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. The Delian brick is a three-dimensional rep-2-tile obtained by bisecting a cuboid with edge lengths in the ratio perpendicular to its longest edges (Pegg 2018).

For polygons, allowing the smaller copies to have unequal sizes gives the more general notion of an irreptile (Richter 2009, p. 110). For example, the two-piece golden bee dissection uses linear scale factors and , where is the golden ratio (Durand et al. 2020).

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.

Panckoucke (1749, p. xxx) illustrates a dissection of an L-shaped triomino into four equal, similar pieces. In modern terminology, this is a rep-4 dissection with linear scale factor .

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.