Binary tiling: Difference between revisions

Revision as of 06:05, 21 September 2024

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In [[geometry]], a '''binary tiling''' (sometimes called the '''Böröczky tiling'''){{r|df}} is a [[tiling of the hyperbolic plane]], resembling a [[quadtree]] over the [[Poincaré half-plane model]] of the hyperbolic plane. The tiles may be convex [[pentagon]]s, or they may be shapes with four sides, alternating between line segments and curved arcs.

There are uncountably many combinatoriallydistinct binary tilings for a given shape of tile. They are all weakly [[aperiodic tiling|aperiodic]], meaning that they can have a one-dimensional [[symmetry group]] 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 {{ill|Károly Böröczky|hu|Böröczky Károly (matematikus, 1964)}}.{{r|radin|agol|bor}} Closely related tilings have been used since the late 1930s in the [[Smith chart]] for radio engineering, and appear in a 1957 print by [[M. C. Escher]].{{r|escher}}