidi-font-family: Arial;">What you tried to say to me<o:p></o:p></span><br />
<span lang="EN-US" style="font-family: "Comic Sans MS"; mso-ansi-language: EN-US; mso-bidi-font-family: Arial;">And how you suffered for your sanity<o:p></o:p></span><br />
<span lang="EN-US" style="font-family: "Comic Sans MS"; mso-ansi-language: EN-US; mso-bidi-font-family: Arial;">And how you tried to set them free<o:p></o:p></span><br />
<span lang="EN-US" style="font-family: "Comic Sans MS"; mso-ansi-language: EN-US; mso-bidi-font-family: Arial;">They would not listen, they did not know how,<o:p></o:p></span><br />
<span lang="EN-US" style="font-family: "Comic Sans MS"; mso-ansi-language: EN-US; mso-bidi-font-family: Arial;">Perhaps they'll listen now<o:p></o:p></span><br />
<br />
<div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 0pt;">
<span lang="EN-US" style="font-family: "Comic Sans MS"; mso-ansi-language: EN-US; mso-bidi-font-family: Arial;">For they could not love you<o:p></o:p></span></div>
<span lang="EN-US" style="font-family: "Comic Sans MS"; mso-ansi-language: EN-US; mso-bidi-font-family: Arial;">But still your love was true<o:p></o:p></span><br />
<span lang="EN-US" style="font-family: "Comic Sans MS"; mso-ansi-language: EN-US; mso-bidi-font-family: Arial;">And when no hope was left inside<o:p></o:p></span><br />
<span lang="EN-US" style="font-family: "Comic Sans MS"; mso-ansi-language: EN-US; mso-bidi-font-family: Arial;">On that starry, starry night<o:p></o:p></span><br />
<span lang="EN-US" style="font-family: "Comic Sans MS"; mso-ansi-language: EN-US; mso-bidi-font-family: Arial;">You took your life as lovers often do<o:p></o:p></span><br />
<span lang="EN-US" style="font-family: "Comic Sans MS"; mso-ansi-language: EN-US; mso-bidi-font-family: Arial;">But I could have told you, Vincent, <o:p></o:p></span><br />
<span lang="EN-US" style="font-family: "Comic Sans MS"; mso-ansi-language: EN-US; mso-bidi-font-family: Arial;">This world was never meant for one as beautiful
as you<o:p></o:p></span><br />
<br />
<div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 0pt;">
<span lang="EN-US" style="font-family: "Comic Sans MS"; mso-ansi-language: EN-US; mso-bidi-font-family: Arial;">Starry, starry night<o:p></o:p></span></div>
<span lang="EN-US" style="font-family: "Comic Sans MS"; mso-ansi-language: EN-US; mso-bidi-font-family: Arial;">Portraits hung in empty halls<o:p></o:p></span><br />
<span lang="EN-US" style="font-family: "Comic Sans MS"; mso-ansi-language: EN-US; mso-bidi-font-family: Arial;">Frameless heads on nameless walls<o:p></o:p></span><br />
<span lang="EN-US" style="font-family: "Comic Sans MS"; mso-ansi-language: EN-US; mso-bidi-font-family: Arial;">With eyes that watch the world and can't forget<o:p></o:p></span><br />
<span lang="EN-US" style="font-family: "Comic Sans MS"; mso-ansi-language: EN-US; mso-bidi-font-family: Arial;">Like strangers that you've met<o:p></o:p></span><br />
<span lang="EN-US" style="font-family: "Comic Sans MS"; mso-ansi-language: EN-US; mso-bidi-font-family: Arial;">The ragged men in ragged clothes<o:p></o:p></span><br />
<span lang="EN-US" style="font-family: "Comic Sans MS"; mso-ansi-language: EN-US; mso-bidi-font-family: Arial;">The silver thorn, the bloody rose, <o:p></o:p></span><br />
<span lang="EN-US" style="font-family: "Comic Sans MS"; mso-ansi-language: EN-US; mso-bidi-font-family: Arial;">Lie crushed and broken on the virgin snow<o:p></o:p></span><br />
<br />
<div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 0pt;">
<span lang="EN-US" style="font-family: "Comic Sans MS"; mso-ansi-language: EN-US; mso-bidi-font-family: Arial;">Now, I think I know<o:p></o:p></span></div>
<span lang="EN-US" style="font-family: "Comic Sans MS"; mso-ansi-language: EN-US; mso-bidi-font-family: Arial;">What you tried to say to me<o:p></o:p></span><br />
<span lang="EN-US" style="font-family: "Comic Sans MS"; mso-ansi-language: EN-US; mso-bidi-font-family: Arial;">And how you suffered for your sanity<o:p></o:p></span><br />
<span lang="EN-US" style="font-family: "Comic Sans MS"; mso-ansi-language: EN-US; mso-bidi-font-family: Arial;">And how you tried to set them free<o:p></o:p></span><br />
<span lang="EN-US" style="font-family: "Comic Sans MS"; mso-ansi-language: EN-US; mso-bidi-font-family: Arial;">They would not listen, they're not listening
still<o:p></o:p></span><br />
