Menger Sponge

Visualization

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$dim=3
A=3^-1*([0,0,0]|[0,0,1]|[0,0,2]|[0,1,0]|[0,1,2]|[0,2,0]|[0,2,1]|[0,2,2]|[1,0,0]|[1,0,2]|[1,2,0]|[1,2,2]|[2,0,0]|[2,0,1]|[2,0,2]|[2,1,0]|[2,1,2]|[2,2,0]|[2,2,1]|[2,2,2])*A

Overview

The Menger sponge is the 3D analog of the Sierpiński carpet, constructed by Karl Menger in 1926 as an example of a universal curve (a compact, connected, locally connected topological space that contains a topological copy of every curve).

Start with a unit cube. Divide it into 27 sub-cubes in a 3×3×3 grid. Remove the center sub-cube and the 6 face-center sub-cubes — the 7 that share a face center with the original. This leaves 20 sub-cubes. Repeat the procedure for each remaining sub-cube indefinitely.

Algebraic Structure

The 20 maps are pure scalings by $1/3$ followed by translations to the 20 retained positions in $\{0,1,2\}^3$. The removed positions are the center $(1,1,1)$ and the six face centers $(1,1,0)$, $(1,1,2)$, $(1,0,1)$, $(1,2,1)$, $(0,1,1)$, $(2,1,1)$.

Each map has the form:

The Hausdorff dimension is:

Properties

• The sponge has zero volume but infinite surface area.
• Every cross-section perpendicular to a main axis is a Sierpiński carpet.
• The sponge is topologically one-dimensional — every loop can be contracted to a point within the sponge — despite its fractal dimension of ≈ 2.727.

Boundary Dimension

The boundary $\partial A$ (the surface of the sponge) has Hausdorff dimension

At each scale step, the surface is divided into $8$ self-similar pieces each scaled by $1/3$, giving the Moran equation $8 \cdot (1/3)^d = 1$.

Each axis-aligned face of the sponge is a Sierpiński carpet lying in a flat Euclidean plane: the face-center sub-cubes are exactly the ones removed at every scale, so the outer surface consists of six planar cross-sections — each a 2D Sierpiński carpet with its own fractal structure and $\dim_H = \log 8 / \log 3 \approx 1.893$. The faces are flat but not straight-edged polygons.

References

• Menger, K. (1926). Allgemeine Räume und Cartesische Räume. Proceedings of the Section of Sciences, 29, 476–482.
• Menger sponge — Wikipedia

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