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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. The binary tilings show that, in the hyperbolic plane, a tiling can have congruent coronas without being isohedral.{{r|ds}}
The [[dual tiling]]s of the binary 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. They is not monohedral: the binary tilings have vertices where three or four tiles meet, and correspondingly the dual tilings have some tiles that are triangles and some tiles that are quadrilaterals. 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==
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