Menger sponge

In mathematics, the Menger sponge is a fractal curve. It is the universal curve, in that it has topological dimension one, and any other curve or graph is homeomorphic to some subset of the Menger sponge. It is sometimes called the Menger-Sierpinski sponge or, incorrectly, the Sierpinski sponge. It is a three-dimensional extension of the Cantor set and Sierpinski carpet. It was first described by Austrian mathematician Karl Menger in 1926.

Construction

Menger sponge, first four levels of the construction.

Construction of a Menger sponge can be visualized as follows: Begin with a cube. (first image) Divide every face of the cube into 9 squares. This will sub-divide the cube into 27 smaller cubes, like a Rubik's Cube Remove the cube at the middle of every face, and remove the cube in the center, leaving 20 cubes (second image). This is a Level 1 Menger sponge. Repeat steps 1-3 for each of the remaining smaller cubes. The second repetition will give you a Level 2 sponge (third image), the third a Level 3 sponge (fourth image), and so on. After an infinite number of iterations, a complete Menger sponge will remain.

The number of cubes increases by : $20^{n}$. Where $n$ is the number of iterations performed on the first cube:

Iters	Cubes	Sum
0	1	1
1	20	21
2	400	421
3	8,000	8,421
4	160,000	168,421
5	3,200,000	3,368,421
6	64,000,000	67,368,421

At the first level, no iterations are performed, (20 ⁰ = 1).

Properties

File:Menger sm.png

An illustration of M₄, the fourth iteration of the construction process

Each face of the Menger sponge is a Sierpinski carpet; furthermore, any intersection of the Menger sponge with a diagonal or medium of the initial cube M₀ is a Cantor set. The Menger sponge is a closed set; since it is also bounded, the Heine-Borel theorem yields that it is compact. Furthermore, the Menger sponge is uncountable and has Lebesgue measure 0.

The topological dimension of the Menger sponge is one; indeed, the sponge was first constructed by Menger in 1926 while exploring the concept of topological dimension. Note that the topological dimension of any curve is one; that is, curves are topologically one-dimensional. Menger showed, in the 1926 construction, that the sponge is a universal curve, in that any possible one-dimensional curve is homeomorphic to a subset of the Menger sponge. Note that by curve we mean any object of Lebesgue covering dimension one; this includes trees and graphs with an arbitrary countable number of edges, vertices and closed loops, connected in arbitrary ways.

In a similar way, the Sierpinski carpet is a universal curve for all curves that can be drawn on the two-dimensional plane. The Menger sponge constructed in three dimensions extends this idea to graphs that are not flat, and might be embedded in any number of dimensions. Thus any geometry of quantum loop gravity can be embedded in a Menger sponge.

The sponge has a Hausdorff dimension of (ln 20) / (ln 3) (approx. 2.726833).

Formal definition

Formally, a Menger sponge can be defined as follows:

$M:=\bigcap _{n\in \mathbb {N} }M_{n}$

where M₀ is the unit cube and

$M_{n+1}:=\left\{{\begin{matrix}(x,y,z)\in \mathbb {R} ^{3}:&{\begin{matrix}\exists i,j,k\in \{0,1,2\}:(3x-i,3y-j,3z-k)\in M_{n}\\{\mbox{and at most one of }}i,j,k{\mbox{ is equal to 1}}\end{matrix}}\end{matrix}}\right\}$

See also

• List of fractals by Hausdorff dimension
• Sierpinski triangle
• Sierpinski tetrahedron

References

• Karl Menger, General Spaces and Cartesian Spaces, (1926) Communications to the Amsterdam Academy of Sciences. English translation reprinted in Classics on Fractals, Gerald A.Edgar, editor, Addison-Wesley (1993) ISBN 0-201-58701-7
• Karl Menger, Dimensionstheorie, (1928) B.G Teubner Publishers, Leipzig.

External links

• An interactive Menger sponge
• Fractal polyhedra (VRML) and interactive Java models
• Puzzle Hunt — Video explaining Zeno's paradoxes using Menger-Sierpinski sponge
• The Business Card Menger Sponge Project
• Menger Sponge Assembly Construction of a Level-3 Menger Sponge from business cards