Making Patterns with Wang Tiles

th

i

54

n

2

r

19

view

tiles

array

17:

Wang tiles (constructed in 1961 by Hao Wang) are a set of 13 squares with the diagonals of each square dividing it into four colored triangles.

Wang packed these squares in the plane in the usual checkerboard pattern, but without rotating or reflecting them, and under the condition that when two squares shared an edge, the colors on opposite sides of the edge had to match. It turned out that these 13 tiles can only tile the plane aperiodically. Wang tiles have applications in areas like DNA computing and textures in computer graphics; they are related to cellular automata and Turing machines.

This Demonstration is a variation on this idea.

The original 13 tiles and all of their rotations and reflections form an ordered list

A

of 60 tiles.

The packing starts with the

th

i

tile of

A

at the top left. A new set

A'

is formed by shifting

A

cyclically by

r

places to the left. The first tile from

A'

that matches is added to the right of the last laid tile or at the beginning of the next row if the last row had

n

tiles. This continues until the packing has size

n×n

.

The number and subset of tiles used are shown below the completed packing.

If "view" is set to "tiles", the packing of the tiles is shown; if "view" is set to "array", each tile of

A

is assigned a different color and there are five times as many tiles.