18. Complex formulas

The following equation relates the well-known mathematical constants e, pi, i, to derive another famous constant:

the golden number φ = 1.61803398874989...

As with Euler's identity, also in this case some of the most important mathematical constants appear simultaneously in the same formula.

To demonstrate the relationship, we must start with the Golden Ratio.

It involves dividing a segment AB into 2 parts (which we will call AC and CB) in such a way that the continuous proportion AB : AC = AC : CB holds.

Euclid used this formula when working on pentagons.

We set the smallest segment CB = 1 and AC = x, from which AB = 1 + x

The required condition is therefore: (1 + x) / x = x / 1

So, we have x² – x – 1 = 0

The solutions of this quadratic equation have as solutions:

The Golden Section was the starting point for the Greek study of regular pentagons, and everything associated with them, such as the decagon, the dodecahedron and the icosahedron.

As we will see later, if we draw a regular pentagon with side 1, then the diagonals have the golden number as their length.

The term Golden Ratio is relatively recent and appears to have been used for the first time by Martin Ohm (brother of the more famous Georg Simon Ohm who gave his name to the law) in his 1835 book.

Before seeing why, let's review some trigonometry notions.

Trigonometry studies right triangles starting from their angles. Its main task consists in calculating the measurements that characterize the elements of the triangle (sides, angles, etc.) by means of special functions starting from already known measurements.

Trigonometric functions (the most important are sine and cosine) are also used independently of geometry, for example in connection with the exponential function.

1) The sine of an angle is the ratio between the length of the opposite side and the length of the hypotenuse.

2) The cosine of an angle is the ratio between the length of the adjacent side and the length of the hypotenuse.

Euler's formula is a formula in the field of complex analysis that shows a profound relationship between trigonometric functions and the complex exponential function. Euler's identity is a special case of Euler's formula.

Euler's formula, named after the mathematician Leonhard Euler, was first proven by Roger Cotes in 1714 and then rediscovered and made famous by Euler in 1748. Neither saw the geometric interpretation of the formula: the vision of complexes numbers like points in the plane arrived only about 50 years later, thanks to Wessel, Argand and Gauss.

The most widespread proof is based on the Taylor series expansion of the exponential function.

Euler's formula also allows the sine and cosine functions to be interpreted as simple variants of the exponential function:

Euler's formula: e^ix = cos x + i sin x

Euler's formula also allows the sine and cosine functions to be interpreted as simple variants of the exponential function: 

sin x = ( e^ix - e^-ix ) / 2i            cos x = ( e^ix + e^-ix ) / 2

As is known, angles can be expressed in different ways, the most used are sexagesimal degrees and radians. Below, depending on the purpose, both will be considered.

Let's now show some notable angles: 30, 36, 45, 60 and 90 degrees.

In particular, the sine (30°) = ½ (whose square is equal to ¼); similarly, the sine squares of 45, 60 and 90 degrees are respectively: 2/4, 3/4 and 4/4.

A regular pentagon with side 1 (e.g. DE) has other particular properties and it can be shown that the diagonals (e.g. AD) have length φ.

The 2 isosceles triangles ADE and DCE are similar so AD : DE = DE : CE

Setting AD = x we ​​have:

x : 1 = 1 : (x – 1)           1 = x² – x           x² – x – 1 = 0

So, in analogy to what was seen above: AD = φ

that is, in a regular pentagon of side 1, the diagonals are equal to φ

For the right triangle ABC we can therefore obtain:

cos 36° = AB / AC = φ / 2 = 0.809016994374945…

Combining this result with the cosine function, we obtain the initial statement:

Below I report other notable formulas:

Calcinator™ Free Online Mobile Web Scientific Calculator: complex numbers, exponential trigonometric statistics hyperbolic and algebraic functions

17. Benford law

Benford's law (binary numbering):

for any number greater than zero,

"1" appears as the first digit in

 100% of cases.

The problem was this: consider the first digit in the decimal expansion of 2ⁿ for n ≥ 0: 1, 2, 4, 8, 1, 3, 6, 1, 2, 5, 1, 2, 4, ...

Does the digit 7 appear in this sequence?

Does 7 or 8 appear more frequently? 

How much more frequently?

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, ...

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, 7, 1, 2, 5, 1, 2, 4, 9, 1, 3, 7, 1, 2, 5, 1, 2, 4, 9, 1, 3, 7, 1, 2, 5, 1, 2, 4, 9, 1, 3, 7, 1, 3, 6, 1, 2, 4, 9, 1, 3, 7, 1, 3, 6, 1, 2, 4, 9, 1, 3, ...

