Details and Options
A pattern is a rectangular array of positive integer values suitable for use in the resource function TilingPatternPlot.
A tile mask is a subset of positions within a rectangular array, such as the a values in {{a,a,a},{_,a,_}}. This particular mask is also known as the Tetris T shape.
A tile is an array that can contain integers or blanks.
All tiles in a tileset fit in an array of the same size, say {a,b}. If all subarrays of that size in a larger pattern matches a tile in the tileset, then that tileset can be used to make the given pattern.
An all-zero tile
![](https://www.wolframcloud.com/obj/resourcesystem/images/9cc/9ccab8a8-7d8a-41ce-a17a-f3e2faa0d035/1-0-0/0cc772b74909fad0.png)
leads to an all white or all zero pattern
![](https://www.wolframcloud.com/obj/resourcesystem/images/9cc/9ccab8a8-7d8a-41ce-a17a-f3e2faa0d035/1-0-0/2f4b6bbfc78083fc.png)
.
An all-one tile
![](https://www.wolframcloud.com/obj/resourcesystem/images/9cc/9ccab8a8-7d8a-41ce-a17a-f3e2faa0d035/1-0-0/1eb56f3b2bb19520.png)
leads to an all black or all one pattern
![](https://www.wolframcloud.com/obj/resourcesystem/images/9cc/9ccab8a8-7d8a-41ce-a17a-f3e2faa0d035/1-0-0/0a48a85bb5da9c3c.png)
.
The two tiles
![](https://www.wolframcloud.com/obj/resourcesystem/images/9cc/9ccab8a8-7d8a-41ce-a17a-f3e2faa0d035/1-0-0/2c72db8dbda41908.png)
lead to a checkboard pattern
![](https://www.wolframcloud.com/obj/resourcesystem/images/9cc/9ccab8a8-7d8a-41ce-a17a-f3e2faa0d035/1-0-0/3adb9d959181e740.png)
. No subset of the tiles will make a larger pattern, so these two tiles produce a minimal tiling.
The tileset
![](https://www.wolframcloud.com/obj/resourcesystem/images/9cc/9ccab8a8-7d8a-41ce-a17a-f3e2faa0d035/1-0-0/3ad4ab08c39fd5e6.png)
will produce the all-white pattern, but the second tile is not necessary. Therefore, this is not a minimal tileset.
The above patterns have a size of 4×4.
The input
tiles should be an array of integer or
Blank (_) values.
Consider
![](https://www.wolframcloud.com/obj/resourcesystem/images/9cc/9ccab8a8-7d8a-41ce-a17a-f3e2faa0d035/1-0-0/08a1b30d8ff6d55f.png)
and
{{_,0,0},{0,1,_}}. These both represent a 4-cell tile within a 6-cell array, where the integers indicate the coloring of the tile and the
Blank (_) values are wildcards.
A tiling with a periodic boundary can be divided into identical pieces. Basically, this gives a tiling on a torus that may be used for an infinite tiling.