a topological study of tilings -...
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IntroductionBesicovitch topology
Cantor topologyTopological games
A Topological Study of Tilings
Grégory Lafitte1 Michaël Weiss2
1Laboratoire d’Informatique Fondamentale de Marseille (LIF), CNRS –Aix-Marseille Université, 39, rue Joliot-Curie, F-13453 Marseille Cedex 13, France
2Centre Universitaire d’Informatique, Université de Genève, Battelle bâtiment A, 7route de Drize, 1227 Carouge, Switzerland
Theory and Applications of Models of Computation, 2008
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
Outline
1 IntroductionWang TilingsMotivation
2 Besicovitch topologyDefinitionThe topological space
3 Cantor topologyDefinitionThe topological space
4 Topological gamesIntroductionGames on tilings
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
Wang TilingsMotivation
Outline
1 IntroductionWang TilingsMotivation
2 Besicovitch topologyDefinitionThe topological space
3 Cantor topologyDefinitionThe topological space
4 Topological gamesIntroductionGames on tilings
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
Wang TilingsMotivation
Wang Tilings
A Wang tile is a unit size square with colored edges.A tile set "τ " is a finite set of Wang tiles.To tile consists in placing the tiles in the grid Z2, respectingthe color matching. A tiling P is associated to a tilingfunction fP : Z2 → τ .
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
Wang TilingsMotivation
Wang Tilings
A Wang tile is a unit size square with colored edges.A tile set "τ " is a finite set of Wang tiles.To tile consists in placing the tiles in the grid Z2, respectingthe color matching. A tiling P is associated to a tilingfunction fP : Z2 → τ .
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
Wang TilingsMotivation
Wang Tilings
A Wang tile is a unit size square with colored edges.A tile set "τ " is a finite set of Wang tiles.To tile consists in placing the tiles in the grid Z2, respectingthe color matching. A tiling P is associated to a tilingfunction fP : Z2 → τ .
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
Wang TilingsMotivation
Wang Tilings
A Wang tile is a unit size square with colored edges.A tile set "τ " is a finite set of Wang tiles.To tile consists in placing the tiles in the grid Z2, respectingthe color matching. A tiling P is associated to a tilingfunction fP : Z2 → τ .
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
Wang TilingsMotivation
Wang Tilings
A Wang tile is a unit size square with colored edges.A tile set "τ " is a finite set of Wang tiles.To tile consists in placing the tiles in the grid Z2, respectingthe color matching. A tiling P is associated to a tilingfunction fP : Z2 → τ .
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
Wang TilingsMotivation
Wang Tilings
A Wang tile is a unit size square with colored edges.A tile set "τ " is a finite set of Wang tiles.To tile consists in placing the tiles in the grid Z2, respectingthe color matching. A tiling P is associated to a tilingfunction fP : Z2 → τ .
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
Wang TilingsMotivation
Wang Tilings
A Wang tile is a unit size square with colored edges.A tile set "τ " is a finite set of Wang tiles.To tile consists in placing the tiles in the grid Z2, respectingthe color matching. A tiling P is associated to a tilingfunction fP : Z2 → τ .
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
Wang TilingsMotivation
Wang Tilings
A Wang tile is a unit size square with colored edges.A tile set "τ " is a finite set of Wang tiles.To tile consists in placing the tiles in the grid Z2, respectingthe color matching. A tiling P is associated to a tilingfunction fP : Z2 → τ .
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
Wang TilingsMotivation
Wang Tilings
A Wang tile is a unit size square with colored edges.A tile set "τ " is a finite set of Wang tiles.To tile consists in placing the tiles in the grid Z2, respectingthe color matching. A tiling P is associated to a tilingfunction fP : Z2 → τ .
Domino ProblemDoes a given tile set tile the plane?
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
Wang TilingsMotivation
Wang Tilings
A Wang tile is a unit size square with colored edges.A tile set "τ " is a finite set of Wang tiles.To tile consists in placing the tiles in the grid Z2, respectingthe color matching. A tiling P is associated to a tilingfunction fP : Z2 → τ .
Domino ProblemDoes a given tile set tile the plane?
Berger’s proofThe problem is undecidable. Tilings areequivalent to Turing machine.