Sloane's On-Line Encyclopedia of Integer Sequences A008952

By definition, position 1 is set to log₁₀ 1 = 0, while position 10 to 1 (log₁₀ 10 = 1) and 100 to 2 (log₁₀ 100 = 2), the intermediate positions are at:

log₁₀ 2 = 0.3010, log₁₀ 3 = 0.4771, log₁₀ 4 = 0.6020,…, log₁₀ 9 = 0.9542

Each interval I (m) of numbers starting with the digit m is between log (m) and log (m + 1), so the various intervals hold:

log (m + 1) - log (m) = log ((m + 1) / m) = log (1 + 1/m)

Returning to the problem, the digit 7 appears log (1 + 1/7) = 0.05799 = 5.8% and to see it appear we have to wait 2⁴⁶ = 70368744177664.

The second question can be answered that it appears more 7, and

for n > 209, the frequency f (7) of the digit 7 is greater than that of 8:

f (7) / f (8) tends to log₁₀ (1 + 1/7) / log₁₀ (1 + 1/8) = 1.133706496

Note: I(1) = I(2) + I(3) = I(4) + I(5) + I(6) + I(7)

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.

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 after the American physicist Frank Benford (1883-1948) who formulated it in 1938, although it had already been highlighted by the American astronomer Simon Newcomb (1835 -1909) in 1881.

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.

This distribution has a characteristic property known as "scale invariance" and is often used to discover "forged" data.

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.

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 "Benford's law".

The distribution of the n-th digit, as n increases, quickly approaches a uniform distribution with 10% for each of the ten digits, as shown below:

Benford Online Bibliography

https://mathworld.wolfram.com/BenfordsLaw.html

Benford's Law (mathpages.com)

Index to OEIS: Section Be - OeisWiki

https://oeis.org/A008952

16. Binary code

If you divide a square of unit area into 2 equal parts, each will have a value of 0.5.

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; ...

In other words, a geometric series is obtained whose sum converges to 1:

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.

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: 

3,14159265358979323846…  = 11,00100100001111110110101...

The whole part will probably be clear to everyone, while the other part will require a minimum of reasoning.

These tables can perhaps help:

 

Here are some examples of pi in various numbering systems.

As an exercise, you can see how simple the transition from base 2 to base 4 and from base 4 to base 16 is.

base-2  (binary code) :  11.0010010000111111011010101000100010000101101000110000100011010011000100110001100110001010001011100000 …

base-4 : 3.0210033312222020201122030020310301030121202202320003130013031010221000210320020202212133030131000020 …

base-16  (exadecimal code): 3.243F6A8885A308D313198A2E03707344A4093822299F31D0082EFA98EC4E6C89452821E638D01377BE5466CF34E90C6CC0AC …

 

base-3 : 10.0102110122220102110021111102212222201112012121212001211001001012220222120120121112101210112002201202 ...

base-5 : 3.0323221430334324112412240414023142111430203100220034441322110104033213440043244401441042334133011323 ...

base-6 : 3.0503300514151241052344140531253211023012144420041152525533142033313113553513123345533410015154344401 ...

base-7 : 3.0663651432036134110263402244652226643520650240155443215426431025161154565220002622436103301443233631 ...

base-8 : 3.1103755242102643021514230630505600670163211220111602105147630720020273724616611633104505120207461615 ...

base-9 : 3.12418812407442788645177761731035828516545353462652301126321450283864034354163303086781327871588 ...

 

Binary number - Wikipedia

Zibaldone Scientifico: pi (zibalsc.blogspot.com)

https://robertlovespi.net/2014/06/09/the-beginning-of-the-number-pi-in-binary-through-hexadecimal-etc/

15. Oloid

Discovered by Paul Schatz in 1929, the Oloid is defined as the convex hull of 2 circumferences of radius R equal to each other, arranged on two orthogonal planes and such that each of the 2 steps to the center of the other.

This grooved surface has many interesting properties (for example, all of its generating lines have the same length).

The Oloid has a shape particularly suitable for the mixing of fluids.

The Oloid surface S is equal to the area of the sphere of radius R;

while by means of numerical calculation, it can estimate the volume V:

The Oloid is the only three-dimensional shape that can rotate on its entire surface.

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.

Prof. Dr. Peter Berger:    http://www.prof-dr-berger.de/_figurint-tork.htm

 The dell'Oloide equation corresponds to an algebraic surface of order 8 :

http://dictionnaire.sensagent.leparisien.fr/Oloide/es-es/

 https://demonstrations.wolfram.com/MovementOfAnOloid/

https://en.wikipedia.org/wiki/Oloid

https://www.youtube.com/watch?v=GM3_JuFgJ2E

http://www.dimnp.unipi.it/gabiccini-m/robotica.html

http://www.ucke.de/christian/physik/ftp/lectures/udine2003.pdf

http://www.heldermann-verlag.de/jgg/jgg01_05/jgg0113.pdf

http://www.mathcurve.com/surfaces.gb/orthobicycle/orthobicycle.shtml

http://www.mathcurve.com/surfaces/orthobicycle/orthobicycle.shtml

14. Geodesics on a Polyhedron

In mathematics, a geodesic is the shortest curve 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.