<span lang="EN-US" style="font-family: "Comic Sans MS"; mso-ansi-language: EN-US; mso-bidi-font-family: Arial;">Perhaps they never will.</span><br />
<div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 0pt;">
<span lang="EN-US" style="font-family: "Comic Sans MS"; mso-ansi-language: EN-US; mso-bidi-font-family: Arial;"><span style="mso-tab-count: 4;"> </span><span style="mso-spacerun: yes;"> </span>Don McLean - Vincent<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 0pt;">
<span style="font-family: "Arial",sans-serif; font-size: 12pt; mso-bidi-font-size: 11.0pt;"><o:p> </o:p></span></div>
<br />
<div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 0pt; text-align: justify;">
<b style="mso-bidi-font-weight: normal;"><span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US; mso-bidi-font-size: 11.0pt;">Vincent van Gogh</span></b><span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US; mso-bidi-font-size: 11.0pt;"> produced more than 900 paintings and 1,100 drawings,
despite the fact that he lived only 37 years. He suffered from a form of
epilepsy and a manic tendency to write and paint. In January 1889 he was
discharged from the hospital in Arles, where he had been hospitalized after the
incident. The precise chain of events that led to the celebrated incident of
van Gogh slicing off his ear is not known reliably in detail.<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 0pt; text-align: justify;">
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US; mso-bidi-font-size: 11.0pt;">In just over a decade he produced more than 2,100 artworks, including
860 oil paintings and more than 1,300 watercolors, drawings, sketches and
prints.<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 0pt; text-align: justify;">
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US; mso-bidi-font-size: 11.0pt;">But the artist felt that he had not fully recovered: his mental health
was not yet integrated. So, in May, he sought asylum at the psychiatric
hospital near Saint Remy in France. <o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 0pt; text-align: justify;">
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US; mso-bidi-font-size: 11.0pt;">It was here that he painted some of his most famous works, such as
"Starry Night".<o:p></o:p></span></div>
<br />
<a href="http://en.wikipedia.org/wiki/Vincent_van_Gogh" target="_blank">http://en.wikipedia.org/wiki/<strong><span style="background-color: yellow; font-size: large;">Vincent_van_Gogh</span></strong></a><br />
<a href="http://en.wikipedia.org/wiki/The_Starry_Night" target="_blank">http://en.wikipedia.org/wiki/<strong><span style="background-color: yellow; font-size: large;">The_Starry_Night</span></strong></a><br />
<br />
<div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 0pt; text-align: justify;">
</div>
<div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 0pt; text-align: justify;">
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US; mso-bidi-font-size: 11.0pt;">"<b style="mso-bidi-font-weight: normal;">Starry Night</b>" is
an oil painting on canvas (92x73 cm), made in 1889 and preserved in the Museum
of Modern Art ( <b style="mso-bidi-font-weight: normal;"><a href="http://en.wikipedia.org/wiki/Museum_of_Modern_Art" target="_blank">MoMA</a></b> ) in New York.<o:p></o:p></span></div>
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US; mso-bidi-font-size: 11.0pt;"><o:p> </o:p></span><br />
<br />
<div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 0pt; text-align: justify;">
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US; mso-bidi-font-size: 11.0pt;">The images shown below, indicate which constellations are supposedly
depicted in the paintings and were taken from the online magazine <b style="mso-bidi-font-weight: normal;"><i style="mso-bidi-font-style: normal;">Elapsus</i></b>:</span></div>
<h3 style="text-align: center;">
<a href="http://www.elapsus.it/home1/index.php/scienza/astronomia/63-tra-arte-e-astronomia-le-stelle-di-van-gogh" target="_blank">http://www.elapsus.it/home1/index.php/scienza/astronomia/63-tra-arte-e-astronomia-le-stelle-di-van-gogh</a></h3>
<div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 0pt; text-align: justify;">
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US; mso-bidi-font-size: 11.0pt;"><o:p> </o:p></span></div>