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
Wang TilingsMotivation
Different kinds of tile sets
A tile set can be:FinitePeriodicAperiodicNon-recursiveComplete. . .
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
Wang TilingsMotivation
Different kinds of tile sets
A tile set can be:FinitePeriodicAperiodicNon-recursiveComplete. . .
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
Wang TilingsMotivation
Different kinds of tile sets
A tile set can be:FinitePeriodicAperiodicNon-recursiveComplete. . .
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
Wang TilingsMotivation
Different kinds of tile sets
A tile set can be:FinitePeriodicAperiodicNon-recursiveComplete. . .
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
Wang TilingsMotivation
Different kinds of tile sets
A tile set can be:FinitePeriodicAperiodicNon-recursiveComplete. . .
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
Wang TilingsMotivation
Different kinds of tile sets
A tile set can be:FinitePeriodicAperiodicNon-recursiveComplete. . .
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
Wang TilingsMotivation
Different kinds of tile sets
A tile set can be:FinitePeriodicAperiodicNon-recursiveComplete. . .
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
Wang TilingsMotivation
Outline
1 IntroductionWang TilingsMotivation
2 Besicovitch topologyDefinitionThe topological space
3 Cantor topologyDefinitionThe topological space
4 Topological gamesIntroductionGames on tilings
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
Wang TilingsMotivation
Motivation
Two commonly-used topologies on cellular automataProximity between tilings and cellular automataAdapt these topologies to tilingsUnderstand the behavior of the space of tilingsStudy computation from a geometrical point of view
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
Wang TilingsMotivation
Motivation
Two commonly-used topologies on cellular automataProximity between tilings and cellular automataAdapt these topologies to tilingsUnderstand the behavior of the space of tilingsStudy computation from a geometrical point of view
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
Wang TilingsMotivation
Motivation
Two commonly-used topologies on cellular automataProximity between tilings and cellular automataAdapt these topologies to tilingsUnderstand the behavior of the space of tilingsStudy computation from a geometrical point of view
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
Wang TilingsMotivation
Motivation
Two commonly-used topologies on cellular automataProximity between tilings and cellular automataAdapt these topologies to tilingsUnderstand the behavior of the space of tilingsStudy computation from a geometrical point of view
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
Wang TilingsMotivation
Motivation
Two commonly-used topologies on cellular automataProximity between tilings and cellular automataAdapt these topologies to tilingsUnderstand the behavior of the space of tilingsStudy computation from a geometrical point of view
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
Wang TilingsMotivation
What we have, what we expect
ResultsHow does the tiling space behave inside the mappingspaceWhat both topologies can say on tilingsTopological games to study meagerness
ExpectationsGames to solve problems as:Which kinds of structures are more likely to appear?What is most common ? Periodic or aperiodic?
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
Wang TilingsMotivation
What we have, what we expect
ResultsHow does the tiling space behave inside the mappingspaceWhat both topologies can say on tilingsTopological games to study meagerness
ExpectationsGames to solve problems as:Which kinds of structures are more likely to appear?What is most common ? Periodic or aperiodic?
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
Wang TilingsMotivation
What we have, what we expect
ResultsHow does the tiling space behave inside the mappingspaceWhat both topologies can say on tilingsTopological games to study meagerness
ExpectationsGames to solve problems as:Which kinds of structures are more likely to appear?What is most common ? Periodic or aperiodic?
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
Wang TilingsMotivation
What we have, what we expect
ResultsHow does the tiling space behave inside the mappingspaceWhat both topologies can say on tilingsTopological games to study meagerness
ExpectationsGames to solve problems as:Which kinds of structures are more likely to appear?What is most common ? Periodic or aperiodic?
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
Wang TilingsMotivation
What we have, what we expect
ResultsHow does the tiling space behave inside the mappingspaceWhat both topologies can say on tilingsTopological games to study meagerness
ExpectationsGames to solve problems as:Which kinds of structures are more likely to appear?What is most common ? Periodic or aperiodic?
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
Wang TilingsMotivation
What we have, what we expect
ResultsHow does the tiling space behave inside the mappingspaceWhat both topologies can say on tilingsTopological games to study meagerness
ExpectationsGames to solve problems as:Which kinds of structures are more likely to appear?What is most common ? Periodic or aperiodic?
