Rep-Tile

A rep--tile, also called an -reptile, is a closed set with nonempty interior that can be dissected into mutually congruent smaller copies, each similar to the original set, where is an integer (Kynčl and Patáková 2017). The definition applies in any dimension, although rep-tiles are often illustrated by polygons. The triangular polygonal spiral is one such example, while certain space-filling tetrahedra are rep-8-tiles.

Repeated subdivision of a rep-tile gives a geometric rule for a substitution tiling with a single tile type. Equivalently, the tile can be enlarged until its component copies have the original size, giving a tile inflation rule (Frank 2008).

The above figure shows the zeroth through fifth iterations obtained by rep-tiling the L-polyomino.

Trott (2007) uses the rep-4 subdivision of the L-shaped triomino for the surface pattern of the Version 6 Spikey. The subdivided pattern is mapped onto the faces of a dodecahedron.