Menger Sponge

5 iterations

At the 6th iteration

Menger sponge after four iterations

Menger sponge is a geometric fractal , one of the three-dimensional analogues of the Sierpinski carpet .

Content

1 Construction
- 1.1 Iterative Method
- 1.2 The game of chaos
2 Properties
3 See also
4 notes
5 Links

Build

Iterative Method

Cube $C_{0}$ ${\ displaystyle C_ {0}}$ with edge 1 is divided by planes parallel to its faces into 27 equal cubes. Out of the cube $C_{0}$ ${\ displaystyle C_ {0}}$ the central cube and all cubes of this unit adjacent to it along two-dimensional faces are deleted. It turns out a lot $C_{1}$ ${\ displaystyle C_ {1}}$ consisting of the 20 remaining closed cubes of the "first rank". By doing exactly the same with each of the cubes of the first rank, we get the set $C_{2}$ ${\ displaystyle C_ {2}}$ consisting of 400 cubes of the second rank. Continuing this process indefinitely, we get an infinite sequence

C_{0}\supset C_{1}\supset \dots \supset C_{n}\supset \dots

{\ displaystyle C_ {0} \ supset C_ {1} \ supset \ dots \ supset C_ {n} \ supset \ dots}

,

the intersection of the members of which is the Menger sponge.

The game of chaos

Menger sponge can also be obtained using a process called the ^[1] ^[2] , which is as follows:

20 attractor points are defined: 8 vertices and 12 midpoints of the edges of the original cube.
Some starting point is specified. $P_{0}$ ${\ displaystyle P_ {0}}$ lying inside the cube.
A sequence of points is constructed in the following cycle:
1. The attractor is randomly selected $A$ ${\ displaystyle A}$ out of 20 possible with equal probability.
2. Point is being built $P_{i}$ ${\ displaystyle P_ {i}}$ with new coordinates: $x_{i}={\frac {x_{i-1}+2x_{A}}{3}};y_{i}={\frac {y_{i-1}+2y_{A}}{3}};z_{i}={\frac {z_{i-1}+2z_{A}}{3}}$ ${\ displaystyle x_ {i} = {\ frac {x_ {i-1} + 2x_ {A}} {3}}; y_ {i} = {\ frac {y_ {i-1} + 2y_ {A}} {3}}; z_ {i} = {\ frac {z_ {i-1} + 2z_ {A}} {3}}}$ where: $x_{i-1},y_{i-1},z_{i-1}$ ${\ displaystyle x_ {i-1}, y_ {i-1}, z_ {i-1}}$ - coordinates of the previous point $P_{i-1}$ ${\ displaystyle P_ {i-1}}$ ; $x_{A},y_{A},z_{A}$ ${\ displaystyle x_ {A}, y_ {A}, z_ {A}}$ - coordinates of the selected attractor.

If you execute the cycle quite a few times (at least 100 thousand) and then discard the first few tens of points, then the remaining points will form a figure close to the Menger sponge.

Properties

Menger sponge in section

Menger sponge consists of 20 identical parts, the similarity coefficient is 1/3.
The orthogonal projections of the Menger sponge represent the Sierpinski carpet .
The Menger sponge has an intermediate (i.e. not integer ) Hausdorff dimension , which is equal to $\ln 20/\ln 3\approx 2,73$ ${\ displaystyle \ ln 20 / \ ln 3 \ approx 2.73}$ since it consists of 20 equal parts, each of which is similar to the entire sponge with a similarity factor of 1/3.
Menger sponge has a topological dimension of 1, moreover
- The Menger sponge is topologically characterized as a one-dimensional connected locally connected metrizable compactum that does not have locally dividing points (i.e., for any connected neighborhood $U$ ${\ displaystyle U}$ any point $x\in M$ ${\ displaystyle x \ in M}$ a bunch of $U\backslash x$ ${\ displaystyle U \ backslash x}$ connected) and not having nonempty open and embeddable subsets of the plane.
The Menger Sponge is a universal Uryson curve , that is, whatever the Uryson curve is $C$ ${\ displaystyle C}$ , there is a subset of Menger sponge $C^{'}$ ${\ displaystyle C '}$ homeomorphic $C$ ${\ displaystyle C}$ .
Menger sponge has zero volume, but infinite face area. The volume is determined by the formula 20/27 per iteration.
Section of a Menger sponge bounded by a cube with side 1 and center at the origin, plane $x+y+z=0$ ${\ displaystyle x + y + z = 0}$ contains hexagrams .

Notes

↑ Michael Barnsley , Louise Barnsley. Fractal transformations // Fractals as an art. Collection of articles / Transl. in English, fr. E.V. Nikolaeva. - SPb. : Sparta, 2015 .-- S. 35 .-- 224 p. - ISBN 9785040137008 .
↑ Dariusz Buraczewski, Ewa Damek, Thomas Mikosch. Stochastic Models with Power-Law Tails: The Equation X = AX + B. - Springer, 2016-07-04. - 325 p. - P. 7. - ISBN 9783319296791 .

Links

[1] Michael Barnsley , Louise Barnsley. Fractal transformations // Fractals as an art. Collection of articles / Transl. in English, fr. E.V. Nikolaeva. - SPb. : Sparta, 2015 .-- S. 35 .-- 224 p. - ISBN 9785040137008 .

[2] Dariusz Buraczewski, Ewa Damek, Thomas Mikosch. Stochastic Models with Power-Law Tails: The Equation X = AX + B. - Springer, 2016-07-04. - 325 p. - P. 7. - ISBN 9783319296791 .