Zeta 137: 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: