Binary tiling: Difference between revisions

Revision as of 04:13, 28 October 2024

Revision as of 04:14, 28 October 2024

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Another application involves the area of tiles in a monohedral tiling.

In defining the binary tilings, there is a free parameter, the distance between the vertical sides of the tiles. All tiles in a single tiling have equal hyperbolic areas, but if this distance changes, the (equal) area of all the tiles will also change, in proportion. Choosing this distance to be arbitrarily small shows that the hyperbolic plane has tilings by congruent tiles of arbitrarily small area.{{r|agol}}

[[Jarkko Kari]] has used a system of colorings of tiles from a binary tiling, analogous to [[Wang tile]]s, to prove that determining whether a given system of hyperbolic [[prototile]]s can tile the hyperbolic plane is an [[undecidable problem]].{{r|kari}} Subdivisions of a binary tiling that replace each tile by a [[grid graph]] have been used to obtain tight bounds on the [[Fine-grained reduction|fine-grained complexity]] of [[graph algorithm]]s.{{r|kmvww}} Recursive [[data structure]]s resembling quadtrees, based on binary tiling, have been used for approximate [[nearest neighbor problem|nearest neighbor queries]] in the hyperbolic plane.{{r|kbvw}}

==Related patterns==