A Penrose tiling is a kind of aperiodic tiling discovered by Roger Penrose. In 1973 he announced a tiling made from a set of six tiles and then, by slicing and re-gluing, was able to reduce the number of tiles to just two.
The most elegant of Penrose tilings use two rhombi, a thick one and a thin one, which are fitted together so that no two tiles are aligned to form a single parallelogram (otherwise, a single rhombus could be used to make a periodic tiling). All angles are multiples of (π/5) radians (36°, or one tenth of a circle). Each tile has four sides with a length of one unit. One tile has four corners with the angles 72°, 72°, 108°, and 108° (2, 2, 3, and 3 multiples of 36°); the other has angles of 36°, 36°, 144°, and 144° (1, 1, 4, and 4 multiples of 36°). On each tile one of the vertices (corners) is colored black and two of the sides are marked with arrows.
The only rules for assembling the tiles to ensure an aperiodic tiling are that two adjacent vertices must be of the same color, and two adjacent edges must have arrows pointing in the same direction or no arrows at all. These rules ensure that, taken over a large enough area of the plane, the pattern of tiles doesn't repeat. In a correct Penrose tiling, the ratio of thick rhombi to thin ones is always the same and is equal to the golden ratio.
Although Penrose tilings started out as nothing more than an interesting mathematical diversion, they have turned out to mimic the arrangements of atoms in some newly discovered materials, known as quasicrystals.
1. Gardner, Martin. Penrose Tiles and Trapdoor Ciphers... and the Return of Dr. Matrix. New York: W. H. Freeman, 1989.