Menger sponge

The Menger sponge is a fractal solid. It is also known as the Menger-Sierpinski sponge or, incorrectly, the Sierpinski sponge. It is a three-dimensional extension of the Cantor set and Sierpinski carpet, with Hausdorff dimension (ln 20) / (ln 3) (approx. 2,726833). It was first described by Austrian mathematician Karl Menger in 1927.

Contents

Construction

Construction of a Menger sponge can be visualized as follows:

Begin with a cube, (first image).
Shrink the cube to $1 / 3$ of its original size and make 20 copies of it.
Place the copies so they will form a new cube of the same size as the original one but lacking the centerparts, (next image).
Repeat the process from step 2 for each of the remaining smaller cubes.

After an infinite number of iterations, a 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 ⁿ⁼⁰ = 1).

Properties

An illustration of M3, the third iteration of the construction process. Image © Paul Bourke, used by permission

An illustration of M₃, the third iteration of the construction process. Image © Paul Bourke, used by permission

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.

As Peitgen, Jürgens and Saupe showed in 1992, the Menger sponge is also a super-object for all compact one-dimensional objects; that is, a topological equivalent of any compact one-dimensional object can be found in the Menger sponge.

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

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

External links

Categories: Fractals