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==Tiles==
A [[Tessellation|tiling]] of a [[surface]] is a covering of the surface by [[geometric shape]]s, called tiles, with no overlaps and no gaps. An example is the familiar tiling of the [[Euclidean plane]] by [[square]]s, meeting edge-to-edge. When all the tiles have the same shape (they are all [[Congruence (geometry)|congruent]]), the tiling is called a [[monohedral tiling]], and the shape of the tiles is called the [[prototile]] of the tiling.{{r|adams}} The binary tilings are monohedral tilings of the [[hyperbolic plane]], a kind of [[non-Euclidean geometry]] with different notions of length, area, congruence, and symmetry than the Euclidean plane.{{r|radin}}
In one version of the tiling, each tile is a subset of the hyperbolic plane that lies between two hyperbolic lines and two [[horocycle]]s that are all asymptotic to the same [[ideal point]], with the horocycles at distance <math>\log 2</math> from each other. The resulting shape has four right angles, like a rectangle, with its sides alternating between segments of hyperbolic lines and arcs of horocycles. The choice of <math>\log 2</math> as the distance between the two horocycles causes one of the two arcs of horocycles (the one farther from the asymptotic point) to be twice as long as the other. These tiles may be packed along their line segment sides to fill out the annular region between the two horocycles, and to pack a nested family of congruent annuli between equally spaced horocycles on either side of them. When these annular packings line up so that each half of the outer horocyclic arc of a tile in one annulus matches up with the inner horocyclic arc of a tile in the next annulus, the result is a binary tiling.{{r|df}}
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== References ==
{{reflist|refs=
<ref name=adams>{{cite book|title=The Tiling Book: An Introduction to the Mathematical Theory of Tilings|first=Colin|last=Adams|publisher=American Mathematical Society|year=2022|isbn=9781470468972|pages=[https://books.google.com/books?id=LvGGEAAAQBAJ&pg=PA21 21–23]}}</ref>
<ref name=agol>{{cite web|first=Ian|last=Agol|authorlink=Ian Agol|title=Smallest tile to tessellate the hyperbolic plane|url=https://mathoverflow.net/q/291453|work=[[MathOverflow]]|date=January 26, 2018}}</ref>
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