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[[File:Binary tiling straight.svg|thumb|Binary tiling with [[pentagonal tiling|convex pentagon tiles]], in the Poincaré half-plane model.]]
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 horocyclic segments, 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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| ___location = Boston
| mr = 1217085
| pages =
| publisher = Jones and Bartlett Publishers
| title = The Poincaré Half-Plane: A Gateway to Modern Geometry
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