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-n2, rep-n3, 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.
Related category TILINGS
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