Rep-Tiles

If a polyomino can be divided into a finite number of congruent copies similar to itself, we say that it is a rep-tile, or more specifically, an -reptile. The so-called P-pentomino is a 4-reptile as we show later. Rep-tiles are self-similar pieces and so can be regarded as fractals. It can be proved that they tile the plane in a nonperiodic way [11], but it was only recently discovered, as reported by Roger Penrose in his fascinating and insightful book Shadows of the Mind, that there are three polyominoes that tile the plane only aperiodically [12].

The following L-triomino is also a 4-reptile and therefore also a -reptile for all .

The value of , given as an argument to the function repL, can be computed from those of and from . Giving it explicitly lets us distort the piece at will.

The Sphinx

Rep-tiles also arise in the shape of polyiamonds. The sphinx is one of the most widely known 4-reptile polyiamonds.

As indicated at the end of the previous section, more challenges are still present in the world of polyominoes. Even the seemingly simple task of finding out the number of tilings of an rectangle using dominoes poses considerable difficulties (e.g., problem 7.51 in [7]). We can only guess as to the difficulty of these problems in the worlds inhabited by polyiamonds, polyhexes, and polykites. The advantages provided by the development of sophisticated languages like Mathematica yield a promising future for further investigations of this fascinating topic.