<table cellpadding="0" cellspacing="0" class="tr-caption-container" style="float: left;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgPPhbJHOHkEin2LY47fDVMwIJtJHUVG8shujbvwBm2wYEBC00wuU59kx2bOYbpcwyUhWNw3CYCR4jl3vzK4lHlCwDkbNZvxjyshpsNhreL-ACIkzP0QRe_VpRZ1pW12sUmOKstB6m7sow/s1600/SP32-20150406-222150.gif" imageanchor="1" style="clear: left; margin-bottom: 1em; margin-left: auto; margin-right: auto;"><img border="0" height="264" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgPPhbJHOHkEin2LY47fDVMwIJtJHUVG8shujbvwBm2wYEBC00wuU59kx2bOYbpcwyUhWNw3CYCR4jl3vzK4lHlCwDkbNZvxjyshpsNhreL-ACIkzP0QRe_VpRZ1pW12sUmOKstB6m7sow/s640/SP32-20150406-222150.gif" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><span style="font-size: large;">Starry Night</span></td></tr>
</tbody></table>
<span style="font-family: "Arial",sans-serif; font-size: 12pt; mso-bidi-font-size: 11.0pt;"><span style="mso-spacerun: yes;"> </span><o:p></o:p></span><br />
<br />
<table cellpadding="0" cellspacing="0" class="tr-caption-container" style="float: left;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEitaw16qEk3_dg0UtPvh_rRMFbkCbL7bkv3YDZVlWHr5hfk7ByUnI6jsgd3nq_Hn-d0P3ID9lrc75KmFtUZxtkup23Bou-at9qrYCaKv-j4XiGH5wtqyYdTSKgIq6Q65vAqyv2Lfl-rQpM/s1600/SP32-20150322-212609.gif" imageanchor="1" style="clear: left; margin-bottom: 1em; margin-left: auto; margin-right: auto;"><img border="0" height="426" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEitaw16qEk3_dg0UtPvh_rRMFbkCbL7bkv3YDZVlWHr5hfk7ByUnI6jsgd3nq_Hn-d0P3ID9lrc75KmFtUZxtkup23Bou-at9qrYCaKv-j4XiGH5wtqyYdTSKgIq6Q65vAqyv2Lfl-rQpM/s640/SP32-20150322-212609.gif" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><strong>Starry Night Over the Rhone</strong></td></tr>
</tbody></table>
<br />
<br />
<br />
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj_3kNbNQrD7i33dNNZF0kuqj25IEkXLtC71HMSvnrzBheh6DWc6JmE8LKOezKChK7pntBGLu86MustAyR_z_NMWFXNJKCFUpkQk7EzpqV7EPKkvY86s-A0GERVHzjR3wEAw3d8EITiI_M/s1600/SP32-20150405-232344.gif" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="498" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj_3kNbNQrD7i33dNNZF0kuqj25IEkXLtC71HMSvnrzBheh6DWc6JmE8LKOezKChK7pntBGLu86MustAyR_z_NMWFXNJKCFUpkQk7EzpqV7EPKkvY86s-A0GERVHzjR3wEAw3d8EITiI_M/s640/SP32-20150405-232344.gif" width="640" /></a><br />
<br />
<i style="mso-bidi-font-style: normal;"><span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US; mso-bidi-font-size: 11.0pt;"></span></i><br />
<i style="mso-bidi-font-style: normal;"><span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US; mso-bidi-font-size: 11.0pt;"></span></i><br />
<i style="mso-bidi-font-style: normal;"><span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US; mso-bidi-font-size: 11.0pt;"></span></i><br />
<i style="mso-bidi-font-style: normal;"><span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US; mso-bidi-font-size: 11.0pt;"></span></i><br />
<i style="mso-bidi-font-style: normal;"><span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US; mso-bidi-font-size: 11.0pt;">Vincent </span></i><span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US; mso-bidi-font-size: 11.0pt;">(by <strong>Don McLean</strong>) is from the album <i style="mso-bidi-font-style: normal;">American Pie</i>
recorded and released in 1971. It's also known as <i style="mso-bidi-font-style: normal;">Starry Starry
Night</i>. <o:p></o:p></span><br />
<br />
<div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 0pt; text-align: justify;">
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US; mso-bidi-font-size: 11.0pt;"></span><br /></div>
<div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 0pt; text-align: justify;">
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US; mso-bidi-font-size: 11.0pt;">The words of the first stanza are a clear reference to the painting <b style="mso-bidi-font-weight: normal;">Starry Night</b>.<o:p></o:p></span></div>
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US; mso-bidi-font-size: 11.0pt;"> It's possible to listen to the
following YouTube link:</span><br />
<br />
<div align="center" class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 0pt; text-align: center;">
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: large; mso-ansi-language: EN-US; mso-bidi-font-size: 11.0pt;"><a href="https://www.youtube.com/watch?v=oxHnRfhDmrk" target="_blank">https://www.youtube.com/watch?v=oxHnRfhDmrk</a></span><span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US; mso-bidi-font-size: 11.0pt;"><o:p></o:p></span></div>
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US; mso-bidi-font-size: 11.0pt;"><o:p> </o:p></span><br />
