What Is... Penrose’s Tiles?

A tiling of a subset A of R is a countable set T of tiles in R such that: (1) ∪T = A, (2) whenever S, T belong to T and are distinct we have intS ∩ intT = ∅. Part of a pentagonal tiling is shown in Figure 2. A patch is a tiling by finitely many tiles of a connected, simply connected subset of R which cannot be disconnected by removing one point. To say that a patch A belongs to a tiling T means that A is a subset of T . Given a tiling T we define an equivalence relation ∼ on T by isometries. A set of representatives for ∼ will be called a set of protiles for T . If a set P of tiles is a set of protiles for some tiling T we say that P admits T . Let T be a tiling of R. An isometry σ of R is called a symmetry of T if it maps every tile of T onto a tile of T .