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{{Short description|Tiling of the hyperbolic plane}}

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[[File:Hyperbolic binary tiling.png|upright=1.2|alt=Binary tiling on Poincare disk|thumb|A binary tiling displayed in the [[Poincaré disk model]] of the [[hyperbolic plane]]. Each sidetile of a tileedge lies on a [[horocycle]] (shown as circles interior to the modeldisk) or a hyperbolic line (shown as arcs of circles perpendicular to the modeldisk boundary). TheseThe horocycles and lines are all asymptotic to a commonan [[ideal point]] located at the right side of the Poincaré disk.]]

In [[geometry]], a '''binary tiling''' (sometimes called a '''Böröczky tiling'''){{r|df}} is a [[tiling of the hyperbolic plane]], resembling a [[quadtree]] over the [[Poincaré half-plane model]] of the hyperbolic plane. The tiles are congruent, each adjoining five others. They may be convex [[pentagon]]s, or non-convex shapes with four sides, alternatingly line segments and [[horocycle|horocyclic]] arcs, meeting at four right angles.

There are uncountably many distinct binary tilings for a given shape of tile. They are all weakly [[aperiodic tiling|aperiodic]], meaningwhich means that they can have a one-dimensional [[symmetry group]] but not a two-dimensional family of symmetries. There exist binary tilings with tiles of arbitrarily small area.

Binary tilings were first studied mathematically in 1974 by {{ill|Károly Böröczky|hu|Böröczky Károly (matematikus, 1964)}}.{{r|radin|agol|bor}} Closely related tilings have been used since the late 1930s in the [[Smith chart]] for radio engineering, and appear in a 1957 print by [[M. C. Escher]].{{r|escher}}

==Tiles==

[[File:1rm60rt0t rtbcwh.jpg|thumb|upright|Square tiles in a bathroom]]

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 closer to 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 inner horocyclic arc of a tile in one annulus matches up with the outer horocyclic arc of a tile in the next annulus, the result is a binary tiling.{{r|df}}

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}}

In one version of thebinary tiling, each tile is a subsetshape ofbounded the hyperbolic plane that lies betweenby two hyperbolic lines and two [[horocycle]]shorocycles. thatThese arefour allcurves should be asymptotic to the same [[ideal point]], with the two horocycles at hyperbolic distance <math>\log 2</math> from each other. TheWith resultingthese shapechoices, the tile 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 closerfarther tofrom the asymptotic point) to behave twice asthe longhyperbolic aslength of the otheropposite arc. TheseCopies tilesof maythis beshape, packedmeeting edge-to-edge along their line segment sides, tocan fill outtile the annularslab or crescent shaped region between the two horocycles,. and to pack aA nested family of congruentthese annulislabs betweenor equallycrescents spacedcan horocyclesthen ontile eitherthe sideentire ofhyperbolic them.plane, When these annular packings linelined up so that each half of the inner horocycliclong arc of aeach tile in one annulusslab matchesis upcovered withby the outershort horocyclic arcarcs of atwo tiletiles in the next annulus,slab. theThe result is a binary tiling.{{r|df}}

[[File:Binary Tiling.png|thumb|A portion of a binary tiling displayed in the [[Poincaré half-plane model]]. The horizontal lines correspond to horocycles in the hyperbolic plane, and the vertical line segments correspond to hyperbolic lines.]]

In the [[Poincaré half-plane model]] of hyperbolic geometry, witheach thetile idealcan pointbe chosenmodeled toas bean aaxis-parallel [[pointsquare ator infinity]]rectangle.{{r|radin|fg}} forIn thethis half-planemodel, hyperbolicrays linesthrough asymptoticthe tovertical thissides pointof area modeledtile asmodel verticalhyperbolic rayslines, and horocycles asymptotic to thisthe point areat modeledinfinity, as horizontaland lines.{{r|rr}} Thisthrough givesthe eachhorizontal tilesides theof overalla shape in thetile model ofhorocycles, anasymptotic axis-parallelto squarethe orsame rectanglepoint.{{r|radin|fgrr}} ForThe thishyperbolic model,length theof hyperbolican distancearc betweenof pointsa withhorizontal the same <math>y</math>-coordinatehorocycle is theirits Euclidean distancelength divided by its <math>y</math>-coordinate, while the hyperbolic distance between points with the same <math>x</math>-coordinate is the [[logarithm]] of the ratio of their <math>y</math>-coordinates.{{r|stahl}} From these facts one can calculate that successive horocycles of a binary tiling, at hyperbolic distance <math>\ln 2</math>, are modeled by horizontal lines whose Euclidean distance from the <math>x</math>-axis doubles at each step, and that the two bottom half-arcs of a binary tile each equal the top arc.

[[File:Binary tiling straight.svg|thumb|Binary tiling with [[pentagonal tiling|convex pentagon tiles]], in the Poincaré half-plane model.]]