<br />
<div align="center" class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 0pt; text-align: center;">
<span style="font-family: "Arial",sans-serif;"><a href="http://fr.wikipedia.org/wiki/Mus%C3%A9e_d%27Orsay" target="_blank">http://fr.wikipedia.org/wiki/Mus%C3%A9e_d%27<strong><span style="background-color: yellow;">Orsay</span></strong></a></span><span style="font-family: "Arial",sans-serif; font-size: 12pt; mso-bidi-font-size: 11.0pt;"><o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 0pt; text-align: justify;">
<span style="font-family: "Arial",sans-serif; font-size: 12pt; mso-bidi-font-size: 11.0pt;"><span style="mso-spacerun: yes;"> </span><o:p></o:p></span></div>
zibhttp://www.blogger.com/profile/17401589710495280112noreply@blogger.com0tag:blogger.com,1999:blog-287967698191537182.post-89161740774182869422015-05-18T14:46:00.003-07:002015-05-18T16:12:10.857-07:003. Rule of 9’s<span lang="EN-US" style="font-family: "Courier New"; font-size: 12pt; mso-ansi-language: EN-US;">The rule of 9% is used to estimate the extent of a
burn. <o:p></o:p></span><br />
<span lang="EN-US" style="font-family: "Courier New"; font-size: 12pt; mso-ansi-language: EN-US;"><o:p> </o:p></span><br />
<br />
<div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 0pt;">
<u><span lang="EN-US" style="font-family: "Courier New"; font-size: 12pt; mso-ansi-language: EN-US;">Percentage of Body Surface in an Adult</span></u><span lang="EN-US" style="font-family: "Courier New"; font-size: 12pt; mso-ansi-language: EN-US;">: <o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 0pt;">
<span lang="EN-US" style="font-family: "Courier New"; font-size: 12pt; mso-ansi-language: EN-US;">Head and Neck<span style="mso-spacerun: yes;"> </span><span style="mso-tab-count: 1;"> </span><span style="mso-spacerun: yes;"> </span>9% <o:p></o:p></span></div>
<span lang="EN-US" style="font-family: "Courier New"; font-size: 12pt; mso-ansi-language: EN-US;">Upper torso<span style="mso-spacerun: yes;"> </span><span style="mso-tab-count: 1;"> </span><span style="mso-spacerun: yes;"> </span>9%<span style="mso-spacerun: yes;"> </span><span style="mso-tab-count: 1;"> </span>(Also
posterior 9%) <o:p></o:p></span><br />
<span lang="EN-US" style="font-family: "Courier New"; font-size: 12pt; mso-ansi-language: EN-US;">Mid torso<span style="mso-spacerun: yes;"> </span><span style="mso-spacerun: yes;"> </span><span style="mso-tab-count: 1;"> </span><span style="mso-tab-count: 1;"> </span>9%<span style="mso-spacerun: yes;"> </span><span style="mso-tab-count: 1;"> </span>(Also
posterior 9%) <o:p></o:p></span><br />
<span lang="EN-US" style="font-family: "Courier New"; font-size: 12pt; mso-ansi-language: EN-US;">Upper limbs<span style="mso-spacerun: yes;"> </span><span style="mso-spacerun: yes;"> </span><span style="mso-tab-count: 1;"> </span><span style="mso-spacerun: yes;"> </span>9%<span style="mso-spacerun: yes;"> </span><span style="mso-tab-count: 1;"> </span>(Each) <o:p></o:p></span><br />
<span lang="EN-US" style="font-family: "Courier New"; font-size: 12pt; mso-ansi-language: EN-US;">Lower limbs<span style="mso-spacerun: yes;"> </span><span style="mso-tab-count: 1;"> </span><span style="mso-tab-count: 1;"> </span>18%<span style="mso-spacerun: yes;"> </span><span style="mso-tab-count: 1;"> </span>(Each)
<o:p></o:p></span><br />
<span lang="EN-US" style="font-family: "Courier New"; font-size: 12pt; mso-ansi-language: EN-US;">Perineum<span style="mso-spacerun: yes;"> </span><span style="mso-spacerun: yes;"> </span><span style="mso-tab-count: 1;"> </span><span style="mso-spacerun: yes;"> </span>1% <o:p></o:p></span><br />
<br />
<div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 0pt;">
<span lang="EN-US" style="font-family: "Courier New"; font-size: 12pt; mso-ansi-language: EN-US;"><o:p> </o:p></span></div>
<br />
<div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 0pt;">
<u><span lang="EN-US" style="font-family: "Courier New"; font-size: 12pt; mso-ansi-language: EN-US;">Percentage of Body Surface in a Child</span></u><span lang="EN-US" style="font-family: "Courier New"; font-size: 12pt; mso-ansi-language: EN-US;">:<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 0pt;">
<span lang="EN-US" style="font-family: "Courier New"; font-size: 12pt; mso-ansi-language: EN-US;">Head and Neck<span style="mso-spacerun: yes;"> </span><span style="mso-spacerun: yes;"> </span><span style="mso-tab-count: 1;"> </span>18% <o:p></o:p></span></div>