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
DefinitionThe topological space
Outline
1 IntroductionWang TilingsMotivation
2 Besicovitch topologyDefinitionThe topological space
3 Cantor topologyDefinitionThe topological space
4 Topological gamesIntroductionGames on tilings
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
DefinitionThe topological space
Definition
Traditional topology on cellular automataLimsup of the different bits of two configurationsHow to compare tile sets?
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
DefinitionThe topological space
Definition
Traditional topology on cellular automataLimsup of the different bits of two configurationsHow to compare tile sets?
0
1 111 100 0 0
1 11 0 1 00 1
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
DefinitionThe topological space
Definition
Traditional topology on cellular automataLimsup of the different bits of two configurationsHow to compare tile sets?
0
1 111 100 0 0
1 11 0 1 00 1
S = 0
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
DefinitionThe topological space
Definition
Traditional topology on cellular automataLimsup of the different bits of two configurationsHow to compare tile sets?
0
1 111 100 0 0
1 11 0 1 00 1
S = 0 , 0
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
DefinitionThe topological space
Definition
Traditional topology on cellular automataLimsup of the different bits of two configurationsHow to compare tile sets?
0
1 111 100 0 0
1 11 0 1 00 1
S = 0 , 0 , 1 / 5
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
DefinitionThe topological space
Definition
Traditional topology on cellular automataLimsup of the different bits of two configurationsHow to compare tile sets?
0
1 111 100 0 0
1 11 0 1 00 1
S = 0 , 0 , 1 / 5 , 2 / 7 , 2 / 9 , . . .
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
DefinitionThe topological space
Definition
Traditional topology on cellular automataLimsup of the different bits of two configurationsHow to compare tile sets?
0
1 111 100 0 0
1 11 0 1 00 1
S = 0 , 0 , 1 / 5 , 2 / 7 , 2 / 9 , . . .
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
DefinitionThe topological space
The metric
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
DefinitionThe topological space
The metric
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
DefinitionThe topological space
The metric
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
DefinitionThe topological space
The metric
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
DefinitionThe topological space
The metric
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
DefinitionThe topological space
The metric
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
DefinitionThe topological space
The metric
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
DefinitionThe topological space
The metric
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
DefinitionThe topological space
The metric
S = 0
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
DefinitionThe topological space
The metric
S = 0 , 2 / 9
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
DefinitionThe topological space
The metric
S = 0 , 2 / 9 , 3 / 2 5
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
DefinitionThe topological space
The metric
S = 0 , 2 / 9 , 3 / 2 5 , 1 0 / 4 9
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
DefinitionThe topological space
The metric
S = 0 , 2 / 9 , 3 / 2 5 , 1 0 / 4 9 , 1 5 / 8 1 . . .
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
DefinitionThe topological space
Outline
1 IntroductionWang TilingsMotivation
2 Besicovitch topologyDefinitionThe topological space
3 Cantor topologyDefinitionThe topological space
4 Topological gamesIntroductionGames on tilings
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
DefinitionThe topological space
Properties
MB: {{1, . . . , s}Z2}s>1 with the Besicovitch metricTB: Tilings with the Besicovitch metric
MotivationUnderstand the behavior of TB in MB
PropertyThe classes of equivalent tilings in TB are still uncountablymany
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
DefinitionThe topological space
Properties
MB: {{1, . . . , s}Z2}s>1 with the Besicovitch metricTB: Tilings with the Besicovitch metric
MotivationUnderstand the behavior of TB in MB
PropertyThe classes of equivalent tilings in TB are still uncountablymany
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
DefinitionThe topological space
Properties
MB: {{1, . . . , s}Z2}s>1 with the Besicovitch metricTB: Tilings with the Besicovitch metric
MotivationUnderstand the behavior of TB in MB
PropertyThe classes of equivalent tilings in TB are still uncountablymany
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
DefinitionThe topological space
Properties
MB: {{1, . . . , s}Z2}s>1 with the Besicovitch metricTB: Tilings with the Besicovitch metric
MotivationUnderstand the behavior of TB in MB
PropertyThe classes of equivalent tilings in TB are still uncountablymany