An alternative and combinatorially equivalent version of the tiling places its vertices at the same points, but connects them by hyperbolic line segments instead of horocyclicarcs segmentsof horocycles, so that each tile becomes a hyperbolic convex pentagon. This makes the tiling a proper [[pentagonal tiling]].{{r|fg|kari}} The hyperbolic lines through the non-vertical sides of these tiles are modeled in the half-plane model by semicircles centered on the <math>x</math>-axis, and the sides form arcs of these semicircles.{{r|stahl}}

If one considers only adjacencies between tiles of different sizes, omitting the side-to-side adjacencies, this adjacency pattern gives the tiles of a binary tiling the structure of a [[binary tree]]. Representative points within each tile, connected according to this adjacency structure, give an embedding of an infinite binary tree as a [[hyperbolic tree]].{{r|kbvw}}

{{-}}

==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 above it) to the red tile (in an outer position).]]

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.{{r|einstein}}

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}}

Some binary tilings have a one-dimensional infinite symmetry group. For instance, when a binary tiling is viewed in the half-plane model, it may be possible to [[Scaling (geometry)|scale]] the model by any [[power of two]] without changing the tiling. When this is possible, the tiling has infinitely many symmetries, one for each power of two.{{r|df}} However, no binary tiling has a two-dimensional symmetry group. This can be expressed mathematically by saying that it is not possible to find a finite set of tiles such that all tiles can be mapped to the finite set by a symmetry of the tiling. More technically, no binary tiling has a [[Cocompact group action|cocompact]] symmetry group.{{r|radin}}

More strongly than having all tiles the same shape, all [[Heesch's problem|first coronas]] of tiles, the set of tiles touching a single central tile, have the same pattern of tiles (up to symmetries of the hyperbolic plane allowing reflections). For tilings of the Euclidean plane, having all first coronas the same implies that the tiling is periodic and [[isohedral tiling|isohedral]] (having all tiles symmetric to each other); binary tilings provide a strong counterexample for the corresponding property in the hyperbolic plane.{{r|ds}}

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}}

Corresponding to the fact that these tilings are non-periodic but monohedral (having only one tile shape), the [[dual tiling]]s of these tilings are non-periodic but ''monocoronal'' (having the same pattern of tiles surrounding each vertex). These dual 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.{{r|fg}}

MoreIn 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, havecoronas are considered the same patternif ofthey tilesare (up to symmetriesreflections of theeach hyperbolic plane allowing reflections)other. For tilings of the Euclidean plane, having all first coronas the same implies that the tiling is periodic and [[isohedral tiling|isohedral]], (havingmeaning that all tiles are symmetric to each other);. The binary tilings provideshow athat, strongin counterexamplethe forhyperbolic theplane, correspondinga propertytiling incan thehave hyperboliccongruent planecoronas without being isohedral.{{r|ds}}

[[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 are 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==

[[File:Binary tiling density.svg|thumb|upright=1.1|Is the average number of red points per tile 1/3 (left) or 2/3 (right)?]]

TheBinary originaltilings applicationwere forfirst whichstudied mathematically in 1974 by {{ill|Károly Böröczky|hu|Böröczky studiesKároly these(matematikus, tilings1964)}}.{{r|radin|agol|bor}} Böröczky was concernsinvestigating 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}}

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}}

==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>

}}

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 trianglestriangle]]s.{{r|escher}} It is one of several Escher prints based on the half-plane model of the hyperbolic plane.{{r|dunham}} When interpreted as Euclidean shapes rather than hyperbolically, the tiles are squares and the subdivided triangles are isosceles right triangles. The print itself replaces each triangle by a stylized lizard.{{r|escher}}

The [[Smith chart]], froma 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}}

The [[Cayley graph]] of the [[Baumslag–Solitar group]] <math>BS(1,2)</math>, canhas bethe decomposedgroup intoelements "sheets"as vertices, planarconnected structuresby withedges arepresenting geometrymultiplication [[Quasi-isometry|quasi-isometric]]by tothis thegroup's standard hyperbolicgenerating planeelements. The CayleyThis graph iscan embeddedbe ontodecomposed each sheetinto as the graph"sheets", ofwhose vertices and edges ofform a binary tiling. At each level of a binary tiling, there are two choices for how to continue the tiling at the next higher level. Any two sheets will coincide for some number of levels until separating from each other by following different choices at one of these levels, giving the sheets the structure of an infinite binary tree.{{r|cfm|as}}

[[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}}]]

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}}

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 subset of a binary tiling that lies inside a horocycle. (For horizontal horocycles in the Poincaré half-plane, the inside is above the horocycle.) These mixed binary and apeirogonal tilings avoid the density paradoxes of the binary tiling.{{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}}

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}}

Omitting the side-to-side connections gives an embedding of an infinite binary tree as a [[hyperbolic tree]].{{r|kbvw}}

{{-}}

==See also==

*[[Tetrapentagonal tiling]], a periodic tiling of the hyperbolic plane by pentagons

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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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| last2 = Schraudner | first2 = Michael

| arxiv = 2012.11037

| journal = Comptes Rendus. Mathématique Acad. Sci. Paris

| mr = 4753921

| pages = 553–580

| title = Tilings of the hyperbolic plane of substitutive origin as subshifts of finite type on Baumslag-SolitarBaumslag–Solitar groups <math>BS(1,2n)</math>

| 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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| ___location = Boston

| mr = 1217085

| pages = 64–6664–67

| publisher = Jones and Bartlett Publishers

| title = The Poincaré Half-Plane: A Gateway to Modern Geometry