<span lang="EN-US" style="font-family: "Courier New"; font-size: 12pt; mso-ansi-language: EN-US;">Upper torso<span style="mso-spacerun: yes;"> </span><span style="mso-spacerun: yes;"> </span><span style="mso-tab-count: 1;"> </span><span style="mso-spacerun: yes;"> </span>9%<span style="mso-spacerun: yes;"> </span><span style="mso-tab-count: 1;"> </span>(Also posterior 9%) <o:p></o:p></span><br />
<span lang="EN-US" style="font-family: "Courier New"; font-size: 12pt; mso-ansi-language: EN-US;">Mid torso<span style="mso-spacerun: yes;"> </span><span style="mso-spacerun: yes;"> </span><span style="mso-tab-count: 1;"> </span><span style="mso-tab-count: 1;"> </span><span style="mso-spacerun: yes;"> </span>9%<span style="mso-spacerun: yes;"> </span><span style="mso-tab-count: 1;"> </span>(Also posterior 9%) <o:p></o:p></span><br />
<span lang="EN-US" style="font-family: "Courier New"; font-size: 12pt; mso-ansi-language: EN-US;">Upper limbs<span style="mso-spacerun: yes;"> </span><span style="mso-spacerun: yes;"> </span><span style="mso-tab-count: 1;"> </span><span style="mso-spacerun: yes;"> </span>9%<span style="mso-spacerun: yes;"> </span><span style="mso-tab-count: 1;"> </span>(Each) <o:p></o:p></span><br />
<span lang="EN-US" style="font-family: "Courier New"; font-size: 12pt; mso-ansi-language: EN-US;">Lower limbs<span style="mso-spacerun: yes;">
</span><span style="mso-tab-count: 1;"> </span>14%<span style="mso-spacerun: yes;"> </span><span style="mso-tab-count: 1;"> </span>(Each)
<o:p></o:p></span><br />
<br />
<div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 0pt;">
<span lang="EN-US" style="font-family: "Courier New"; font-size: 12pt; mso-ansi-language: EN-US;"><o:p> </o:p></span></div>
<br />
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<u><span lang="EN-US" style="font-family: "Courier New"; font-size: 12pt; mso-ansi-language: EN-US;">Rule of Palm</span></u><span lang="EN-US" style="font-family: "Courier New"; font-size: 12pt; mso-ansi-language: EN-US;">:<o:p></o:p></span></div>
<br />
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<span lang="EN-US" style="font-family: "Courier New"; font-size: 12pt; mso-ansi-language: EN-US;">Each palm is approximately 1% of Body Surface.</span></div>
<div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 0pt;">
<span lang="EN-US" style="font-family: "Courier New"; font-size: 12pt; mso-ansi-language: EN-US;"></span> </div>
<div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 0pt;">
<span lang="EN-US" style="font-family: "Courier New"; font-size: 12pt; mso-ansi-language: EN-US;"></span> </div>
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<span lang="EN-US" style="font-family: "Courier New"; font-size: 12pt; mso-ansi-language: EN-US;"></span><br /></div>
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<span lang="EN-US" style="font-family: "Courier New"; font-size: 12pt; mso-ansi-language: EN-US;"><o:p> </o:p></span></div>
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<span lang="EN-US" style="font-family: "Courier New"; font-size: 12pt; mso-ansi-language: EN-US;"><a href="http://en.wikipedia.org/wiki/Burn"><span style="color: blue;">http://en.wikipedia.org/wiki/Burn</span></a><o:p></o:p></span></div>
<span lang="EN-US" style="font-family: "Courier New"; font-size: 12pt; mso-ansi-language: EN-US;"><a href="http://en.wikipedia.org/wiki/Total_body_surface_area"><span style="color: blue;">http://en.wikipedia.org/wiki/Total_body_surface_area</span></a><o:p></o:p></span><br />
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<span lang="EN-US" style="font-family: "Courier New"; font-size: 12pt; mso-ansi-language: EN-US;"><o:p> </o:p></span></div>
zibhttp://www.blogger.com/profile/17401589710495280112noreply@blogger.com0tag:blogger.com,1999:blog-287967698191537182.post-19004343608075393452015-05-18T14:03:00.003-07:002015-05-18T14:06:04.126-07:002. How It Works: The Sewing Machine<div style="text-align: justify;">
</div>
<div style="text-align: justify;">
</div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEihbRrvZfBMBSllxS36W1BAhRA7vDsZguvDYMWvSlztcWgad3cEoq3ybIMeSLbDo9Uyr56HriDEer_jbQgVZD8kiZuCTrFUHwXphyphenhyphenDKcUoJFllCd0ls2SKLv4xte8TwVrZFD7PvDa1gYkw/s1600/macchina_per_cucire.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEihbRrvZfBMBSllxS36W1BAhRA7vDsZguvDYMWvSlztcWgad3cEoq3ybIMeSLbDo9Uyr56HriDEer_jbQgVZD8kiZuCTrFUHwXphyphenhyphenDKcUoJFllCd0ls2SKLv4xte8TwVrZFD7PvDa1gYkw/s400/macchina_per_cucire.gif" width="382" /></a></div>
