Binary tiling: Difference between revisions

Revision as of 07:23, 27 October 2024

Revision as of 07:43, 27 October 2024

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==Enumeration and aperiodicity==

In the square tiling of the Euclidean plane, every two tiles are positioned in the same way: there is a symmetry of the whole tiling (a [[translation (geometry)|translation]]) that takes one tile to the other. But a binary tiling does not have symmetries that take every tile to every other tile. For instance, for the four tiles two levels below any given tile, no symmetry takes a middle tile to an outer tile. Further, there is only one way of tiling the Euclidean plane by square tiles that meet edge-to-edge, but there are uncountably many edge-to-edge binary tilings. The prototile of the binary tiling can be modified to force the tiling to be edge-to-edge, by adding small protrusions to some sides and matching indentations to others.{{r|df}}

The tiles of a binary tiling are not all symmetric to each other; for instance, for the four tiles two levels below any given tile, no symmetry takes a middle tile to an outer tile. There are uncountably many different tilings of the hyperbolic plane by these tiles, even when they are modified by adding protrusions and indentations to force them to meet edge-to-edge. None of these different tilings are periodic (having a [[Cocompact group action|cocompact]] symmetry group),{{r|radin}} although some (such as the one in which there exists a line that is completely covered by tile edges) have a one-dimensional infinite symmetry group.{{r|df}} As a tile all of whose tilings are not fully periodic, the [[prototile]] of a binary tiling solves an analogue of the {{not a typo|<!-- lowercase is intentional -->[[einstein problem]]}} in the hyperbolic plane. However, it is only ''weakly aperiodic'', meaning that no tiling has a two-dimensional group of symmetries. Because they can have one-dimensional symmetries, the binary tilings are not ''strongly aperiodic''.{{r|einstein}}

Some binary tilings have a one-dimensional infinite symmetry group. For instance, when a binary tiling is viewed in the half-plane model, it may be possible to [[Scaling (geometry)|scale]] the model by any [[power of two]] without changing the tiling; when this is possible, the tiling has infinitely many symmetries, one for each power of two.{{r|df}} However, no binary tiling has a two-dimensional symmetry group. This can be expressed mathematically by saying that it is not possible to find a finite set of tiles such that all tiles can be mapped to the finite set by a symmetry of the tiling. More technically, no binary tiling has a [[Cocompact group action|cocompact]] symmetry group.{{r|radin}}

As a tile all of whose tilings are not fully periodic, the [[prototile]] of a binary tiling solves an analogue of the {{not a typo|<!-- lowercase is intentional -->[[einstein problem]]}} in the hyperbolic plane. This problem asks for a single prototile that tiles only aperiodically; long after the discovery of the binary tilings, it was solved in the Euclidean plane by the discovery of the "hat" and "spectre" tilings. However, the binary tilings are only ''weakly aperiodic'', meaning that no tiling has a two-dimensional group of symmetries. Because they can have one-dimensional symmetries, the binary tilings are not ''strongly aperiodic''.{{r|einstein}}

In binary tilings, more strongly than having all tiles the same shape, all [[Heesch's problem|first coronas]] of tiles have the same shape. The first corona is the set of tiles touching a single central tile. Here, coronas are considered the same if they are reflections of each other. For tilings of the Euclidean plane, having all first coronas the same implies that the tiling is periodic and [[isohedral tiling|isohedral]], meaning that all tiles are symmetric to each other. Binary tilings provide a strong counterexample for the corresponding property in the hyperbolic plane.{{r|ds}}