Fractal Reptiles Based on Polyforms

A self-replicating tile, or "reptile", is a tile that can be tiled by smaller copies of itself. Polyominoes, polyhexes, and polyiamonds are shapes comprised of squares, hexagons, and equilateral triangles, respectively, joined in edge-to-edge fashion. These polyforms can be arranged in ways that result in fractal reptiles in the limit of an infinite number of iterations. This process is described in detail in a 2025 paper dealing with polyominoes, by Robert Fathauer. Other polyforms, including polyiamonds and polyhexes, are dealt with in a 2026 paper. The method is described more briefly here.

For iterated tiles that neck down to single polygons or have single-polygon-wide gaps, there is a subtle problem with infinity. In the limit, these bridges and gaps go to zero, making the tile no longer a closed topological disk. At the same time, the tile only becomes self-replicating in the limit. So in some sense, these necking-down iterated polyforms are not fractal reptiles, though one can make an arbitrarily-good approximation to a fractal reptile.

The polyforms shown as buttons below link to pages with fractal reptiles based on them.