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

And here’s one that’s almost but not quite like it.

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

And here are variations of the Cross Menger 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, with a tetrahedron base.

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

Figures created with Wolfram Mathematica versions 10 and 11.

Robert Dickau, 2016–2018.

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