Wang Tiles

 

edge and corner matched tilesets

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Overall 4x4 Patterns

One goal is to arrange a complete tileset in a square array so that it can seamlessly cover any area. This requires stage borders to wrap correctly top to bottom and left to right.

2-corner Wang Tileset

Complete set of Wang 2-corner 'Patch' tiles
tile tile tile tile tile tile tile tile tile tile tile tile tile tile tile tile
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

The 2-corner tileset has a unique solution. The 16 tiles are arranged in a Brigids Cross pattern. It can tile an area seamlessly. Corner tilesets in the gallery are all use this pattern.

Brigid's Cross

The pattern also has a mirror image form.
Inverting the tiles does not produce a new pattern. It results in the same pattern shifted two tiles diagonally.

Brigid's Cross
6 14 8 4
1 7 12 5
2 15 11 13
3 3 8 1
(mirror image)
3 4 7 13
8 11 12 8
1 14 5 15
3 2 1 6

2-edge Wang Tileset

Using a 2-edge tileset results in very many possible solutions.

Complete set of Wang 2-edge 'Path' tiles
tile tile tile tile tile tile tile tile tile tile tile tile tile tile tile tile
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

In the gallery, tilesets are shown as Layout 0. This is the simplist layout, but it has three disconnected paths. To reduce the number of possible solutions we can restict the layouts to connected mazes only.

Here are mazes with a '0' stage border. Note that tile-15 is placed in the same position in each layout. This removes similar rotation solutions. Each layout can be reflected though in the xy axis for a second solution.

Layout 0
tile tile tile tile
tile tile tile tile
tile tile tile tile
tile tile tile tile
Layout 1
tile tile tile tile
tile tile tile tile
tile tile tile tile
tile tile tile tile
Layout 2
tile tile tile tile
tile tile tile tile
tile tile tile tile
tile tile tile tile
Layout 3
tile tile tile tile
tile tile tile tile
tile tile tile tile
tile tile tile tile
Layout 4
tile tile tile tile
tile tile tile tile
tile tile tile tile
tile tile tile tile
Layout 5
tile tile tile tile
tile tile tile tile
tile tile tile tile
tile tile tile tile
Layout 6
tile tile tile tile
tile tile tile tile
tile tile tile tile
tile tile tile tile
Layout 7
tile tile tile tile
tile tile tile tile
tile tile tile tile
tile tile tile tile
Layout 8
tile tile tile tile
tile tile tile tile
tile tile tile tile
tile tile tile tile
Layout 9
tile tile tile tile
tile tile tile tile
tile tile tile tile
tile tile tile tile
Layout 10
tile tile tile tile
tile tile tile tile
tile tile tile tile
tile tile tile tile
Layout 11
tile tile tile tile
tile tile tile tile
tile tile tile tile
tile tile tile tile
Layout 12
tile tile tile tile
tile tile tile tile
tile tile tile tile
tile tile tile tile
Layout 13
tile tile tile tile
tile tile tile tile
tile tile tile tile
tile tile tile tile
Layout 14
tile tile tile tile
tile tile tile tile
tile tile tile tile
tile tile tile tile
Layout 15
tile tile tile tile
tile tile tile tile
tile tile tile tile
tile tile tile tile

If the maze paths are not confined and can wrap the stage borders then there are a few symmetrical solutions. Edge 3 and Edge 4 layouts produce a connected maze, as paths connect top - bottom and left - right stage borders.

The last solution is both symmetrical and connected. It is similar to the unique Brigid's Cross solution for a 2-corner tileset. It also has a mirror image by inverting the tiles.

Patt 1
tile tile tile tile
tile tile tile tile
tile tile tile tile
tile tile tile tile
Patt 2
tile tile tile tile
tile tile tile tile
tile tile tile tile
tile tile tile tile
Patt 3
tile tile tile tile
tile tile tile tile
tile tile tile tile
tile tile tile tile
Patt 4
tile tile tile tile
tile tile tile tile
tile tile tile tile
tile tile tile tile

Patt-1 and Patt-4 overall patterns are shown below. Patt-1 paths are limited to a diagonal pattern, while Patt-4 produces a fully connected maze.

Stage: Patt 1 and Patt 4 overall patterns

Overall edge patterns

Wang Tiles | cr31.co.uk | 2017