The smallest aperiodic set of Wang tiles hal-01166053 E. Jeandel and M. Rao Loria (Nancy), LIP (Lyon) January 18 E. Jeandel and M. Rao, The smallest aperiodic set of Wang tiles, hal-01166053 1/71
Wang tiles Each tile can be used as much as you want. The goal is to tile the entire plane, s.t. two adjacent tiles match on their common edge. E. Jeandel and M. Rao, The smallest aperiodic set of Wang tiles, hal-01166053 2/71
Example E. Jeandel and M. Rao, The smallest aperiodic set of Wang tiles, hal-01166053 3/71
Plan Introduction 1 10 tiles is not sufficient 2 A set with 11 tiles 3 E. Jeandel and M. Rao, The smallest aperiodic set of Wang tiles, hal-01166053 4/71
Trichotomy theorem You see, in this world, there’s three kinds of tilesets, my friend: Those who cannot tile a square of size n for some n They do not tile the plane Those who can tile a square of size n for some n with the same colors on the borders This gives a periodic tiling of the plane Those who tile the plane, but cannot do it periodically. These are aperiodic r tilesets. E. Jeandel and M. Rao, The smallest aperiodic set of Wang tiles, hal-01166053 5/71
Example 1 E. Jeandel and M. Rao, The smallest aperiodic set of Wang tiles, hal-01166053 6/71
Example 1 A 1 × 1 square: E. Jeandel and M. Rao, The smallest aperiodic set of Wang tiles, hal-01166053 7/71
Example 1 A 2 × 2 square: E. Jeandel and M. Rao, The smallest aperiodic set of Wang tiles, hal-01166053 8/71
Example 1 A 4 × 4 square: E. Jeandel and M. Rao, The smallest aperiodic set of Wang tiles, hal-01166053 9/71
Example 1 A 10 × 10 square E. Jeandel and M. Rao, The smallest aperiodic set of Wang tiles, hal-01166053 10/71
Example 1 A 30 × 30 periodic square E. Jeandel and M. Rao, The smallest aperiodic set of Wang tiles, hal-01166053 11/71
Example 2 E. Jeandel and M. Rao, The smallest aperiodic set of Wang tiles, hal-01166053 12/71
Example 2 A 4 × 4 square E. Jeandel and M. Rao, The smallest aperiodic set of Wang tiles, hal-01166053 13/71
Example 2 A 16 × 16 square: E. Jeandel and M. Rao, The smallest aperiodic set of Wang tiles, hal-01166053 14/71
Example 2 A 256 × 256 square: (Actual result may differ from picture shown) E. Jeandel and M. Rao, The smallest aperiodic set of Wang tiles, hal-01166053 15/71
Example 2 No 512 × 512 square E. Jeandel and M. Rao, The smallest aperiodic set of Wang tiles, hal-01166053 16/71
Example 3.1 E. Jeandel and M. Rao, The smallest aperiodic set of Wang tiles, hal-01166053 17/71
E. Jeandel and M. Rao, The smallest aperiodic set of Wang tiles, hal-01166053 18/71
Main difficulty - Undecidability There is no algorithm to decide in which case of the trichotomy a given tileset falls. In particular there is no systematic way to prove that some tileset is aperiodic (tiles, but not periodically) E. Jeandel and M. Rao, The smallest aperiodic set of Wang tiles, hal-01166053 19/71
Aperiodic tilesets Aperiodic tilesets are the cornerstone of nearly any result in tiling theory. A lot of different constructions of aperiodic tilesets in the literature, some with very specific properties. E. Jeandel and M. Rao, The smallest aperiodic set of Wang tiles, hal-01166053 20/71
History of aperiodic tilesets Berger, 1964 20426 tiles Berger, 1964 104 tiles Lauchli, 1966 40 Robinson, 1967 52 Knuth, 1968 92 Robinson, 1969 56 Robinson, 1971 35 Penrose, 1976 34 (32,24) Ammann, 1978 24 (16) Kari, 1996 14 Culik, 1996 13 5 colors Discrete geometry Planar geometry Arithmetic E. Jeandel and M. Rao, The smallest aperiodic set of Wang tiles, hal-01166053 21/71
Lower bounds for aperiodic tilesets Robinson (< 1980 ?) > 4 tiles Hu, Lin, 2010 > 2 colors Chen, Hu, Lai, Lin, 2012 > 3 colors E. Jeandel and M. Rao, The smallest aperiodic set of Wang tiles, hal-01166053 22/71
Contribution Theorem (J.-Rao, 2015) There is an aperiodic set of 11 tiles using 4 colors and this is optimal. E. Jeandel and M. Rao, The smallest aperiodic set of Wang tiles, hal-01166053 23/71
Difficulties We are trying to find a needle in a haystack E. Jeandel and M. Rao, The smallest aperiodic set of Wang tiles, hal-01166053 24/71
Difficulties We are trying to find an undecidable needle in a haystack E. Jeandel and M. Rao, The smallest aperiodic set of Wang tiles, hal-01166053 25/71
Plan Introduction 1 10 tiles is not sufficient 2 A set with 11 tiles 3 E. Jeandel and M. Rao, The smallest aperiodic set of Wang tiles, hal-01166053 26/71
