In the mathematics of tessellations, a non-periodic tiling is a tiling that does not have any translational symmetry. An aperiodic set of prototiles is a set of tile-types that can tile, but only non-periodically. The tilings produced by one of these sets of prototiles may be called aperiodic tilings.

The Penrose tilings are a well-known example of aperiodic tilings.

In March 2023, four researchers, David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss, announced the proof that the tile discovered by David Smith is an aperiodic monotile, i.e., a solution to the einstein problem, a problem that seeks the existence of any single shape aperiodic tile. In May 2023 the same authors published a chiral aperiodic monotile with similar but stronger constraints.

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 arbitrarily large periodic parts.

A tiling is called aperiodic if its hull contains only non-periodic tilings. The hull of a tiling ${\displaystyle T\subset \mathbb {R} ^{d}}$ 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 tilings 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 ${\displaystyle \varepsilon }$).

To give an even simpler example than above, consider a one-dimensional tiling T of the line that looks like ...aaaaaabaaaaa... where a represents an interval of length one, b represents an interval of length two. Thus the tiling T consists of infinitely many copies of a and one copy of b (with centre 0, say). Now all translates of T are the tilings with one b somewhere and as else. The sequence of tilings where b is centred at ${\displaystyle 1,2,4,\ldots ,2^{n},\ldots }$ converges – in the local topology – to the periodic tiling consisting of as only. Thus T is not an aperiodic tiling, since its hull contains the periodic tiling ...aaaaaa....