Binary tiling: Difference between revisions

Revision as of 05:16, 24 September 2024

Revision as of 20:59, 6 October 2024

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[[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.]]

In the [[Poincaré half-plane model]] of hyperbolic geometry, with the ideal point chosen to be a [[point at infinity]] for the half-plane, hyperbolic lines asymptotic to this point are modeled as vertical rays, and horocycles asymptotic to this point are modeled as horizontal lines.{{r|rr}} This gives each tile the overall shape in the model of an axis-parallel square or rectangle.{{r|radin|fg}} For this model, the hyperbolic distancelength betweenof pointsan witharc theof samea <math>y</math>-coordinatehorizontal horocycle is theirits Euclidean distancelength divided by its <math>y</math>-coordinate, while the hyperbolic distance between points with the same <math>x</math>-coordinate is the [[logarithm]] of the ratio of their <math>y</math>-coordinates.{{r|stahl}} From these facts one can calculate that successive horocycles of a binary tiling, at hyperbolic distance <math>\ln 2</math>, are modeled by horizontal lines whose Euclidean distance from the <math>x</math>-axis doubles at each step, and that the two bottom half-arcs of a binary tile each equal the top arc.

[[File:Binary tiling straight.svg|thumb|Binary tiling with [[pentagonal tiling|convex pentagon tiles]], in the Poincaré half-plane model.]]