<br />
<div style="text-align: justify;">
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;">It's a topological
question: when the needle passes the tissue below, it carries the wire in the
loop. On the other side there is a system that puts the wire of the second
spool (the little one in the machine) in this loop, then the needle rises,
taking with it the second wire. <o:p></o:p></span></div>
<div style="text-align: justify;">
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;">In this way a series of
knots are formed along the seam between the spool higher, the big one, and the
lower, the small one.<o:p></o:p></span></div>
<div style="text-align: justify;">
</div>
<div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 0pt; text-align: justify;">
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;">The first practical and
widely used sewing machine was invented by <strong>Barthélemy Thimonnier</strong>, a French
tailor, in 1829.</span><br />
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"><o:p></o:p></span> </div>
<div style="text-align: justify;">
</div>
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<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"><a href="http://en.wikipedia.org/wiki/Sewing_machine"><span style="color: blue;">http://en.wikipedia.org/wiki/Sewing_machine</span></a></span></div>
<div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 0pt; text-align: justify;">
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"><a href="http://www.howitworksdaily.com/">http://www.howitworksdaily.com/</a></span></div>
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<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"></span> </div>
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<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"></span> </div>
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<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"></span> </div>
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zibhttp://www.blogger.com/profile/17401589710495280112noreply@blogger.com0tag:blogger.com,1999:blog-287967698191537182.post-68226646394721721612015-05-17T15:15:00.000-07:002015-05-17T15:33:55.602-07:001. Nights without stars<div style="text-align: justify;">
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;">Imagine how it
would be our idea of the Universe if the sky was always covered by clouds. </span><br />
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;">At a
certain hour of the morning we would begin to have a diffuse light and after a
while the dark returns. We would not have a reference that will show us the
time during day or night (Sun, Moon or stars) and would not know even the
shadows of the various objects exposed to the sun. </span><br />
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;">In addition to not know the
charm of some sunsets, sundials would not have been invented. <o:p></o:p></span></div>
<div style="text-align: justify;">
</div>
<div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 0pt; text-align: justify;">
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;">We would have the same
clues that could provide some element: magnetic materials that orients
according to a certain direction, the pendulums that tend to have a rotary
motion and the relation between the periods of the year where the days are
warmer, with the fact that they are also brighter and longer. <o:p></o:p></span></div>
<div style="text-align: justify;">
</div>
<div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 0pt; text-align: justify;">
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;">Thanks to different
position of stars at different latitudes, Greek philosophers were able to understand
curvature of the Earth. <o:p></o:p></span></div>
<div style="text-align: justify;">
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;">Or, as a result, the
observation of the shadow of the Earth projected onto the Moon during a lunar
eclipse. <o:p></o:p></span></div>
<div style="text-align: justify;">
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;">And, finally, on the
horizon that seems bent, with ships that disappear at a certain distance from
the shore, providing a clear indication of the curvature of the Earth, perhaps
