Re: Fractal Sponges of Arbitrary Dimension 2 =< D =< 3

In message <1185846755.877311.95520@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>, ranecurl@xxxxxxxxx writes
I seek descriptions and generators for fully connected deterministic
fractal sponges other than the Menger Sponge. They should exist with
fractal dimensions from 2 to 3.

There's a tetrahedron. (Repeatedly remove the smaller tetrahedra defined by drawing lines between the midpoints of edges). Similar figures are composed of 10, 20 etc, rather than 4, self similar parts.

Presumably similar figures can be produced from square pyramids.

You can take the 27 part dissection of a cube, and remove the central part, and so on. (Not that you can see the sponge-like nature of this figure from the outside.)

You can take a cube, dissected into 125 (rather than 27) parts, and remove the 13 (rather that 7 parts) joining the midpoints of the faces. The same can be done with dissections into 7^3, 9^3 etc parts.

You can take the 125 part dissection of the cube, and replace the groups of 8 parts at each corner with a single larger cube, giving a dissection into 8 larger, and 61 smaller parts.

You can take a dissection of a cube into 64 parts, and then remove the 32 parts which touch the lines connecting the midpoints of the face.

Similarly with dissections into 6^3, 8^3 etc parts.

More figures arise when the pieces removed do not lie on the lines connecting the midpoints of the faces, and yet more when any requirement for symmetry is relaxed. (For an example of the former, take the 125 part dissection, and remove 4 symmetrically placed columns from each face.)
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Stewart Robert Hinsley
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