Aperiodic set of prototiles

Definition

Let $n \ge 1$ and let $S$ be a finite set of subsets of $\R^n$ , called prototiles, possibly together with matching rules on how they may meet. A tiling of $\R^n$ by $S$ is a tiling in which every tile is congruent to a member of $S$ and the matching rules hold. The set $S$ is aperiodic if:

there is a tiling of $\R^n$ by $S$ ;
no tiling of $\R^n$ by $S$ is periodic.

Remarks

Example: the two rhombi of the Penrose tiling, with their matching rules, form an aperiodic set of prototiles in $\R^2$ .

Non-example: the set $S = \set{[0,1]^2}$ consisting of the unit square is not aperiodic. Clause 1 holds, but the standard grid of unit squares is a periodic tiling by $S$ , so clause 2 fails.

A tiling with no nonzero period is called non-periodic. A set of prototiles is aperiodic exactly when it admits tilings and all of them are non-periodic. The first aperiodic set, found by Robert Berger in 1966, consisted of a very large number of square tiles with coloured edges.

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Tag	`KRX3JWDV1B`
URI	`m2://KRX3JWDV1B`
Kind	definition
Writing	W0 stub
MSC	52C23, 52C20, 05B45
Defines	Aperiodic set of prototiles
Uses	Tessellation, Periodic tiling

Aperiodic set of prototiles

Definition

Remarks

Reading path

Further reading

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