Math 2.0

Mathematical logic and foundations Theorem

Wang tile undecidability

Builds on Decidable languageTuring machine

Statement

A Wang tile is a unit square whose four edges carry colours from a finite set, and tiles may not be rotated or reflected. A finite set WW of Wang tiles tiles the plane if there is a map τ ⁣:ℤ2→W\tau \colon \Z^2 \to W with two properties for all (i,j)∈ℤ2(i, j) \in \Z^2. The right edge of τ(i,j)\tau(i, j) has the colour of the left edge of τ(i+1,j)\tau(i + 1, j). The top edge of τ(i,j)\tau(i, j) has the colour of the bottom edge of τ(i,j+1)\tau(i, j + 1). Then the domino problem is undecidable: the language of codes of finite sets WW of Wang tiles that tile the plane is not decidable.

Remarks

The proof encodes the run of a Turing machine on blank tape into tilings, so that a tile set tiles the plane exactly when the machine never halts. Then a decision procedure for tilings would solve the halting problem. The hard part is to force the computation to appear in every tiling, and this is done with an aperiodic tile set that builds a hierarchy of nested squares. Wang had observed that, if every tile set that tiles the plane also had a periodic tiling, the problem would be decidable. Berger proved the theorem in 1966, and thereby showed that aperiodic sets of Wang tiles exist. By a compactness argument, WW tiles the plane if and only if it tiles every finite square {0,…,m}2\set{0, \dots, m}^2. So the set of tile sets that do not tile the plane is recursively enumerable, and the set of tile sets that do tile the plane is not. Example: a single tile with all four edges of one colour tiles the plane, while a single tile whose left and right edges have different colours does not.

Going deeper

For a student

The theorem is a standard example of undecidability outside logic, in geometry and combinatorics. It is proved by a many-one reduction from the complement of the halting problem for machines started on a blank tape. It implies that tiling sets of many other kinds, and the emptiness of two-dimensional subshifts of finite type, are undecidable. By contrast, the analogous problem in one dimension is decidable, since it reduces to finding a cycle in a finite graph.

Reading path

To understand this page from the ground up, read these 29 pages in order. The foundations come first.

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Element, definition, no evidence levelElementdefinitionFunction, concept, no evidence levelFunctionconceptMonoid, concept, no evidence levelMonoidconceptProposition, concept, no evidence levelPropositionconceptSet, concept, no evidence levelSetconceptSet, definition, no evidence levelSetdefinitionTuring machine, definition, no evidence levelTuring machinedefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionSubset, definition, no evidence levelSubsetdefinitionUniqueness quantifier, definition, no evidence levelUniqueness quantifierdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionFunction, definition, no evidence levelFunctiondefinitionCodomain, definition, no evidence levelCodomaindefinitionInjective function, definition, no evidence levelInjective functiondefinitionFree monoid, definition, no evidence levelFree monoiddefinitionSurjective function, definition, no evidence levelSurjective functiondefinitionBijective function, definition, no evidence levelBijective functiondefinitionEquinumerous sets, definition, no evidence levelEquinumerous setsdefinitionFinite set, definition, no evidence levelFinite setdefinitionAlphabet, definition, no evidence levelAlphabetdefinitionString, definition, no evidence levelStringdefinitionFormal language, definition, no evidence levelFormal languagedefinitionDecidable language, definition, no evidence levelDecidable languagedefinitionWang tile undecidability, statement, no evidence levelWang tile undecidabil…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Element definition
  2. Function concept
  3. Monoid concept
  4. Proposition concept
  5. Set concept
  6. Set definition
  7. Turing machine definition
  8. Implication definition
  9. Predicate definition
  10. Ordered pair definition
  11. Existential quantifier definition
  12. Universal quantifier definition
  13. Cartesian product definition
  14. Subset definition
  15. Uniqueness quantifier definition
  16. Binary relation definition
  17. Function definition
  18. Codomain definition
  19. Injective function definition
  20. Free monoid definition
  21. Surjective function definition
  22. Bijective function definition
  23. Equinumerous sets definition
  24. Finite set definition
  25. Alphabet definition
  26. String definition
  27. Formal language definition
  28. Decidable language definition
  29. Wang tile undecidability statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Wang_tile
About this page
TagAJH5HPD2B7
URIm2://AJH5HPD2B7
Kindstatement, theorem
WritingW0 stub
MSC03D35, 05B45, 52C20
StatusThis copy of the site has no build data. Run m2 status AJH5HPD2B7.
AboutWang tile
UsesDecidable language, Turing machine
Changed3 November 2026

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