we couldn’t observe it so frequently, since few people would venture offshore
without the help of the stars for orientation. <o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 0pt; text-align: justify;">
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;">The French physicist
<strong>Gaspard Gustave de Coriolis</strong> in 1835 showed that an observer placed on a rotating
frame of reference will have to deal with forces (Coriolis) which depend not
only by angular velocity, but also by direction and body speed. <o:p></o:p></span></div>
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;">These fictitious forces
are the basis of cyclonic or anti-cyclonic atmosphere systems formation.<o:p></o:p></span><br />
<br />
<div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 0pt; text-align: justify;">
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;">To remember the sense
of rotation of the phenomenon you can use this simple mnemonic scheme (valid
only in the northern hemisphere): <o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 0pt; text-align: justify;">
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;">·<span style="mso-spacerun: yes;"> </span>Anticyclone (high pressure)<span style="mso-spacerun: yes;"> </span>>>><span style="mso-spacerun: yes;"> </span>Clockwise <o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 0pt; text-align: justify;">
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;">·<span style="mso-spacerun: yes;"> </span>Cyclone<span style="mso-spacerun: yes;"> </span>(Low pressure)<span style="mso-spacerun: yes;"> </span><span style="mso-tab-count: 1;"> </span>>>><span style="mso-spacerun: yes;"> </span>Counterclockwise <o:p></o:p></span></div>
<br />
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<div class="separator" style="clear: both; text-align: center;">
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<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"><o:p> </o:p></span></div>
<br />
<div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 0pt; text-align: justify;">
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;">The effects of this
force are also manifested in atomic physics (in polyatomic molecules whose
molecular motion can be described as a rigid rotation with in addition a
vibration of the parts around the position of equilibrium) and this produces a
particular confusion, in the molecular spectrum, between the rotational and vibration
levels.<o:p></o:p></span></div>
<div style="text-align: justify;">
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;">Flies (Diptera) and
some moths (Lepidoptera) exploit the Coriolis effect in flight with specialized
appendages and organs that relay information about the angular velocity of
their bodies. Coriolis forces resulting from linear motion of these appendages
are detected within the rotating frame of reference of the insects' bodies. In
the case of flies, their specialized appendages are dumbbell shaped organs
located just behind their wings called halteres. The halteres oscillate in a
plane at the same beat frequency as the main wings so that any body rotation
results in lateral deviation of the haltered from their plane of motion. In
moths, their antennae are responsible for the sensing of Coriolis forces in the
similar manner as with the halters in flies. In both flies and moths, a
collection of mechanic-sensors at the base of the appendage are sensitive to
deviations at the beat frequency, correlating to rotation in the pitch and roll
planes, and at twice the beat frequency, correlating to rotation in the yaw
plane.<o:p></o:p></span></div>
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"><o:p> </o:p></span><br />
<br />
<div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 0pt; text-align: justify;">
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;">The best known example
is the <strong>Foucault pendulum</strong>. <o:p></o:p></span></div>
<br />
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<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"><o:p> </o:p></span></div>
<div style="text-align: center;">
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"><a href="http://en.wikipedia.org/wiki/Coriolis_effect"><span style="color: blue;">http://en.wikipedia.org/wiki/Coriolis_effect</span></a><o:p></o:p></span></div>
<div style="text-align: center;">
