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Two common models for the hyperbolic plane are the [[Poincaré disk model]] and [[Poincaré half-plane model]]. In these, the points of the hyperbolic plane are modeled by points in the Euclidean plane, in an open disk or the half-plane above a horizontal line respectively. Hyperbolic lines are modeled by those Euclidean circles and lines that cross the model's boundary [[perpendicular]]ly. The boundary points of the model are called [[ideal point]]s, and a hyperbolic line through an ideal point is said to be ''asymptotic'' to it. The half-plane model has one more ideal point, the [[point at infinity]], asymptotic to all vertical lines. Another important class of hyperbolic curves, called [[horocycle]]s, are modeled as circles that are tangent to the boundary of the model, or as horizontal lines in the half-plane model. Horocycles are asymptotic to their point of tangency, or to the point at infinity if they have none.{{r|rr|stahl}}
[[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.]]▼
[[File:Binary tiling straight.svg|thumb|Binary tiling with [[pentagonal tiling|convex pentagon tiles]], in the Poincaré half-plane model.]]▼
In one version of binary tiling, each tile is a shape bounded by two hyperbolic lines and two horocycles. These four curves should be asymptotic to the same ideal point, with the two horocycles at hyperbolic distance <math>\log 2</math> from each other. With these choices, the tile 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 farther from the asymptotic point) to have twice the hyperbolic length of the opposite arc. Copies of this shape, meeting edge-to-edge along their line segment sides, can tile the slab or crescent shaped region between two horocycles. A nested family of these slabs or crescents can then tile the entire hyperbolic plane, lined up so that the long arc of each tile in one slab is covered by the short arcs of two tiles in the next slab. 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.]]
In the [[Poincaré half-plane model]] of hyperbolic geometry, each tile can be modeled as an axis-parallel square or rectangle.{{r|radin|fg}} In this model, rays through the vertical sides of a tile model hyperbolic lines, asymptotic to the point at infinity, and lines through the horizontal sides of a tile model horocycles, asymptotic to the same point.{{r|rr}} The hyperbolic length of an arc of a horizontal horocycle is its Euclidean length 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.]]
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}}
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