Wang Tiles

 

edge and corner matched tilesets

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Reduced Tilesets

The mathematical problem of proving if an infinite plane can be tiled with a limited set of Wang tiles was proposed by mathematician Hao Wang.

As an example, tileset 1 is capable of filling the grid, but tileset 2 is not. A vacant cell cannot be filled.

Tileset 1
6 14 8
Coverable
14 6 8
8 14 6
6 8 14
Tileset 2
6 14 8
Non coverable
6 8 14
14 8 qmark
6 8 14

Wang investigated the minimum number of tiles needed to tile a plane. However, he was concerned with the mathematical proofs of non periodic tile patterns on an infinite plane. We can explore pleasing reduced tilesets without the maths.

Tilesets must include enough tiles to ensure random mazes can develop. But not too many unnecessary tiles that the resulting paths looks messy.

For instance, a 2-edge Wang tileset consists of 16 tiles. Sometimes we want to use less tiles. For aesthetic reasons, efficiency and maybe manufacturing costs.

Tileset

Often, the blank (tile-0) and the crossroads (tile-15) are removed. It produces a maze with a more even distribution of paths. Also, the four dead end paths (tiles 1, 2, 4, and 8) can be removed as they are too easily dismissed when solving a maze. This gives a reduced tileset of 10 tiles.

All 16 combinations of two edges
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
A reduced set of 10 tiles
tile tile tile tile tile tile tile tile tile tile
3 5 6 7 9 10 11 12 13 14

See Puzzle Tilesets for reduced tile sets used in tile matching puzzles.

Minimum Wang Tileset

If the grid is tiled from the top-left corner to the bottom-right corner (as natural text flow) then we only need four different tiles. As each tile is placed, it only needs to match the tile above and to the left. So we need only four different combinations of top (North) and left (West) edges.

Minimum Tileset
tile tile tile tile
tile tile tile tile
tile tile tile tile
tile tile tile tile
0s 1s 8s 9s
Random tiling of 4 tiles
tile tile tile tile tile tile
tile tile tile tile tile tile
tile tile tile tile tile tile
tile tile tile tile tile tile

The top row shows the tiles used as the tileset to build the random layout. Each tile (6, 7 14 and 9) has a different combination of top and left edge. So they can fill any grid, (the grid cannot wrap). Each column holds alternative tiles.

The first placed tile is chosen at random. The tiles in the top row and left column of the grid only have two possible choices. All other grid tiles have no choice. The tile above and to the left determines the tile selection. Therefore the grid top and left tiles act as a kind of 'seed' for the whole grid.

Labyrinth Tiles

Labyrinth or finger mazes have no branches, only one long path made up of straights and bends.

Labyrinth set of 6 tiles
tile tile tile tile tile tile
3 5 6 9 10 12

Stage: Labyrinth

Maze Tiles

Vine Tileset

Vine set of 4 tiles
tile tile tile
0 1 2 3

Stage: Ceramic Vine Tiles

Maze Tiles

Wang Tiles | cr31.co.uk | 2017