A small strongly aperiodic tile set in H 2 Jarkko Kari Mathematics Department, University of Turku, Finland
In the Euclidean plane E 2 , the term aperiodicity refers to the property that a tile set admits valid tilings, but no such tiling is periodic , i.e., invariant under a (non-trivial) translation.
In the Euclidean plane E 2 , the term aperiodicity refers to the property that a tile set admits valid tilings, but no such tiling is periodic , i.e., invariant under a (non-trivial) translation. A seemingly weaker requirement would be that no valid tiling is doubly periodic , i.e., invariant under two non-collinear translations. However, it is well known that in E 2 a tile set admits a periodic tiling if and only if it admits a doubly periodic tilings, so the two concepts of aperiodicity coincide.
In the hyperbolic plane H 2 the situation is different: one has two different concepts of aperiodicity. We follow the terminology of Chaim Goodman-Strauss: • A tile set that admits valid tilings is called strongly aperiodic if it does not admit a tiling whose symmetry group contains an infinite cyclic subgroup,
In the hyperbolic plane H 2 the situation is different: one has two different concepts of aperiodicity. We follow the terminology of Chaim Goodman-Strauss: • A tile set that admits valid tilings is called strongly aperiodic if it does not admit a tiling whose symmetry group contains an infinite cyclic subgroup, • A tile set that admits valid tilings is called weakly aperiodic if it does not admit a tiling whose symmetry group has a compact fundamental domain.
In H 2 a single prototile can be weakly aperiodic [Penrose 1978] . First strongly aperiodic protoset was constructed in [Goodman-Strauss 2004] . In this talk we give a new strongly aperiodic set that consists of 15 tiles. Our construction is analogous to our construction in E 2 of an aperiodic set of 14 Wang tiles [Kari 1996] .
In H 2 a single prototile can be weakly aperiodic [Penrose 1978] . First strongly aperiodic protoset was constructed in [Goodman-Strauss 2004] . In this talk we give a new strongly aperiodic set that consists of 15 tiles. Our construction is analogous to our construction in E 2 of an aperiodic set of 14 Wang tiles [Kari 1996] . The existence of strongly aperiodic tile sets is related to the undecidability of the tiling problem. The tiling problem in H 2 was proved undecidable independently in [Kari 2007] and [Margenstern 2007] . The construction presented in this talk can be viewed as a simplification of our undecidability proof.
An aperiodic Wang tile set in E 2 Since our construction is analogous to the aperiodic Wang tile set in E 2 , we first review the Euclidean construction.
A Wang tile is a unit square tile with colored edges. A tile set T is a finite collection of such tiles. A valid tiling is an assignment Z 2 − → T of tiles on infinite square lattice so that the abutting edges of adjacent tiles have the same color.
A Wang tile is a unit square tile with colored edges. A tile set T is a finite collection of such tiles. A valid tiling is an assignment Z 2 − → T of tiles on infinite square lattice so that the abutting edges of adjacent tiles have the same color. For example, consider Wang tiles A D B C
With copies of the given four tiles we can properly tile a 5 × 5 square. . . A D B C C A D B C C D B A C C A D B C C B A D C C . . . and since the colors on the borders match this square can be repeated to form a doubly periodic tiling of the plane.
Note: Wang tiles are abstract tiles, but one can effective transform them into equivalent concrete shapes (e.g. polygons with rational coordinates). For example, we can make each Wang tile into a unit square tile whose left and upper edges have a bump and the right and lower edge has a dent. The shape of the bump/dent depends on the color of the edge. Each color has a unique shape associated with it (and different shapes are used for horizontal and vertical colors). A D B C A B D C
The colors in our Wang tiles are real numbers, for example 1 1 0 1 0 0 0 -1 -1 -1 0 -1 2 1 2 1
The colors in our Wang tiles are real numbers, for example 1 1 0 1 0 0 0 -1 -1 -1 0 -1 2 1 2 1 We say that tile n e w s multiplies by number q ∈ R if qn + w = s + e. (The ”input” n comes from the north, and the ”carry in” w from the west is added to the product qn . The result is split between the ”output” s to the south and the ”carry out” e to the east.)
The colors in our Wang tiles are real numbers, for example 1 1 0 1 0 0 0 -1 -1 -1 0 -1 2 1 2 1 We say that tile n e w s multiplies by number q ∈ R if qn + w = s + e. The four sample tiles above all multiply by q = 2 .
Suppose we have a correctly tiled horizontal segment where all tiles multiply by the same q . n n n n 1 2 k 3 e w k 1 s s s s 2 k 1 3 It easily follows that q ( n 1 + n 2 + . . . + n k ) + w 1 = ( s 1 + s 2 + . . . + s k ) + e k . To see this, simply sum up the equations qn 1 + w 1 = s 1 + e 1 qn 2 + w 2 = s 2 + e 2 . . . qn k + w k = s k + e k , taking into account that always e i = w i +1 .
