Binary tiling: Difference between revisions

Revision as of 07:14, 27 October 2024

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==Enumeration and aperiodicity==

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,. ratherBecause thanthey "stronglycan aperiodic",have whichone-dimensional wouldsymmetries, meanthe thatbinary notilings tilingare hasnot an''strongly infinite group of symmetriesaperiodic''.{{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}}