Binary tiling: Difference between revisions

Revision as of 07:15, 24 October 2024

Revision as of 07:17, 24 October 2024

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An alternative and combinatorially equivalent version of the tiling places its vertices at the same points, but connects them by hyperbolic line segments instead of arcs of horocycles, so that each tile becomes a hyperbolic convex pentagon. This makes the tiling a proper [[pentagonal tiling]].{{r|fg|kari}} The hyperbolic lines through the non-vertical sides of these tiles are modeled in the half-plane model by semicircles centered on the <math>x</math>-axis, and the sides form arcs of these semicircles.{{r|stahl}}

If one considers only adjacencies between tiles of different sizes, omitting the side-to-side adjacencies, this adjacency pattern gives the tiles of a binary tiling the structure of a [[binary tree]]. Representative points within each tile, connected according to this adjacency structure, give an embedding of an infinite binary tree as a [[hyperbolic tree]].{{r|kbvw}}

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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}}

Omitting the side-to-side connections gives an embedding of an infinite binary tree as a [[hyperbolic tree]].{{r|kbvw}}

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