Binary tiling: Difference between revisions

Revision as of 03:48, 28 October 2024

Revision as of 03:59, 28 October 2024

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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 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]], 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 partthe subset of a binary tiling, thethat tilinglies ofinside a [[horoball]]horocycle. above a(For horizontal linehorocycles in the Poincaré half-plane, the inside is above the modelhorocycle.) These mixed binary and apeirogonal tilings avoid the density paradoxes of the binary tiling.{{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}}