# rep-tile

A rep-tile is a repetitive tiling: a shape with the
property that it tiles a larger version of itself, using identical copies
of itself. A simple example is a square because four copies of any square
tile a larger square. Any triangle also is a rep-tile, because four copies
of it tile a larger version of this triangle. Rep-tiles that require *n* tiles to build a larger version of themselves are said to be rep-*n*;
thus a square is rep-4. Since any of these larger replicas can be combined
to give an even larger, second-generation copy, a rep-*n* tile is
also rep-*n*^{2}, rep-*n*^{3}, and so on.
Often tiles have several rep-numbers. If a tile is rep-*n* and rep-*m*,
it is also rep-*mn*, since replicas can be built with *n* tiles, then combined, *m* at a time, to give a yet larger version.

The set of rep-tiles is a subset of the set of **irreptiles**.
An irreptile is any shape that tiles a larger version of itself using either
differently sized or identical copies of itself. The problem to find all
irreptiles in the Euclidean plane has been studied but not yet completely
solved. A related set of problems is to find for each irreptile the minimum
number of smaller copies needed to tile the original shape; in many cases
it is difficult to prove such a minimality. The name "rep-tile" was coined
by Simon Golomb.