Fractal Sponge Variants

This page is a holding area for miscellaneous variants of fractal carpets and sponges such as the Menger sponge and the Jerusalem/Cross Menger sponge.

For example, here’s one variant of the Sierpinski carpet with non-square holes cut out at each iteration.

Affine Sierpinski carpet variants, stages 0-4

And here’s another variant with different scaling factors from the previous set.

Different Sierpinski carpet variants, stages 0-4

This Menger sponge variant is based on Fig. 6.6.3 in Gerald A. Edgar, Measure, Topology, and Fractal Geometry, Springer-Verlag, 1990, p. 187.

Menger sponge affine variants, stages 0-3

Menger sponge affine variant, stage 4

The idea is that the respective iterations are self-affine but not self-similar. Something about calculating the Hausdorff dimension doesn’t work the same way. Gerald A. Edgar labels the figure “Homeomorph of the Menger sponge”.

For that matter, there’s no need for the scales to be symmetric.

Asymmetical Menger sponge affine variants, stages 0-3

Asymmetrical Menger sponge affine variant, stage 4

Complements of levels 1, 2, and 3, if you’re interested.

These were first drafts of the previous, before I noticed I’d missed the point.

Different variant, stages 1-4 Different variant, stage 5

Non-sponge variant called the Mosely snowflake. Similar to the Menger sponge, start with a cube, and divide it into 33 = 27 smaller cubes. At each stage, remove the 8 outer corners of the starting cube instead of the center cubes as done with the Menger sponge. This version is called the heavier version of the Mosely snowflake.

Mosely snowflake, heavier version, stages 1 through 3

And similar with the center removed, called the lighter version of the Mosely snowflake (lighter as if you were weighing it on a scale, I assume). Picking an isometric viewpoint helps the figure look more like a snowflake, but also look a bit less three-dimensional.

Mosely snowflake, lighter version, stages 1 through 3

And here are variations of the Cross Menger (Jerusalem) carpet using a triangle base.

triangle version of Cross Menger fractal, stages 0 through 2
triangle version of Cross Menger fractal, stages 3 through 5

triangle version of Cross Menger fractal variant, stages 0 through 2
triangle version of Cross Menger fractal variant, stages 3 through 5

In Kigami and Lapidas’s paper “Weyl’s Problem for the Spectral Distribution of Laplacians on P.C.F. Self-Similar Fractals”, a similar figure (Fig. 2) is called a Modified Sierpinski gasket. (Thanks to Roger Bagula for the reference.)

Ditto, 3D extension with a tetrahedron base. (Not aware of a name for this one. Nowhere to go but up from “Cross Menger tetrahedron”? The holes aren’t cross-shaped.)

tetrahedral version of Cross Menger fractal, stages 0 through 2
tetrahedral version of Cross Menger fractal, stages 3 through 5

Complements of the previous.

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And some lopsided versions of the triangle carpet.

lopsided triangle variants

Similar, with interior rotations:

another lopsided triangle variant

And a rotationally symmetric variant:

rotationally symmetric triangle variants

Figures created with Wolfram Mathematica versions 10, 11, 12, and 14.

Robert Dickau, 2016–2026.

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