Aperiodic tiling

An aperiodic tiling is a non-periodic tiling with the additional property that it does not contain arbitrarily large periodic patches. A set of tile-types (or prototiles) is aperiodic if copies of these tiles can form only non-periodic tilings. The Penrose tilings are the best-known examples of aperiodic tilings.

Aperiodic tilings serve as mathematical models for quasicrystals, physical solids that were discovered in 1982 by Dan Shechtman who subsequently won the Nobel prize in 2011. However, the specific local structure of these materials is still poorly understood.

Several methods for constructing aperiodic tilings are known.

Consider a periodic tiling by unit squares (it looks like infinite graph paper). Now cut one square into two rectangles. The tiling obtained in this way is non-periodic: there is no non-zero shift that leaves this tiling fixed. But clearly this example is much less interesting than the Penrose tiling. In order to rule out such boring examples, one defines an aperiodic tiling to be one that does not contain arbitrary large periodic parts.

A tiling is called aperiodic if its hull contains only non-periodic tilings. The hull of a tiling ${\displaystyle T\in \mathbb {R} ^{d}}$ contains all translates T+x of T, together with all tilings that can be approximated by translates of T. Formally this is the closure of the set ${\displaystyle \{T+x\,:\,x\in \mathbb {R} ^{d}\}}$ in the local topology. In the local topology (resp. the corresponding metric) two tiles are ${\displaystyle \varepsilon }$-close if they agree in a ball of radius ${\displaystyle 1/\varepsilon }$ around the origin (possibly after shifting one of the tilings by an amount less than ${\displaystyle \varepsilon }$).

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