Menger Sponges 2

A Menger Sponge hangs suspended above the surface of a body of water as two of its central Level 2 iterations float away from it. In mathematics, the... more

by Walter Neal

Title

Menger Sponges 2

Artist

Walter Neal

Medium

Digital Art

Description

A Menger Sponge hangs suspended above the surface of a body of water as two of its central Level 2 iterations float away from it. In mathematics, the Menger sponge is a fractal curve. It is a universal curve, in that it has topological dimension one, and any other curve (more precisely: any compact metric space of topological dimension 1) is homeomorphic to some subset of it. It is sometimes called the Menger-Sierpinski sponge or the Sierpinski sponge, because each face of the Menger sponge is a Sierpinski carpet. It is a three-dimensional extension of the Cantor set (any intersection of the Menger sponge with a diagonal or medium of the initial cube M0) and Sierpinski carpet. The Menger sponge simultaneously exhibits an infinite surface area and encloses zero volume. In spite of this, there exists a homeomorphism of the cube having finite distortion that "squeezes the sponge" in the sense that the holes in the sponge go to a Cantor set of zero measure. It was first described by Karl Menger (1926) while exploring the concept of topological dimension.

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June 3rd, 2013

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