Binary tiling: Difference between revisions

Revision as of 20:32, 21 September 2024

Revision as of 20:34, 21 September 2024

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==Tiles==

In one version of the tiling, each tile is a subset of the hyperbolic plane that lies between two hyperbolic lines and two [[horocycle]]s that are all asymptotic to the same [[ideal point]], with the horocycles at distance <math>\log 2</math> from each other. The resulting shape has four right angles, like a rectangle, with its sides alternating between segments of hyperbolic lines and arcs of horocycles. The choice of <math>\log 2</math> as the distance between the two horocycles causes one of the two arcs of horocycles (the one closerfarther tofrom the asymptotic point) to be twice as long as the other. These tiles may be packed along their line segment sides to fill out the annular region between the two horocycles, and to pack a nested family of congruent annuli between equally spaced horocycles on either side of them. When these annular packings line up so that each half of the inner horocyclic arc of a tile in one annulus matches up with the outer horocyclic arc of a tile in the next annulus, the result is a binary tiling.{{r|df}}

[[File:Binary Tiling.png|thumb|A portion of a binary tiling displayed in the [[Poincaré half-plane model]]. The horizontal lines correspond to horocycles in the hyperbolic plane, and the vertical line segments correspond to hyperbolic lines.]]