What are the shortest paths on polyhedron?

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).

Prolonging this segment, beyond its extremes, we can join all the pairs of points on the surface of the polyhedron.

A cubic-edged planet with an edge equal to the diameter of the Earth (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 Mount Everest (8,844 m) and the Mariana Trench (-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.

 

On the Carlos Furuti website, you can find many Cartographic Projections:

http://www.progonos.com/furuti/MapProj/Normal/ProjPoly/Foldout/Cube/cube.html

and you can even print a cubic Earth.

The great British mathematician Henry Ernest Dudeney (1857 - 1930) invented hundreds of mathematical games a couple will be reported dealing with geodesics on polyhedron.

THE JOURNEY OF THE MOSCOW

A fly, starting from point A, can travel the 4 sides of the base of a cubic block in 4 minutes. How long will it take to get from A to the opposite corner B?

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).

THE SPIDER AND THE FLY

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?

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.

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).

http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Dudeney.html

http://mathworld.wolfram.com/SpiderandFlyProblem.html
https://pbbmath.weebly.com/blog/the-logic-puzzles-of-henry-dudeney
http://www.ilovephilosophy.com/viewtopic.php?f=4&t=181293
https://math.stackexchange.com/questions/292495/gentle-introduction-to-quasi-geodesics
http://www.progonos.com/furuti/MapProj/Normal/ProjPoly/projPoly3.html

13. Squaring the Circle in n-Dimensions

In the previous post we saw how, starting from the Wallis sieve, we can arrive at a 3D-cubic fractal-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.

For 2D we have a square divided into 9 equal parts:

The number of elements (or cardinality) of the Cartesian product of 2 sets is the product of the number of elements of the 2 sets. Generalizing, the number of elements of the Cartesian product of n sets is the product of the number of elements of each set.

We can give a tabular representation, which consists in writing all the possible pairs enclosed in braces:

{(1, 1); (1, 2); (1, 3); (2, 1); (2, 2); (2, 3); (3, 1); (3, 2); (3, 3)}

or by means of a double entry table, 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:

As can be seen, the only square removed is (2, 2).

In the case of 3D fractal-like we can do a similar reasoning and the 7 cubes removed will be:

(1, 2, 2); (3, 2, 2); (2, 1, 2); (2, 2, 2); (2, 3, 2); (2, 2, 1); (2, 2, 3)

The following table summarizes (for each dimension) how many groups of numbers 2 we can find; eg in 3D we have 8 triples with zero 2, 12 with only 2, 6 with 2 elements equal to 2 and 1 with all the numbers equal to 2 (which corresponds to the central cube):

the combinations with at least 2 elements equal to 2 are 6 + 1 = 7, on a total of 27 cubes (shown in the last column) and for a cube of side 3.

We can do the same reasoning for a cube of side 5:

last column shows the number of cubes remaining.

Note: the coefficients that appear in the table can be easily calculated using the formula of the Newton binomial - in this case (4 + 1) n -

Now, in the previous post we saw that in 2D for an initial value of 4, multiplied by 8/9, 24/25, 48/49, 80/81, ..., (n² - 1) / n²

we have the formula derived from the Wallis product:

which can be summarized as follows:

While in 3D we have this result:

where the numbers in gray represent the various denominators.

I omit the passages and report what I calculated with Wolfram|Alpha:

which can be summarized:

In the formulas the Euler Gamma function 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.

Here I am only interested in noting that the right part of the formulas containing the index n tends to 1 when tending n to infinity.

So what remains of the 3 formulas is pi and the number is in the denominator. Remembering that each formula must be multiplied by the number of "quadrants", the first (2D) 2² = 4, the second (3D) for 2³ = 8 and the third (4D) for 16.

Thus obtaining the "Volumes" of the n-dimensional spheres:

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.

https://community.wolfram.com/groups/-/m/t/822984

https://www.wolframalpha.com/input/?i=product+(2k%2F(2k%2B1))%5E4+((2k%2B5)%2F(2k%2B1)),+k%3D1+to+infinity

https://math.stackexchange.com/questions/1783732/is-a-hypersphere-of-non-integer-dimension-a-fractal

https://math.stackexchange.com/search?q=fractal+sphere+gamma

http://zeta137.blogspot.com/2019/06/12-wallis-and-squaring-of-circle.html?_sm_au_=iQW2QM3DR7VWvD1j

http://zeta137.blogspot.com/2015/06/6-wallis-and-pippenger-infinite-products.html?_sm_au_=iQW2QM3DR7VWvD1j