New trichotomy There are three kinds of tilesets Those that cannot tile a horizontal strip of height n for some n Those that can tile a horizontal strip of height n for some n with the same colors on the two borders. Those that can tile the plane, but not periodically. E. Jeandel and M. Rao, The smallest aperiodic set of Wang tiles, hal-01166053 27/71
How to find the smallest aperiodic tileset Test all tilesets with 11 tiles or less For all tilesets: Test for all n how they tile a strip of height n , until they fall in the first two cases, or memory is exhausted In the last case, time to work! E. Jeandel and M. Rao, The smallest aperiodic set of Wang tiles, hal-01166053 28/71
Question How to test efficiently how a tileset tiles a strip of height n ? E. Jeandel and M. Rao, The smallest aperiodic set of Wang tiles, hal-01166053 29/71
Main idea 0 0 1 3 3 | 0 0 1 E. Jeandel and M. Rao, The smallest aperiodic set of Wang tiles, hal-01166053 30/71
Main idea 0 2 1 1 0 2 1 2 1 0 0 0 1 2 1 0 2 1 0 1 0 0 2 1 0 0 2 0 1 0 1 0 2 2 1 0 E. Jeandel and M. Rao, The smallest aperiodic set of Wang tiles, hal-01166053 31/71
Main idea A tileset is the same as a transducer. A tiling of an entire row is a biinfinite path on the transducer It exists iff the underlying graph contains a cycle If we keep only the transitions where input=output, we are looking at periodic tilings of the row E. Jeandel and M. Rao, The smallest aperiodic set of Wang tiles, hal-01166053 32/71
Strips How do we interpret strips of height 2 ? E. Jeandel and M. Rao, The smallest aperiodic set of Wang tiles, hal-01166053 33/71
Strips 4 0 | 4 0 2 0 2 0 0 3 | 0 0 1 0 1 3 E. Jeandel and M. Rao, The smallest aperiodic set of Wang tiles, hal-01166053 34/71
Strips 4 0 | 4 0 2 0 2 0 0 3 | 0 0 1 0 1 3 E. Jeandel and M. Rao, The smallest aperiodic set of Wang tiles, hal-01166053 34/71
Strips 4 0 2 3 | 4 0 0,0 2,1 0 0 1 3 E. Jeandel and M. Rao, The smallest aperiodic set of Wang tiles, hal-01166053 35/71
Strips Strips of height 2 are obtained by composing the transducer with itself output of the first transducer must match input of the second one. E. Jeandel and M. Rao, The smallest aperiodic set of Wang tiles, hal-01166053 36/71
Algorithm We see a tileset as a transducer T . For each n : We compute T n = T n − 1 ◦ T If T n does not contain any cycle, the tileset does not tile a strip of height n If T n contains a cycle where output=input, the tileset tiles a strip of height n periodically Otherwise, we test the next value of n E. Jeandel and M. Rao, The smallest aperiodic set of Wang tiles, hal-01166053 37/71
Main idea 0 2 1 1 0 2 1 2 1 0 0 0 1 2 1 0 2 1 0 1 0 0 2 1 0 0 2 0 1 0 1 0 2 2 1 0 E. Jeandel and M. Rao, The smallest aperiodic set of Wang tiles, hal-01166053 38/71
n = 1 There is a cycle, a strip of height 1 can be obtained E. Jeandel and M. Rao, The smallest aperiodic set of Wang tiles, hal-01166053 39/71
n = 1 There is no cycle, a periodic strip of height 1 cannot be obtained E. Jeandel and M. Rao, The smallest aperiodic set of Wang tiles, hal-01166053 40/71
n = 2 There is a cycle, a strip of height 2 can be obtained E. Jeandel and M. Rao, The smallest aperiodic set of Wang tiles, hal-01166053 41/71
n = 2 There is no cycle, a periodic strip of height 2 cannot be obtained E. Jeandel and M. Rao, The smallest aperiodic set of Wang tiles, hal-01166053 42/71
n = 3 There is a cycle, a strip of height 3 can be obtained E. Jeandel and M. Rao, The smallest aperiodic set of Wang tiles, hal-01166053 43/71
n = 3 There is no cycle, a periodic strip of height 3 cannot be obtained E. Jeandel and M. Rao, The smallest aperiodic set of Wang tiles, hal-01166053 44/71
n = 4 etc, etc. E. Jeandel and M. Rao, The smallest aperiodic set of Wang tiles, hal-01166053 45/71
Optimizations We can be smarter E. Jeandel and M. Rao, The smallest aperiodic set of Wang tiles, hal-01166053 46/71
n = 3 E. Jeandel and M. Rao, The smallest aperiodic set of Wang tiles, hal-01166053 47/71
n = 3 E. Jeandel and M. Rao, The smallest aperiodic set of Wang tiles, hal-01166053 48/71
How to be smarter At each step, we can delete states that have no incoming (outgoing) edges. At each step, we can delete edges that cannot be part of a cycle. i.e. whose ends belong to different strongly connected components. This optimization is powerful enough to treat quickly almost all tilesets of 10 tiles or less E. Jeandel and M. Rao, The smallest aperiodic set of Wang tiles, hal-01166053 49/71
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