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{{Short description|Tiling of the hyperbolic plane}}
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{{Use dmy dates|cs1-dates=ly|date=September 2024}}
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==Tiles==
[[File:1rm60rt0t rtbcwh.jpg|thumb|upright|Square tiles in a bathroom]]
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,{{r|adams}} as seen for instance in many bathrooms.{{sfnp|Adams|2022|p=232}} When all the tiles have the same shape and size (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}}
Two common models for the hyperbolic plane are the [[Poincaré disk model]] and [[Poincaré half-plane model]]. In these, the points of the hyperbolic plane are modeled by points in the Euclidean plane, in an open disk or the half-plane above a horizontal line respectively. Hyperbolic lines are modeled by those Euclidean circles and lines that cross the model's boundary [[perpendicular]]ly. The boundary points of the model are called [[ideal point]]s, and a hyperbolic line through an ideal point is said to be ''asymptotic'' to it. The half-plane model has one more ideal point, the [[point at infinity]], asymptotic to all vertical lines. Another important class of hyperbolic curves, called [[horocycle]]s, are modeled as circles that are tangent to the boundary of the model, or as horizontal lines in the half-plane model. Horocycles are asymptotic to their point of tangency, or to the point at infinity if they have none.{{r|rr|stahl}}
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[[File:Binary-tiling-dual.svg|thumb|A binary tiling (red outline) and its dual tiling (yellow curved triangles and blue and green curved quadrilaterals)]]
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
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Binary tilings were first studied mathematically in 1974 by {{ill|Károly Böröczky|hu|Böröczky Károly (matematikus, 1964)}}.{{r|radin|agol|bor}} Böröczky was investigating the density of a discrete planar point set, the average number of points per unit area. This quantity is used, for instance, to study [[Danzer set]]s. For points placed one per tile in a [[monohedral tiling]] of the Euclidean plane, the density is inverse to the tile area. But for the hyperbolic plane, paradoxical issues ensue.{{r|radin|bor}} The tiles of a binary tiling can be grouped into three-tile subunits, with each subunit consisting of one tile above two more (as viewed in the Poincaré half-plane model). Points centered within the upper tile of each subunit have one point per subunit, for an apparent density equal to one third of the area of a binary tile. However, the same points and the same binary tiling can be regrouped in a different way, with two points per subunit, centered in the two lower tiles of each subunit, with two times the apparent density. This example shows that it is not possible to determine the density of a hyperbolic point set from tilings in this way.{{r|bor|bowen}}
Another application involves the area of tiles in a monohedral tiling.
Adjusting the distance between the two vertical sides of the tiles in a binary tiling causes their area to vary, proportional to this distance. By making this distance arbitrarily small, this tiling can be used to show that the hyperbolic plane has tilings by congruent tiles of arbitrarily small area.{{r|agol}} [[Jarkko Kari]] has used a system of colorings of tiles from a binary tiling, analogous to [[Wang tile]]s, to prove that determining whether a given system of hyperbolic [[prototile]]s can tile the hyperbolic plane is an [[undecidable problem]].{{r|kari}} Subdivisions of a binary tiling that replace each tile by a [[grid graph]] have been used to obtain tight bounds on the [[Fine-grained reduction|fine-grained complexity]] of [[graph algorithm]]s.{{r|kmvww}} Recursive [[data structure]]s resembling quadtrees, based on binary tiling, have been used for approximate [[nearest neighbor problem|nearest neighbor queries]] in the hyperbolic plane.{{r|kbvw}}▼
In defining the binary tilings, there is a free parameter, the distance between the vertical sides of the tiles. All tiles in a single tiling have equal hyperbolic areas, but if this distance changes, the (equal) area of all the tiles will also change, in proportion. Choosing this distance to be arbitrarily small shows that the hyperbolic plane has tilings by congruent tiles of arbitrarily small area.{{r|agol}}
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==Related patterns==
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|image3=Baumslag-Solitar Cayley 3D.svg|caption3=Four sheets from the [[Cayley graph]] of the [[Baumslag–Solitar group]] <math>BS(1,2)</math>
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A 1957 print by [[M. C. Escher]], ''Regular Division of the Plane VI'', has this tiling as its underlying structure, with each square tile of a binary tiling (as seen in its quadtree form) subdivided into three [[isosceles right
The [[Smith chart]], a graphical method of visualizing parameters in [[radio engineering]], resembles a binary tiling on the [[Poincaré disk model]] of the hyperbolic plane, and has been analyzed for its connections to hyperbolic geometry and to Escher's hyperbolic tilings.{{r|gupta}} It was first developed in the late 1930s by Tōsaku Mizuhashi,{{r|mizu}} [[Phillip Hagar Smith]],{{r|smith}} and Amiel R. Volpert.{{r|volpert}}
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[[File:H2-I-3-dual.svg|thumb|Each face in this [[order-3 apeirogonal tiling]] (shown in the Poincaré disk model) can be replaced by part of a binary tiling as modified by Radin.{{r|radin}}]]
A related tiling of the hyperbolic plane by [[Roger Penrose]] can be interpreted as being formed by adjacent pairs of binary tiles, one above the other, whose unions form L-shaped tiles. Like binary tiling, it is weakly aperiodic.{{r|penrose}} [[Charles Radin]] describes another modification to the binary tiling in which an angular bump is added to the two lower sides of each tile, with a matching indentation cut from the upper side of each tile. These modified tiles can form the usual binary tilings, but they can also be used to form different tilings that replace each face of an [[apeirogonal tiling]] by
The [[dual graph]] of a binary tiling has a vertex for each tile, and an edge for each pair of tiles that share an edge. It takes the form of an infinite [[binary tree]] (extending infinitely both upwards and downwards, without a root) with added side-to-side connections between tree nodes at the same level as each other.{{r|df}} An analogous structure for finite [[complete binary tree]]s, with the side-to-side connections at each level extended from paths to cycles, has been studied as a [[network topology]] in [[parallel computing]], the ''ringed tree''.{{r|xtree}} Ringed trees have also been studied in terms of their [[Hyperbolic metric space|hyperbolic metric properties]] in connection with [[small-world network]]s.{{r|cfhm}}
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| last2 = Schraudner | first2 = Michael
| arxiv = 2012.11037
| journal = Comptes Rendus. Mathématique
| mr = 4753921
| pages = 553–580
| title = Tilings of the hyperbolic plane of substitutive origin as subshifts of finite type on
| volume = 362
| year = 2024| doi = 10.5802/crmath.571
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| volume = 293
| year = 2024| doi-access = free
| isbn = 978-3-95977-316-4
}}</ref>
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