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==Enumeration and aperiodicity==
[[File:Binary-tiling-asymmetry.svg|thumb|No symmetry of the binary tiling takes the blue tile (in a middle position relative to the yellow tile two levels higher) to the red tile (in an outer position relative to the same yellow tile)]]
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|radin}}
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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}}
[[File:Binary-tiling-dual.svg|thumb|A binary tiling (red outline) and its dual tiling (yellow curved triangles and blue and green curved quadrilaterals)]]
In binary tilings, more strongly than having all tiles the same shape, all [[Heesch's problem|first coronas]] of tiles have the same shape (possibly after a reflection). 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}}
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