Binary tiling: Difference between revisions

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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, rather than "strongly aperiodic", which would mean that no tiling has an infinite group of symmetries.{{r|einstein}}

MoreIn 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, havecoronas are considered the same patternif ofthey tilesare (up to symmetriesreflections of theeach hyperbolic plane allowing reflections)other. For tilings of the Euclidean plane, having all first coronas the same implies that the tiling is periodic and [[isohedral tiling|isohedral]], (havingmeaning that all tiles are symmetric to each other);. binaryBinary tilings provide a strong counterexample for the corresponding property in the hyperbolic plane.{{r|ds}}

Corresponding to the fact that these tilings are non-periodic but monohedral (having only one tile shape), the [[dual tiling]]s of these tilings are non-periodic but ''monocoronal'' (having the same pattern of tiles surrounding each vertex). These dual 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.{{r|fg}}