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
DefinitionThe topological space
Results
Metric spacesThere is a countable infinite set A of Wang tilings such thatany two tilings of A are almost at distance 1There is no countable set of Wang tilings dense in TB
TB is path-connected
Topological spacesTB is meager in MB
TB is a Baire space
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
DefinitionThe topological space
Results
Metric spacesThere is a countable infinite set A of Wang tilings such thatany two tilings of A are almost at distance 1There is no countable set of Wang tilings dense in TB
TB is path-connected
Topological spacesTB is meager in MB
TB is a Baire space
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
DefinitionThe topological space
Results
Metric spacesThere is a countable infinite set A of Wang tilings such thatany two tilings of A are almost at distance 1There is no countable set of Wang tilings dense in TB
TB is path-connected
Topological spacesTB is meager in MB
TB is a Baire space
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
DefinitionThe topological space
Results
Metric spacesThere is a countable infinite set A of Wang tilings such thatany two tilings of A are almost at distance 1There is no countable set of Wang tilings dense in TB
TB is path-connected
Topological spacesTB is meager in MB
TB is a Baire space
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
DefinitionThe topological space
Results
Metric spacesThere is a countable infinite set A of Wang tilings such thatany two tilings of A are almost at distance 1There is no countable set of Wang tilings dense in TB
TB is path-connected
Topological spacesTB is meager in MB
TB is a Baire space
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
DefinitionThe topological space
Outline
1 IntroductionWang TilingsMotivation
2 Besicovitch topologyDefinitionThe topological space
3 Cantor topologyDefinitionThe topological space
4 Topological gamesIntroductionGames on tilings
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
DefinitionThe topological space
Introduction
Another traditional cellular automata metricConsider the biggest common sequence of twoconfigurationsFor tilings, the biggest common pattern
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
DefinitionThe topological space
Introduction
Another traditional cellular automata metricConsider the biggest common sequence of twoconfigurationsFor tilings, the biggest common pattern
0
1 111 100 0 0
1 11 0 1 00 1
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
DefinitionThe topological space
Introduction
Another traditional cellular automata metricConsider the biggest common sequence of twoconfigurationsFor tilings, the biggest common pattern
0
1 111 100 0 0
1 11 0 1 00 1
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
DefinitionThe topological space
Introduction
Another traditional cellular automata metricConsider the biggest common sequence of twoconfigurationsFor tilings, the biggest common pattern
0
1 111 100 0 0
1 11 0 1 00 1
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
DefinitionThe topological space
Introduction
Another traditional cellular automata metricConsider the biggest common sequence of twoconfigurationsFor tilings, the biggest common pattern
0
1 111 100 0 0
1 11 0 1 00 1
nd = 2- n
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
DefinitionThe topological space
Introduction
Another traditional cellular automata metricConsider the biggest common sequence of twoconfigurationsFor tilings, the biggest common pattern
0
1 111 100 0 0
1 11 0 1 00 1
nd = 2- n
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
DefinitionThe topological space
Definition
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
DefinitionThe topological space
Definition
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
DefinitionThe topological space
Definition
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
DefinitionThe topological space
Definition
i = 2
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
DefinitionThe topological space
Definition
i = 3
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
DefinitionThe topological space
Definition
i = 9
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
DefinitionThe topological space
Definition
i = 9
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
DefinitionThe topological space
Outline
1 IntroductionWang TilingsMotivation
2 Besicovitch topologyDefinitionThe topological space
3 Cantor topologyDefinitionThe topological space
4 Topological gamesIntroductionGames on tilings
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
DefinitionThe topological space
Properties
MC : {{1, . . . , s}Z2}s>1 with the Cantor metricTC : Tilings with the Cantor metric
MotivationHave a better understanding of the local structure
PropertyTC is a 0-dimensional space
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
DefinitionThe topological space
Properties
