Binary tiling: Difference between revisions

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

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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 isare 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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| 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>