Enzymind Labs

Recurrent dynamic systems can be easily extended to result in 3-D patterns.

The Menger Sponge (Wikipedia) / Menger Sponge (PlanetMath) can be considered as a 3-dimensional generalization of the Cantor set, a 3x3x3 two symbol system with the symbols usually represented as filled or unfilled spaces for rendering purposes.

[To Do: give more examples ]

There is a physical puzzle, made with LiveCube elements, that is based on recurrent 3-D construction in a rectilinear space. Apparently the solution is a 3-D finite tiling using a singe non-trivial tile shape. This video demonstrates the 3-D sequencential construction. Note that the shapes are not cubic (nxn matrices), but are formed from equilateral chiral crosses (roughly a swastika) and the rules are not the same (uniformly recurrent) at every iteration.

"Each identical shape is made by 16 cubes. The center 4 cubes are 2-stud cubes. (4 2-stud cubes 12 4-stud cubes) 12 shapes can build 6x6x6 puzzle, 60 shapes build 10x10x10 puzzle, this video uses 168 shapes to build 14x14x14 puzzle."

I don't know the rules; it is not as easy to infer the rules with non-cubic rules and when 3-D rotation and translation is involved, even if the set of possible unique rotations (3) is small. Note that the coloring of the cross is irrelevant to the rules, and apparently only serves to form a nice alternating superficial pattern. Is there a finite set of rules for constructing an infinite tiling?