Binary tiling: Difference between revisions

Revision as of 07:54, 27 October 2024

Revision as of 18:48, 27 October 2024

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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. BinaryThe binary tilings provideshow athat, strongin counterexamplethe forhyperbolic theplane, correspondinga propertytiling incan thehave hyperboliccongruent planecoronas without being isohedral.{{r|ds}}

Corresponding to the fact that these tilings are non-periodic but monohedral (having only one tile shape), theThe [[dual tiling]]s of these tilings are non-periodic but ''monocoronal'' (having the same pattern of tiles surrounding each vertex). These dualbinary tilings are formed by choosing a reference point within each tile of a binary tiling, and connecting pairs of reference points of tiles that share an edge with each other. The fact that the binary tilings are non-periodic but monohedral (having only one tile shape) translates an equivalent fact about the dual tilings: they are non-periodic but ''monocoronal'', having the same pattern of tiles surrounding each vertex.{{r|fg}}

==Applications==