MC : {{1, . . . , s}Z2}s>1 with the Cantor metricTC : Tilings with the Cantor metric
MotivationHave a better understanding of the local structure
PropertyTC is a 0-dimensional space
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
DefinitionThe topological space
Properties
MC : {{1, . . . , s}Z2}s>1 with the Cantor metricTC : Tilings with the Cantor metric
MotivationHave a better understanding of the local structure
PropertyTC is a 0-dimensional space
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
DefinitionThe topological space
Properties
MC : {{1, . . . , s}Z2}s>1 with the Cantor metricTC : Tilings with the Cantor metric
MotivationHave a better understanding of the local structure
PropertyTC is a 0-dimensional space
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
DefinitionThe topological space
Results
Metric spacesThere exists an open ball of MC that does not contain anyWang tilingThere exists a countable set of Wang tilings dense in TC
Topological spacesTC is a Baire space
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
DefinitionThe topological space
Results
Metric spacesThere exists an open ball of MC that does not contain anyWang tilingThere exists a countable set of Wang tilings dense in TC
Topological spacesTC is a Baire space
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
DefinitionThe topological space
Results
Metric spacesThere exists an open ball of MC that does not contain anyWang tilingThere exists a countable set of Wang tilings dense in TC
Topological spacesTC is a Baire space
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
IntroductionGames on tilings
Outline
1 IntroductionWang TilingsMotivation
2 Besicovitch topologyDefinitionThe topological space
3 Cantor topologyDefinitionThe topological space
4 Topological gamesIntroductionGames on tilings
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
IntroductionGames on tilings
Motivation
Banach-MazurGames with perfect information with two playersMakes possible the study of meagerness of subsets of atopological space
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
IntroductionGames on tilings
Motivation
Banach-MazurGames with perfect information with two playersMakes possible the study of meagerness of subsets of atopological space
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
IntroductionGames on tilings
Motivation
Banach-MazurGames with perfect information with two playersMakes possible the study of meagerness of subsets of atopological space
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
IntroductionGames on tilings
Banach-Mazur Games
A topological space X and a subset C of XPlayer I chooses an open set U1 of XPlayer II chooses an open set U2 ⊂ U1, and so on. . .If
⋂Ui ⊂ C, Player II wins
If Player II has a winning strategy, X \ C is meager in X
X
C
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
IntroductionGames on tilings
Banach-Mazur Games
A topological space X and a subset C of XPlayer I chooses an open set U1 of XPlayer II chooses an open set U2 ⊂ U1, and so on. . .If
⋂Ui ⊂ C, Player II wins
If Player II has a winning strategy, X \ C is meager in X
X
C U1
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
IntroductionGames on tilings
Banach-Mazur Games
A topological space X and a subset C of XPlayer I chooses an open set U1 of XPlayer II chooses an open set U2 ⊂ U1, and so on. . .If
⋂Ui ⊂ C, Player II wins
If Player II has a winning strategy, X \ C is meager in X
X
C U2
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
IntroductionGames on tilings
Banach-Mazur Games
A topological space X and a subset C of XPlayer I chooses an open set U1 of XPlayer II chooses an open set U2 ⊂ U1, and so on. . .If
⋂Ui ⊂ C, Player II wins
If Player II has a winning strategy, X \ C is meager in X
X
C
L i m U
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
IntroductionGames on tilings
Banach-Mazur Games
A topological space X and a subset C of XPlayer I chooses an open set U1 of XPlayer II chooses an open set U2 ⊂ U1, and so on. . .If
⋂Ui ⊂ C, Player II wins
If Player II has a winning strategy, X \ C is meager in X
X
C
L i m U
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
IntroductionGames on tilings
Outline
1 IntroductionWang TilingsMotivation
2 Besicovitch topologyDefinitionThe topological space
3 Cantor topologyDefinitionThe topological space
4 Topological gamesIntroductionGames on tilings
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
IntroductionGames on tilings
Games on the Cantor space
An open set X of tilings, and a subset C of XPlayer I chooses a pattern mPlayer II chooses a pattern m′ such that m ∈ m′, and soon. . .Player II wins if the final tiling is in CX \ C is meager if Player II has a winning strategy
TheoremTC is meager in MC
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
IntroductionGames on tilings
Games on the Cantor space
An open set X of tilings, and a subset C of XPlayer I chooses a pattern mPlayer II chooses a pattern m′ such that m ∈ m′, and soon. . .Player II wins if the final tiling is in CX \ C is meager if Player II has a winning strategy
