Binary tiling


In geometry, the binary tiling is a tiling of the hyperbolic plane, resembling a quadtree over the Poincaré half-plane model of the hyperbolic plane. It was first studied in 1974 by.

Tiles

The tiles are shapes bounded by three horocyclic segments, and two line segments. All tiles are congruent. Although they are modeled by squares or rectangles of the Poincaré model, the tiles have five sides rather than four, and are not hyperbolic polygons, because their horocyclic edges are not straight. Alternatively, a combinatorially equivalent tiling uses hyperbolic pentagons that connect the same vertices in the same pattern. In this form of the tiling, the tiles do not appear as rectangles in the halfplane model, and the horocycles formed by sequences of edges are replaced by apeirogons.

Enumeration and aperiodicity

There are uncountably many different tilings of the hyperbolic plane by these tiles, even when they are modified by adding protrusions and indentations to force them to meet edge-to-edge. None of these different tilings are periodic, although some have a one-dimensional infinite symmetry group.

Application

This tiling can be used to show that the hyperbolic plane has tilings by congruent tiles of arbitrarily small area.