# 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 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). Shrink the cube to $\displaystyle 1/3$ 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 : $\displaystyle 20^n$ . Where $\displaystyle n$ 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

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 gasket 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, as a humorous example, any geometry of quantum loop gravity can be embedded in a Menger sponge.

## Formal definition

Formally, a Menger sponge can be defined as follows:

$\displaystyle M := \bigcap_{n\in\mathbb{N}} M_n$

where M0 is the unit cube and

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