Wang Tiles

 

edge and corner matched tilesets

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3-edge Wang Tiles

A Wang tile with 3 different types of edge has 3^4 or 81 different tiles in a tileset. This is quite a lot and so is not often used to just produce a maze.

Here is the complete 3-edge Wang tileset.

tile 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 16
tile tile tile tile tile tile tile tile tile tile tile tile tile tile tile tile tile
17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33
tile tile tile tile tile tile tile tile tile tile tile tile tile tile tile tile tile
34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50
tile tile tile tile tile tile tile tile tile tile tile tile tile tile tile tile tile
51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67
tile tile tile tile tile tile tile tile tile tile tile tile tile
68 69 70 71 72 73 74 75 76 77 78 79 80

Tile Index

The following method creates a unique value for each tile:
Ignore blue edges. Add up yellow and green edges with the following tertiary weighting:

Yellow North edge = 1
Yellow East edge = 3
Yellow South edge = 9
Yellow West edge = 27
Green North edge = 2
Green East edge = 6
Green South edge = 18
Green West edge = 54
Blue edges are ignored.

3-edge weightings
  North
1 or 2
 
West
27 or 54
tile East
3 or 6
  South
9 or 18
 

This tile has 'Yellow North' + 'Green West', or 1 + 54, giving an index of '55'.

Tileset Layout

Here are 2 possible 9x9 layouts of the complete Wang 3-edge tileset. In a packing layout each tile is used once only.

Recursive (fractal) Layout

Here is a possible 9x9 layout of the complete Wang 3-edge tileset which is recursive as it can be extended in a similar manner to include higher tile orders.

You can see the order 2 tileset in the lower left quadrant.

See 4-order for a 4-edge layout.

 

Recursive layout of 3-edge Wang tiles
Wang tiles 3 edge recursive packing

Symmetrical Layout

Here is a possible 9x9 layout of the complete Wang 3-edge tileset which is symmetrical about the xy axis and can be extended in a similar manner to include any tile order. You can see the order 2 tileset in the lower left quadrant.

The bottom row and left column of tiles are repeated throughout the array. Just add up the index values. So the top right tile is 54+18 = 72.

Symmetrical layout of 3-edge Wang tiles
Wang tiles 3 edge symmetrical packing

See Path Tiles for more info and images.
See Stage for random tile arrays.

Stage: Wang 3-edge Tiles

Maze Tiles

Tile Rotation Symmetry

By removing rotations, we can also show the 3-edge Wang tileset like this:

3-edge Wang Tileset
tile tile tile tile tile tile tile tile tile tile tile tile
0 40 80 10
30
20
60
50
70
1
3
9
27
2
6
18
54
39
37
31
13
41
43
49
67
78
74
62
26
79
77
71
53
tile tile tile tile tile tile tile tile tile tile tile tile
4
12
36
28
8
24
72
56
44
52
76
68
5
15
45
55
7
21
63
29
38
34
22
66
42
46
58
14
73
59
17
51
75
65
35
25
16
48
64
32
23
69
47
61
33
19
57
11

Where the numbers below each tile denote a rotation of 90° clockwise. Wang tiles are never rotated but this is a compact way of showing the complete tileset. It is also useful when producing tiles in Photoshop as you can rotate and save images.
For more bitwise maths see 'Flip' and 'Rotate' in the Glossary.
We can create a tile matching puzzle game from a set of these tiles. See Puzzles.

3-edge 'Flow' Tiles

Another way of depicting 3 edges is to use 'flow' or 'directional' paths. Paths can only flow one way across an edge, either into or out of the tile.

tile tile tile tile tile tile tile tile tile tile tile tile tile tile tile tile tile
tile 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 16
tile tile tile tile tile tile tile tile tile tile tile tile tile tile tile tile tile
tile tile tile tile tile tile tile tile tile tile tile tile tile tile tile tile tile
17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33
tile tile tile tile tile tile tile tile tile tile tile tile tile tile tile tile tile
tile tile tile tile tile tile tile tile tile tile tile tile tile tile tile tile tile
34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50
tile tile tile tile tile tile tile tile tile tile tile tile tile tile tile tile tile
tile tile tile tile tile tile tile tile tile tile tile tile tile tile tile tile tile
51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67
tile tile tile tile tile tile tile tile tile tile tile tile tile
tile tile tile tile tile tile tile tile tile tile tile tile tile
68 69 70 71 72 73 74 75 76 77 78 79 80

We still have three different types of edge; blue, yellow and green.

1/ Blue edges have no path.
2/ Yellow edges have a path which flows South or West, (towards the origin).
3/ Green edges have a path which flows North or East, (away from the origin).

Note that we can no longer flip or rotate the tile index value mathematically, as the path direction may need to change. So a yellow edge may need to change into a green edge etc on rotation.

We can use this tileset to create the following 'flow' maze.
See Path Tiles for more info and images.
See Stage for random flow tile arrays.

Stage: Flow Tiles (3-edge)

Maze Tiles

Reduced Flow Tileset

For a perfect tree maze, we can reduce the number of tiles by only using tiles which have one and only one path flowing into the tile. This produces a set of 34 tiles, and here they are, with each tiles possible rotations.

Reduced 3-edge Flow Tileset
tile tile tile tile tile tile tile tile tile tile
0 1
3
18
54
5
24
63
28
7
12
45
56
10
30
20
60
16
39
47
62
26
69
37
32
14
51
65
34
41
53
71
43
44

Tile-0 may not be needed. Tile-44 is used as a special start tile at the center of the tree maze. See Flow tiles.

Reduced (sub) Tilesets

If the tiles are limited to just 2 colors then there are 3 posssible reduced tilesets of 16 tiles. The following table shows a Blue and Yellow, Yellow and Green and a Blue and Green tileset.

A complete 2-edge Wang tileset
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
3-edge Blue and Yellow only
tile tile tile tile tile tile tile tile tile tile tile tile tile tile tile tile
0 1 3 4 9 10 12 13 27 28 30 31 36 37 39 40
3-edge Yellow and Green only
tile tile tile tile tile tile tile tile tile tile tile tile tile tile tile tile
40 41 43 44 49 50 52 53 67 68 70 71 76 77 79 80
3-edge Blue and Green only
tile tile tile tile tile tile tile tile tile tile tile tile tile tile tile tile
0 2 6 8 18 20 24 26 54 56 60 62 72 74 78 80

Renaming the tiles from 0 to 15 and placing in the usual 2-edge layout produces the following three tilesets.

Blue and Yellow
tile tile tile tile
tile tile tile tile
tile tile tile tile
tile tile tile tile
Yellow and Green
tile tile tile tile
tile tile tile tile
tile tile tile tile
tile tile tile tile
Blue and Green
tile tile tile tile
tile tile tile tile
tile tile tile tile
tile tile tile tile

Yellow and Green tileset has an index of +40 Blue and Yellow.
Blue and Green has an index of twice Blue and Yellow.

The remaining 33 tiles all contain a mixture of all 3 terrain levels.

See Twin Tiles for maze creation with 34 tiles from the above 3-edge tileset.

Stage: Celtic 3-edge

Celtic 3-edge

Wang Tiles | cr31.co.uk | 2017