TheoremTC is meager in MC
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
IntroductionGames on tilings
Games on the Cantor space
An open set X of tilings, and a subset C of XPlayer I chooses a pattern mPlayer II chooses a pattern m′ such that m ∈ m′, and soon. . .Player II wins if the final tiling is in CX \ C is meager if Player II has a winning strategy
TheoremTC is meager in MC
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
IntroductionGames on tilings
Games on the Cantor space
An open set X of tilings, and a subset C of XPlayer I chooses a pattern mPlayer II chooses a pattern m′ such that m ∈ m′, and soon. . .Player II wins if the final tiling is in CX \ C is meager if Player II has a winning strategy
TheoremTC is meager in MC
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
IntroductionGames on tilings
Games on the Cantor space
An open set X of tilings, and a subset C of XPlayer I chooses a pattern mPlayer II chooses a pattern m′ such that m ∈ m′, and soon. . .Player II wins if the final tiling is in CX \ C is meager if Player II has a winning strategy
TheoremTC is meager in MC
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
IntroductionGames on tilings
Games on the Cantor space
An open set X of tilings, and a subset C of XPlayer I chooses a pattern mPlayer II chooses a pattern m′ such that m ∈ m′, and soon. . .Player II wins if the final tiling is in CX \ C is meager if Player II has a winning strategy
TheoremTC is meager in MC
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
IntroductionGames on tilings
Games on the Besicovitch space
An open set X of tilings, and a subset C of XPlayer I chooses an open set U1 of XPlayer II chooses an open set U2 such that U2 ⊂ U1, andso on. . .Player II wins if the final tiling is in CX \ C is meager if Player II has a winning strategy
TheoremTB is meager in MB
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
IntroductionGames on tilings
Games on the Besicovitch space
An open set X of tilings, and a subset C of XPlayer I chooses an open set U1 of XPlayer II chooses an open set U2 such that U2 ⊂ U1, andso on. . .Player II wins if the final tiling is in CX \ C is meager if Player II has a winning strategy
TheoremTB is meager in MB
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
IntroductionGames on tilings
Games on the Besicovitch space
An open set X of tilings, and a subset C of XPlayer I chooses an open set U1 of XPlayer II chooses an open set U2 such that U2 ⊂ U1, andso on. . .Player II wins if the final tiling is in CX \ C is meager if Player II has a winning strategy
TheoremTB is meager in MB
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
IntroductionGames on tilings
Games on the Besicovitch space
An open set X of tilings, and a subset C of XPlayer I chooses an open set U1 of XPlayer II chooses an open set U2 such that U2 ⊂ U1, andso on. . .Player II wins if the final tiling is in CX \ C is meager if Player II has a winning strategy
TheoremTB is meager in MB
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
IntroductionGames on tilings
Games on the Besicovitch space
An open set X of tilings, and a subset C of XPlayer I chooses an open set U1 of XPlayer II chooses an open set U2 such that U2 ⊂ U1, andso on. . .Player II wins if the final tiling is in CX \ C is meager if Player II has a winning strategy
TheoremTB is meager in MB
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
IntroductionGames on tilings
Games on the Besicovitch space
An open set X of tilings, and a subset C of XPlayer I chooses an open set U1 of XPlayer II chooses an open set U2 such that U2 ⊂ U1, andso on. . .Player II wins if the final tiling is in CX \ C is meager if Player II has a winning strategy
TheoremTB is meager in MB
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
IntroductionGames on tilings
Further works
Games seems a promising tools for tilingsStudy the meagerness of subsets of tilingsObtain strong results on the probability of a certainstructure to appear
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
IntroductionGames on tilings
Further works
Games seems a promising tools for tilingsStudy the meagerness of subsets of tilingsObtain strong results on the probability of a certainstructure to appear
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
IntroductionGames on tilings
Further works
Games seems a promising tools for tilingsStudy the meagerness of subsets of tilingsObtain strong results on the probability of a certainstructure to appear
Lafitte, Weiss A Topological Study of Tilings
IntroductionBesicovitch topology
Cantor topologyTopological games
IntroductionGames on tilings
Shen’s question
Tilings with errorsFinite holeAperiodic tile set stabilityConsider any aperiodic tile set, can we always remove afinite border of the hole to complete the tiling?
Lafitte, Weiss A Topological Study of Tilings
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