Wang Tiles

 

edge and corner matched tilesets

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3-corner Wang Tileset

The complete set contains (3^4) = 81 tiles, colored blue, yellow and green. (This can be thought of as increasing height of water, sand and grass terrain). The outer edge of each tile is marked, but not the boundary between similar colors on the same tile.

Here is the complete 3-corner 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

Tileset Packing Layout

In a packing layout each tile is used once only. For 81 tiles, a common layout is a 9x9 square array. Here are 2 possible 9x9 layouts of the complete Wang 3-edge tileset.

Recursive (fractal) Layout

The tiles can be arranged in a unique recursive layout. From the lower left corner you can see a 4x4 tile square array showing all 2-corner tiles in a 'Brigid's Cross' layout. (See all 2-corner tiles here).

The whole array can be extended to 16x16 tiles to show all Wang 4-color corners etc. So this is a fractal arrangement, which can be extended for any order of Wang corner tiles.

Recursive layout of 3-corner Wang tiles
Wang tiles recursive array

Symmetrical Layout

Here is a left-right symmetrical layout of all 81 tiles.

All borders wrap, so both arrays can cover any area seemlessly.

See 4-order tilesets for 4-edge layoust.

Symmetrical layout of 3-corner Wang tiles
Wang tiles symmetrical array

Bitwise Tile Index

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

Yellow NorthEast corner = 1
Yellow SouthEast corner = 3
Yellow SouthWest corner = 9
Yellow NorthWest corner = 27

Green NorthEast corner = 2
Green SouthEast corner = 6
Green SouthWest corner = 18
Green NorthWest corner = 54

Blue corners are ignored.

3-corner weightings
NorthWest
Yellow 27 or Green 54
  NorthEast
Yellow 1 or Green 2
  tile  
SouthWest
Yellow 9 or Green 18
  SouthEast
Yellow 3 or Green 6

This tile has 'Green SouthEast' + 'Yellow NorthWest', so, 6 + 27, giving a tile index of '33'.

Tile Rotation Symmetry

By removing self similar rotations, we reduce the 3-corner Wang tileset to 24 tiles. Wang tiles are never rotated but this is a compact way of showing the complete tileset.

3-corner 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

The index numbers below each tile denote a rotation of 90° clockwise. This can be useful when creating tilesets in Photoshop. Rotate and SaveAs the next tile number.

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.

Stage: Wang 3-corner Tiles

Campus

Campus

From the above Wang tiles, a simple design can be constructed where each Wang tile corner color determines a different height of terrain.
Blue - middle/ground, Yellow - high level and Green - low level.

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

See Terrain Tiles for more info and images.
See Stage for random terrain arrays.

Campus symmetrical layout

This is how a campus building would look if built symetrically using 3 levels. As the edges wrap, it can tile any area seamlessly.

Campus 9x9 symetrical layout

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 & Yellow, Yellow & Green and a Blue & Green tileset.

A complete 2-corner 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-corner Blue & 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-corner Yellow & 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-corner Blue & 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-corner layout produces the following three tilesets.

Blue & Yellow
tile tile tile tile
tile tile 12 5
2 15 11 13
3 tile tile tile
Yellow & Green
tile tile tile tile
tile tile tile tile
tile tile tile tile
tile tile tile tile
Blue & Green
tile tile tile tile
tile tile tile tile
tile tile tile tile
tile tile tile tile

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

The remaining 36 tiles (81 - 3x15) all contain a mixture of all 3 terrain levels.

36 tiles containing 3 different levels
tile tile tile tile tile tile tile tile tile tile tile tile
5 7 11 14 15 16 17 19 21 22 23 25
tile tile tile tile tile tile tile tile tile tile tile tile
29 32 33 34 35 38 42 45 46 47 48 51
tile tile tile tile tile tile tile tile tile tile tile tile
55 57 58 59 61 63 64 65 66 69 73 75

Stage: Campus

Campus

Wang Tiles | cr31.co.uk | 2017