Binary tiling: Difference between revisions

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In [[geometry]], a '''binary tiling''' (sometimes called a '''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 are congruent, each adjoining five others. They may be convex [[pentagon]]s, or non-convex shapes with four sides, alternatingly line segments and [[horocycle|horocyclic]] arcs, meeting at four right angles.

There are uncountably many distinct binary tilings for a given shape of tile. They are all weakly [[aperiodic tiling|aperiodic]], meaningwhich means 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)}}. 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]].