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→Enumeration and aperiodicity: I think the idea about protrusions and indentations is better expressed by Radin |
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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.{{r|df}} 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|
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}}
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