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"><a href="http://en.wikipedia.org/wiki/Focault_pendulum"><span style="color: blue;">http://en.wikipedia.org/wiki/Focault_pendulum</span></a><o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 0pt; text-align: justify;">
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;">It is a pendulum free
to swing in any direction for many hours. The Foucault pendulum was first
presented to the public in 1851 and consisted of a sphere of 28 kg suspended on
the dome of the Pantheon in Paris with a 67 m long wire. In an inertial system,
it would track lines always in the same direction, but it didn’t. <o:p></o:p></span></div>
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;">At different latitudes
of the Earth, except along the equator, it’s observed that the oscillation
plane of the pendulum slowly rotates. In particular at North and South Pole the
rotation is in a sidereal day: the swing plane remains stationary while the
Earth rotates, in accordance with the law of motion of Newton. <o:p></o:p></span><br />
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;">At other latitudes the
oscillation plane rotates with an inversely period proportional to the sine of
the same latitude; at 45° the rotation occurs every 1.4 days, at 30° every 2 days
and so on. <o:p></o:p></span><br />
<br />
<div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 0pt; text-align: justify;">
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;">The Coriolis force is
also used in modern gyroscopes that use solid state switches with MEMS
(Micro-Electro-Mechanical Systems). <o:p></o:p></span></div>
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"><o:p> </o:p></span><br />
<br />
<div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 0pt; text-align: justify;">
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;">What is certain is that
without the help of the stars the world-view of the Universe would be definitely
different. <o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"><a href="http://en.wikipedia.org/wiki/Nightfall_(Asimov_short_story_and_novel)"><span style="color: blue;">http://en.wikipedia.org/wiki/Nightfall_(Asimov_short_story_and_novel)</span></a><o:p></o:p></span></div>
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"><a href="http://en.wikipedia.org/wiki/Horizon"><span style="color: blue;">http://en.wikipedia.org/wiki/Horizon</span></a><o:p></o:p></span><br />
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"><a href="http://en.wikipedia.org/wiki/Spherical_Earth"><span style="color: blue;">http://en.wikipedia.org/wiki/Spherical_Earth</span></a></span><br />
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"><a href="http://zibalsc.blogspot.fr/2015/05/186-notti-senza-stelle.html">http://zibalsc.blogspot.fr/2015/05/186-notti-senza-stelle.html</a><o:p></o:p></span><br />
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<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;"><o:p></o:p></span><br />zibhttp://www.blogger.com/profile/17401589710495280112noreply@blogger.com0tag:blogger.com,1999:blog-287967698191537182.post-51854843381495823722015-05-17T13:53:00.001-07:002015-05-17T13:58:29.388-07:00Intro<div style="text-align: justify;">
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;">What you will find in
this blog is the result of reflections on scientific topics known and less-known,
of links and references ... so a <i style="mso-bidi-font-style: normal;">hotchpotch</i>
of ideas. <o:p></o:p></span></div>
<div style="text-align: justify;">
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;">Many descriptions and
sources quoted in the posts are taken from <i style="mso-bidi-font-style: normal;">Wikipedia,
The Free Encyclopedia</i>; in any case, the authors of quotes and links can ask
me to be removed them from the blog or, if you do need them, to quote freely
this site. <o:p></o:p></span><br />
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; mso-ansi-language: EN-US;">The various posts are
only intended to present the most various disparate topics that will be explored
via the links in the text. <o:p></o:p></span><br />
<span lang="EN-US" style="font-family: "Arial",sans-serif; font-size: 12pt; line-height: 115%; mso-ansi-language: EN-US; mso-bidi-language: AR-SA; mso-fareast-font-family: Calibri; mso-fareast-language: EN-US;">In summary a free flow of ideas.</span></div>
zibhttp://www.blogger.com/profile/17401589710495280112noreply@blogger.com0