Binary tiling: Difference between revisions

Revision as of 20:37, 22 September 2024

Revision as of 17:45, 23 September 2024

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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 tile of a binary tiling (as seen in its quadtree form) subdivided into three right triangles.{{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}}

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}}