Menger Sponge

The Menger sponge is a fractal which is the three-dimensional analog of the Sierpiński carpet.

Menger (1926) proved that the Menger sponge is universal for all compact 1-dimensional topological spaces, meaning that any compact topological space of dimension 1 has a homeomorphic copy as a subspace of the Menger sponge (Peitgen et al. 1992, Broden et al. 2024).

The th iteration of the Menger sponge is implemented in the Wolfram Language as MengerMesh[n, 3].

Let be the number of filled boxes, the length of a side of a hole, and the fractional volume after the th iteration, then

			(1)
			(2)
			(3)

The capacity dimension is therefore

			(4)
			(5)
			(6)
			(7)

(OEIS A102447).

The Menger sponge, in addition to being a fractal, is also a super-object for all compact one-dimensional objects, i.e., the topological equivalent of all one-dimensional objects can be found in a Menger sponge (Peitgen et al. 1992).

Broden et al. (2024) proved that all knots can be embedded into the Menger sponge (Barber 2024).

The image above shows a metal print of the Menger sponge created by digital sculptor Bathsheba Grossman (http://www.bathsheba.com/).