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The [[Cayley graph]] of the [[Baumslag–Solitar group]] <math>BS(1,2)</math>, has the group elements as vertices, connected by edges representing multiplication by this group's standard generating elements. This graph can be decomposed into "sheets", whose vertices and edges form 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}}▼
[[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 part of a binary tiling, the tiling of a [[horoball]] above a horizontal line in the half-plane model. 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}}
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