Binary tiling: Difference between revisions

Revision as of 04:13, 28 October 2024

Latest revision as of 02:18, 13 June 2025

(5 intermediate revisions by 4 users not shown)

Line 1:

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

{{good article}}

{{Use dmy dates|cs1-dates=ly|date=September 2024}}

{{Use list-defined references|date=September 2024}}

Line 13 ⟶ 14:

==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}}

Line 37 ⟶ 38:

[[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}}

{{-}}

Line 45 ⟶ 46:

Another application involves the area of tiles in a monohedral tiling.

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}}

[[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==

Line 80 ⟶ 83:

| 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

Line 240 ⟶ 243:

| volume = 293

| year = 2024| doi-access = free

| isbn = 978-3-95977-316-4

}}</ref>