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The [[Cayley graph]] of the [[Baumslag–Solitar group]] <math>BS(1,2)</math> can be decomposed into "sheets", planar structures with a geometry [[Quasi-isometry|quasi-isometric]] to the hyperbolic plane. The Cayley graph is embedded onto each sheet as the graph of vertices and edges of a binary tiling. At each level of the 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 the 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}}
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 the binary tiling, it is weakly aperiodic.{{r|penrose}}
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