Following excerpt from Wikipedia Wang tiles (or Wang dominoes), first proposed by Hao Wang in 1961, are a class of formal systems consisting of equal-sized squares with a color on each edge which can be arranged side by side (on a regular square grid) so that abutting edges of adjacent tiles have the same color; the tiles cannot be rotated or reflected. The following shows an example set of 13 Wang tiles:
The basic question about sets of Wang tiles is proving whether they can tile the plane or not. This means that copies of the tiles can be arranged one by one to fill an infinite plane, without any grid position where no tile in the set can match the side colors of already laid out adjacent tiles. In 1961, Wang proposed an algorithm to take any finite set of tiles and decide whether they tiled the plane. To prove his algorithm worked, he assumed that any set that could tile the plane would be able to tile the plane periodically (with a pattern that repeats, like standard wallpaper). However, in 1966 Robert Berger proved Wang's conjecture was wrong. He presented a set of Wang tiles that could only tile the plane aperiodically. This meant it could fill the plane without holes, but the tiling couldn't be a simple repetition of a finite pattern. This is similar to a Penrose tiling, or the arrangement of atoms in a quasicrystal. Although Berger's original set contained 20,426 tiles, he hypothesized that smaller sets would work, including subsets of his set. In later years, increasingly smaller sets were found. For example, the set of 13 tiles given above is an aperiodic set published by Karel Culik. It can tile the plane, but not periodically. Karel Culik's set of 13 tiles can be transformed into polyomino tiles by replacing the coloured quarter faces and outside edges with a variable edge. Instead of adjacent squares having the same colour, adjacent tiles must have fitting edges. The edges need have no more variability than is required. A binary string can be used to create the different edges corresponding to the different colours a 1 corresponds to a 1 unit extrusion away from the tile a 0 corresponds to a 1 unit intrusion into the tile the tiles have two squares at the beginning and end of each side (otherwise the corners could be pitched off by intrusions meeting at the corners) 8 3 bit possibilities 000 001 010 011 100 101 110 111 these can be paired