[PPT] - The smallest aperiodic set of Wang tiles hal-01166053 E. Jeandel PowerPoint Presentation

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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

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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

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Example

E. Jeandel and M. Rao,

The smallest aperiodic set of Wang tiles, hal-01166053 3/71

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Plan

1

Introduction

2

10 tiles is not sufficient

3

A set with 11 tiles

E. Jeandel and M. Rao,

The smallest aperiodic set of Wang tiles, hal-01166053 4/71

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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 aperiodicr tilesets.

E. Jeandel and M. Rao,

The smallest aperiodic set of Wang tiles, hal-01166053 5/71

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Example 1

E. Jeandel and M. Rao,

The smallest aperiodic set of Wang tiles, hal-01166053 6/71

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Example 1

A 1 × 1 square:

E. Jeandel and M. Rao,

The smallest aperiodic set of Wang tiles, hal-01166053 7/71

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Example 1

A 2 × 2 square:

E. Jeandel and M. Rao,

The smallest aperiodic set of Wang tiles, hal-01166053 8/71

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Example 1

A 4 × 4 square:

E. Jeandel and M. Rao,

The smallest aperiodic set of Wang tiles, hal-01166053 9/71

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Example 1

A 10 × 10 square

E. Jeandel and M. Rao,

The smallest aperiodic set of Wang tiles, hal-01166053 10/71

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Example 1

A 30 × 30 periodic square

E. Jeandel and M. Rao,

The smallest aperiodic set of Wang tiles, hal-01166053 11/71

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Example 2

E. Jeandel and M. Rao,

The smallest aperiodic set of Wang tiles, hal-01166053 12/71

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Example 2

A 4 × 4 square

E. Jeandel and M. Rao,

The smallest aperiodic set of Wang tiles, hal-01166053 13/71

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Example 2

A 16 × 16 square:

E. Jeandel and M. Rao,

The smallest aperiodic set of Wang tiles, hal-01166053 14/71

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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

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Example 2

No 512 × 512 square

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The smallest aperiodic set of Wang tiles, hal-01166053 16/71

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Example 3.1

E. Jeandel and M. Rao,

The smallest aperiodic set of Wang tiles, hal-01166053 17/71

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E. Jeandel and M. Rao,

The smallest aperiodic set of Wang tiles, hal-01166053 18/71

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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

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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

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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

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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

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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

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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

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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

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Plan

1

Introduction

2

10 tiles is not sufficient

3

A set with 11 tiles

E. Jeandel and M. Rao,

The smallest aperiodic set of Wang tiles, hal-01166053 26/71

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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

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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

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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

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Main idea

1 3 1 3|0

E. Jeandel and M. Rao,

The smallest aperiodic set of Wang tiles, hal-01166053 30/71

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Main idea

1 2 1 1 2 2 2 1 1 2 2 1 1 1 1 2 1 2 2 1 1

E. Jeandel and M. Rao,

The smallest aperiodic set of Wang tiles, hal-01166053 31/71

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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

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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

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Strips

1 3 2 4 1 3|0 2 0|4

E. Jeandel and M. Rao,

The smallest aperiodic set of Wang tiles, hal-01166053 34/71

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Strips

1 3 2 4 1 3|0 2 0|4

E. Jeandel and M. Rao,

The smallest aperiodic set of Wang tiles, hal-01166053 34/71

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Strips

1 3 2 4 0,0 2,1 3|4

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The smallest aperiodic set of Wang tiles, hal-01166053 35/71

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Strips

Strips of height 2 are obtained by composing the transducer with itself

utput 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

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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

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Main idea

1 2 1 1 2 2 2 1 1 2 2 1 1 1 1 2 1 2 2 1 1

E. Jeandel and M. Rao,

The smallest aperiodic set of Wang tiles, hal-01166053 38/71

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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

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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

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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

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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

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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

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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

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n = 4

etc, etc.

E. Jeandel and M. Rao,

The smallest aperiodic set of Wang tiles, hal-01166053 45/71

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Optimizations

We can be smarter

E. Jeandel and M. Rao,

The smallest aperiodic set of Wang tiles, hal-01166053 46/71

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n = 3

E. Jeandel and M. Rao,

The smallest aperiodic set of Wang tiles, hal-01166053 47/71

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n = 3

E. Jeandel and M. Rao,

The smallest aperiodic set of Wang tiles, hal-01166053 48/71

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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

f 10 tiles or less
E. Jeandel and M. Rao,

The smallest aperiodic set of Wang tiles, hal-01166053 49/71

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We can do better

The exact transducer does not matter, only the mapping (from input to

utput) it encodes

We can minimize the transducer at each step Warning: minimizing a nondeterministic transducer is PSPACE-hard, we merely reduce it using bisimilarity.

E. Jeandel and M. Rao,

The smallest aperiodic set of Wang tiles, hal-01166053 50/71

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Example

E. Jeandel and M. Rao,

The smallest aperiodic set of Wang tiles, hal-01166053 51/71

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Results

Using transducers, and simplifying transducers at each time, we are able to test all tilesets with (strictly) less than 11 tiles in a few months We do not generate all tilesets, but only those that could be minimal aperiodic (e.g. every state of the transducer should have incoming/outgoing edges).

E. Jeandel and M. Rao,

The smallest aperiodic set of Wang tiles, hal-01166053 52/71

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Results

Among all tilesets, 200 of them did exhaust the available memory. 199 of the remaining were treated using more memory. One tileset remained. We had to examine it by hand, and prove it does not tile the plane.