Suppose we have a correctly tiled horizontal segment where all tiles multiply by the same q . n n n n 1 2 k 3 e w k 1 s s s s 2 k 1 3 If, moreover, the segment begins and ends in the same color ( w 1 = e k ) then q ( n 1 + n 2 + . . . + n k ) = ( s 1 + s 2 + . . . + s k ) .
For example, using our three sample tiles that multiply by q = 2 we can form the segment 1 1 0 -1 -1 2 1 1 in which the sum of the bottom labels is twice the sum of the top labels.
1 0 1 1 0 0 -1 0 -1 -1 0 -1 2 2 1 1 2 2 2 2 2 - - 2 1 1 1 2 1 1 0* 0* 0* 3 3 3 3 3 3 3 1 2 1 1 2 1 1 1 1 1 - - 1 2 2 1 1 1 1 0* 0* 0* 3 3 3 3 3 3 3 1 0 0 1 1 Our aperiodic tile set consists of the four tiles that multiply by 2, together with another family of 10 tiles that all multiply by 2 3 .
T 2 1 0 1 1 0 0 -1 0 -1 -1 0 -1 2 2 1 1 T 2/3 2 2 2 2 2 - - 2 1 1 1 2 1 1 0* 0* 0* 3 3 3 3 3 3 3 1 2 1 1 2 1 1 1 1 1 - - 1 2 2 1 1 1 1 0* 0* 0* 3 3 3 3 3 3 3 1 0 0 1 1 Let us call these two tile sets T 2 and T 2 / 3 . Vertical edge colors of the two parts are made disjoint, so any properly tiled horizontal row comes entirely from one of the two sets.
Let us prove that no periodic tiling exists. Suppose the contrary: A rectangle can be tiled whose top and bottom rows match and left and right sides match. n 1 n 2 n 3 n k n k+1 Denote by n i the sum of the numbers on the i ’th horizontal row (counted from top to bottom). Let the tiles of the i ’th row multiply by q i ∈ { 2 , 2 3 } .
Let us prove that no periodic tiling exists. Suppose the contrary: A rectangle can be tiled whose top and bottom rows match and left and right sides match. n 1 n 2 n 3 n k n k+1 Denote by n i the sum of the numbers on the i ’th horizontal row (counted from top to bottom). Let the tiles of the i ’th row multiply by q i ∈ { 2 , 2 3 } . From our previous discussion we know that n i +1 = q i n i , for all i .
Let us prove that no periodic tiling exists. Suppose the contrary: A rectangle can be tiled whose top and bottom rows match and left and right sides match. n 1 n 2 n 3 n k n k+1 So we have q 1 q 2 q 3 . . . q k n 1 = n k +1
Let us prove that no periodic tiling exists. Suppose the contrary: A rectangle can be tiled whose top and bottom rows match and left and right sides match. n 1 n 2 n 3 n k n k+1 So we have q 1 q 2 q 3 . . . q k n 1 = n k +1 = n 1 .
Let us prove that no periodic tiling exists. Suppose the contrary: A rectangle can be tiled whose top and bottom rows match and left and right sides match. n 1 n 2 n 3 n k n k+1 So we have q 1 q 2 q 3 . . . q k n 1 = n k +1 = n 1 . Clearly n 1 > 0, so we have q 1 q 2 q 3 . . . q k = 1. But this is not possible since 2 and 3 are relatively prime: No product of numbers 3 and 2 3 can equal 1.
Next step: We still need to show that a valid tiling of the plane exists using our tiles. For this purpose we introduce sturmian or balanced representations of real numbers as bi-infinite sequences of two closest integers. The representation of any α ∈ R is the sequence B ( α ) whose k ’th element is B k ( α ) = ⌊ kα ⌋ − ⌊ ( k − 1) α ⌋ . For example, B ( 1 3 ) = . . . 0 0 1 0 0 1 0 0 1 0 0 1 . . .
Next step: We still need to show that a valid tiling of the plane exists using our tiles. For this purpose we introduce sturmian or balanced representations of real numbers as bi-infinite sequences of two closest integers. The representation of any α ∈ R is the sequence B ( α ) whose k ’th element is B k ( α ) = ⌊ kα ⌋ − ⌊ ( k − 1) α ⌋ . For example, B ( 1 3 ) = . . . 0 0 1 0 0 1 0 0 1 0 0 1 . . . B ( 7 5 ) = . . . 1 1 2 1 2 1 1 2 1 2 1 1 . . .
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