Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.
Concept
Aperiodic tiling
Summary
An aperiodic tiling is a non-periodic tiling with the additional property that it does not contain arbitrarily large periodic regions or 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 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.
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 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 in the local topology. In the local topology (resp. the corresponding metric) two tilings are -close if they agree in a ball of radius around the origin (possibly after shifting one of the tilings by an amount less than ).
To give an even simpler example than above, consider a one-dimensional tiling T of the line that looks like where a represents an interval of length one, b represents an interval of length two.
Official source
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.