11. Section of a Menger sponge

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:

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.

 https://youtu.be/5vTS--uuErA

Below we will consider some fractal objects and the section of a fractal object corresponding to the cube.

A Menger sponge is defined as follows:

1. we start from a cube,

2. divide the cube into 27 cubes (as in the Rubik cube),

3. the central cube and the 6 central cubes are removed from each face (20 cubes remain),

4. repeat steps 1 - 3 on each new cube.

At each iteration the number of holes increases, as shown in the figure:

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

Menger sponge is the space that is obtained as the limit of these operations.

In its construction of 1926, Karl Menger showed that the Hausdorff dimension of the sponge is log 20 / log 3 approximately 2,726833 ...

Well, if you take a sponge from Menger and dissect it like the cube seen above, here is what you will get:

These figures are taken from the site of  George W. Hart

     http://www.georgehart.com/index.html

http://www.georgehart.com/rp/half-menger-sponge.html

In the Seeing Stars article it is shown that this section has Hausdorff dimension:

The Cantor set, introduced by the German mathematician Georg Cantor, is a subset of the interval [0, 1] of real numbers, defined recursively, removing at each step an open central segment from each interval.

The Cantor set 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 ...

Sierpinski carpet is also a fractal similar to the Cantor collection obtained from a square,                     by the Polish mathematician Wacław Sierpiński in 1916.

Its fractal dimension is ln 8 / ln 3, equal to = 1.8927 ... (it is interesting to note that it is equal to 3 times the Hausdorff dimension of the Cantor set). This value is slightly higher than the one shown above.

The three-dimensional version of the carpet is the Menger sponge.

The figure shows an extract from the list of fractals by Hausdorff dimension reported in Wikipedia:

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

http://blog.zacharyabel.com/tag/menger-sponge/

http://mathworld.wolfram.com/MengerSponge.html

http://blog.zacharyabel.com/2012/02/seeing-stars/

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

http://www.georgehart.com/rp/half-menger-sponge.html
http://www-history.mcs.st-and.ac.uk/HistTopics/fractals.html

10. Black Holes

A few weeks ago it was announced that the Event Horizon Telescope, a group of eight radio telescopes (on a planetary scale), provided the first direct visual evidence ever obtained of a Black Hole positioned in the heart of Messier 87, a huge galaxy located in the nearby Virgo cluster. In a series of six articles in "The Astrophysical Journal Letters", 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.

When we talk about a Black Hole of this mass what are we talking about?

For example the mass of the planet Earth is 5.97 × 10²⁴ kg (diameter of 12,750 km), while that of the Sun is 1.99 × 10³⁰ kg (diameter 1.39 × 10⁶ km); the relationship between the values of the 2 masses is easy to remember: 333,333.

The mass of Sgr A* (Black Hole at the center of our galaxy) is 4.31 × 10⁶ M_o and finally that of the BH of M87 is equal to 6.5 × 10⁹ M_o (1.29 × 10⁴⁰ kg ).

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_o) we obtain the radius of the BH.

This is because the Schwarzschild Radius is directly proportional to its mass:

The Black Hole of M87 which has a mass 6.5 billion solar masses therefore has a R_S of about 20 billion km (127 Astronomical Units).

Pluto aphelion is located at about 49.3 AU (from the Sun), so the Solar System could be conveniently contained in the Black Hole of M87.

The Schwarzschild Radius R_S is the distance at which the escape velocity is c (speed of light).

Titius-Bode 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 Astronomical Units) and is expressed with the simple formula:

                                                   (3n + 4) / 10

where n takes the values ​​0, 1, 2, 4, 8, 16, ...

This simple geometric progression has a nice graphic representation using logarithmic scale:

where a 1 AU we find by definition the Earth and about 10 AU Saturn. As mentioned above, at 127 we can position the event horizon of the Black Hole of M87.

But what is the average density of a Black Hole?

For Black Holes of "small" dimensions, the average density within the event horizon 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 massive Black Holes. A BH of 387 million solar masses would have the average water density and would be comparable to a giant water balloon extending from the Sun to almost Jupiter. A BH of 11 billion solar masses would have the average air density 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.

So you can live in a large low density BH without even realizing it.

I stop here and let the reader fantasize ….

AU Astronomical Unit (Earth - Sun distance) : 149,597,870 km = 8.5 minutes light

Light year (distance traveled by light in a year) : 9,460 billion km = 63,300 AU

Parsec : 3.262 light years = 30,860 billion km = 206,000 AU

Speed ​​of light : 299,792 km/sec

https://commons.wikimedia.org/w/index.php?curid=74584660