Binary tiling
In geometry, the binary tiling (sometimes called the Böröczky 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 mathematically in 1974 by . However, a closely related tiling was used earlier in a 1957 print by M. C. Escher. Tiles. The tiles are shapes bounded by three horocyclic segments (two of which are part of the same horocycle), 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 (having a cocompact symmetry group), although some (such as the one in which there exists a line that is completely covered by tile edges) 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. A 1957 print by M. C. Escher, "Regular Division of the Plane VI", has this tiling as its underlying structure, with each tile of the binary tiling (as seen in its quadtree form) subdivided into three right triangles. When interpreted as Euclidean shapes rather than hyperbolically , the tiles are squares and the subdivided triangles are isosceles right triangles. The print itself replaces each triangle by a stylized lizard.