Binary tiling

In <a href="/facts/Geometry/wTQf5ffQ">geometry</a>, a binary tiling (sometimes called a Böröczky tiling) is a <a href="/facts/Tiling_of_the_hyperbolic_plane/khwIB6I1">tiling of the hyperbolic plane</a>, resembling a <a href="/facts/Quadtree/6b23FQoD">quadtree</a> over the <a href="/facts/Poincar%C3%A9_half-plane_model/4mg3Mj2b">Poincaré half-plane model</a> of the hyperbolic plane. The tiles are congruent, each adjoining five others. They may be convex <a href="/facts/Pentagon/jeHEARHH">pentagons</a>, or non-convex shapes with four sides, alternatingly line segments and <a href="/facts/Horocycle/sBR9wnuu">horocyclic</a> arcs, meeting at four right angles.
There are uncountably many distinct binary tilings for a given shape of tile. They are all weakly <a href="/facts/Aperiodic_tiling/WFJMWDg4">aperiodic</a>, which means that they can have a one-dimensional <a href="/facts/Symmetry_group/CNqeseMp">symmetry group</a> but not a two-dimensional family of symmetries. There exist binary tilings with tiles of arbitrarily small area.
Binary tilings were first studied mathematically in 1974 by Károly Böröczky [hu]. Closely related tilings have been used since the late 1930s in the <a href="/facts/Smith_chart/jdkHh9Ju">Smith chart</a> for radio engineering, and appear in a 1957 print by <a href="/facts/M._C._Escher/6uuAGlDx">M. C. Escher</a>.

Binary tiling open-in-new

Binary tiling