E. Jeandel and M. Rao,

The smallest aperiodic set of Wang tiles, hal-01166053 53/71

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Example 2

E. Jeandel and M. Rao,

The smallest aperiodic set of Wang tiles, hal-01166053 54/71

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Plan

1

Introduction

2

10 tiles is not sufficient

3

A set with 11 tiles

E. Jeandel and M. Rao,

The smallest aperiodic set of Wang tiles, hal-01166053 55/71

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11 tiles

We did the same for tilesets with 11 tiles, and 26 of them remain 9 of them likely do not tile the plane One of the others did catch our eyes

E. Jeandel and M. Rao,

The smallest aperiodic set of Wang tiles, hal-01166053 56/71

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The set

1 3 1 2 1 2 2 1 1 2 4 1 3 3 2 3 1 1 3 1 1 1 3 1 2 2 3 3 1 3 2 2 4 1 2 2 2 1 2 3 1|0 2|1 2|2 4|2 2|3 1|1 1|1,2|2 3|1 1|4 0|2

E. Jeandel and M. Rao,

The smallest aperiodic set of Wang tiles, hal-01166053 57/71

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Main result

Theorem

The previous tileset is aperiodic. Remainder: there is no general method to prove that a tileset is aperiodic.

E. Jeandel and M. Rao,

The smallest aperiodic set of Wang tiles, hal-01166053 58/71

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Idea of the proof

One “remarks” that every tiling of the plane can be divided into strips of height 8, 10 and 14 which can be described by nice transducers T0, T1, T2. We forget about the initial tileset, and we focus on these three transducers. We have no explanation for this.

E. Jeandel and M. Rao,

The smallest aperiodic set of Wang tiles, hal-01166053 59/71

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The 3 transducers

ǫ|ǫ 05 |(100)12 05+3 |13(000)12 02(111)03|15+3 02(110) |12+3 05(110)02|12(100)15

E. Jeandel and M. Rao,

The smallest aperiodic set of Wang tiles, hal-01166053 60/71

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The 3 transducers

15−3|05−3 13+3 |(110)03 18+3 |05(111)03 13(000)15|08+3) 13(100) |03+3 18(100)13|03(110)08

E. Jeandel and M. Rao,

The smallest aperiodic set of Wang tiles, hal-01166053 61/71

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The 3 transducers

08−3|18−3 05+3 |(100)15 013+3 |18(000)15 05(111)08 |113+3 05(110) |15+3 013(110)05|15(100)113

E. Jeandel and M. Rao,

The smallest aperiodic set of Wang tiles, hal-01166053 62/71

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Idea of the proof

We have a tiling using the three transducers T0, T1, T2. One can prove that in any tiling of the plane, the transducers T0 should be bordered by two transducers T1 Define T3 = T1 ◦ T0 ◦ T1. Then the tiling of the plane can be decomposed into the three transducers T1, T2, T3. T3 is very similar to the previous ones.

E. Jeandel and M. Rao,

The smallest aperiodic set of Wang tiles, hal-01166053 63/71

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The transducer T3

113−3|013−3 18+3 |(110)08 121+3 |013(111)08 18(000)113|021+3) 18(100) |08+3 121(100)18|08(110)021

E. Jeandel and M. Rao,

The smallest aperiodic set of Wang tiles, hal-01166053 64/71

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Idea of the proof

One can generalize the transducers into a family Tn and prove inductively the following: Assume that a tiling of the plane can be decomposed into the three transducers Tn, Tn+1, Tn+2. Then in any tiling of the plane, the transducers Tn should be bordered by two transducers Tn+1 Tn+3 = Tn+1 ◦ Tn ◦ Tn+1. Then the tiling of the plane can be decomposed into the three transducers Tn+1, Tn+2, Tn+3. This proves that this tiles the plane, and also that any tiling should be aperiodic (but substitutive)

E. Jeandel and M. Rao,

The smallest aperiodic set of Wang tiles, hal-01166053 65/71

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Tn for n odd

(g(n) is Fibonacci) 1g(n+2)−3|0g(n+2)−3 1g(n+1)+3 |(110)0g(n+1) 1g(n+3)+3 |0g(n+2)(111)0g(n+1) 1g(n+1)(000)1g(n+2)|0g(n+3)+3 1g(n+1)(100) |0g(n+1)+3 1g(n+3)(100)1g(n+1)|0g(n+1)(110)0g(n+3)

E. Jeandel and M. Rao,

The smallest aperiodic set of Wang tiles, hal-01166053 66/71

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Tn for n even

0g(n+2)−3|1g(n+2)−3 0g(n+1)+3 |(100)1g(n+1) 0g(n+3)+3 |1g(n+2)(000)1g(n+1) 0g(n+1)(111)0g(n+2)|1g(n+3)+3 0g(n+1)(110) |1g(n+1)+3 0g(n+3)(110)0g(n+1)|1g(n+1)(100)1g(n+3)

E. Jeandel and M. Rao,

The smallest aperiodic set of Wang tiles, hal-01166053 67/71

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Main result

Theorem

There exists a tileset of 11 tiles and 5 colors which is aperiodic.

Theorem

There exists a tileset of 11 tiles and 4 colors which is aperiodic. (Same tileset with colors 4 and 0 merged)

E. Jeandel and M. Rao,

The smallest aperiodic set of Wang tiles, hal-01166053 68/71

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Conclusion

We have found the smallest (actually 2) aperiodic tilesets 24 others are possible candidates: One of them has probably the same structure 9 of them are variants of the Kari-Culik family, and should not be aperiodic The 14 other tilesets are the most interesting.

They do not belong to well-known families.

E. Jeandel and M. Rao,

The smallest aperiodic set of Wang tiles, hal-01166053 69/71

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E. Jeandel and M. Rao,

The smallest aperiodic set of Wang tiles, hal-01166053 70/71

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E. Jeandel and M. Rao,

The smallest aperiodic set of Wang tiles, hal-01166053 71/71