Binary tiling: Difference between revisions

Revision as of 06:06, 21 September 2024

Revision as of 06:12, 21 September 2024

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[[File:Hyperbolic binary tiling.png|upright=1.2|alt=Binary tiling on Poincare disk|thumb|A binary tiling displayed in the [[Poincaré disk model]] of the [[hyperbolic plane]]. Each side of a tile lies on a [[horocycle]] (shown as circles interior to the model) or a hyperbolic line (shown as arcs of circles perpendicular to the model boundary). These horocycles and lines are all asymptotic to a common [[ideal point]] located at the right side of the Poincaré disk.]]

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. Each tile adjoins five others. The tiles may be convex [[pentagon]]s, or they may be non-convex shapes with four sides, alternating betweenalternatingly line segments and curved[[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]], 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.

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==See also==

*[[Tetrapentagonal tiling]], a periodic tiling of the hyperbolic plane by pentagons

== References ==