Binary tiling

They may be convex pentagons, or non-convex shapes with four sides, alternatingly line segments and horocyclic arcs, meeting at four right angles.

Binary tilings were first studied mathematically in 1974 by Károly Böröczky [hu].

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.

An example is the familiar tiling of the Euclidean plane by squares, meeting edge-to-edge,[2] as seen for instance in many bathrooms.

[2] The binary tilings are monohedral tilings of the hyperbolic plane, a kind of non-Euclidean geometry with different notions of length, area, congruence, and symmetry than the Euclidean plane.

These four curves should be asymptotic to the same ideal point, with the two horocycles at hyperbolic distance

With these choices, the tile has four right angles, like a rectangle, with its sides alternating between segments of hyperbolic lines and arcs of horocycles.

[6] From these facts one can calculate that successive horocycles of a binary tiling, at hyperbolic distance

-axis doubles at each step, and that the two bottom half-arcs of a binary tile each equal the top arc.

An alternative and combinatorially equivalent version of the tiling places its vertices at the same points, but connects them by hyperbolic line segments instead of arcs of horocycles, so that each tile becomes a hyperbolic convex pentagon.

[4] Some binary tilings have a one-dimensional infinite symmetry group.

More technically, no binary tiling has a cocompact symmetry group.

Because they can have one-dimensional symmetries, the binary tilings are not strongly aperiodic.

[7] Binary tilings were first studied mathematically in 1974 by Károly Böröczky [hu].

This example shows that it is not possible to determine the density of a hyperbolic point set from tilings in this way.

[8] Subdivisions of a binary tiling that replace each tile by a grid graph have been used to obtain tight bounds on the fine-grained complexity of graph algorithms.

[14] Recursive data structures resembling quadtrees, based on binary tiling, have been used for approximate nearest neighbor queries in the hyperbolic plane.

[16] It is one of several Escher prints based on the half-plane model of the hyperbolic plane.

[16] The Smith chart, a graphical method of visualizing parameters in radio engineering, resembles a binary tiling on the Poincaré disk model of the hyperbolic plane, and has been analyzed for its connections to hyperbolic geometry and to Escher's hyperbolic tilings.

[18] It was first developed in the late 1930s by Tōsaku Mizuhashi,[19] Phillip Hagar Smith,[20] and Amiel R.

This graph can be decomposed into "sheets", whose vertices and edges form a binary tiling.

It takes the form of an infinite binary tree (extending infinitely both upwards and downwards, without a root) with added side-to-side connections between tree nodes at the same level as each other.

[1] An analogous structure for finite complete binary trees, with the side-to-side connections at each level extended from paths to cycles, has been studied as a network topology in parallel computing, the ringed tree.

[25] Ringed trees have also been studied in terms of their hyperbolic metric properties in connection with small-world networks.

Binary tiling on Poincare disk
A binary tiling in the Poincaré disk model of the hyperbolic plane . Each tile edge lies on a horocycle (shown as circles interior to the disk) or a hyperbolic line (arcs perpendicular to the disk boundary). The horocycles and lines are asymptotic to an ideal point located at the right side of the Poincaré disk.
Square tiles in a bathroom
A portion of a binary tiling displayed in the Poincaré half-plane model . The horizontal lines correspond to horocycles in the hyperbolic plane, and the vertical line segments correspond to hyperbolic lines.
Binary tiling with convex pentagon tiles , in the Poincaré half-plane model.
No symmetry of the binary tiling takes the blue tile (in a middle position relative to the yellow tile two levels above it) to the red tile (in an outer position).
A binary tiling (red outline) and its dual tiling (yellow curved triangles and blue and green curved quadrilaterals)
Is the average number of red points per tile 1/3 (left) or 2/3 (right)?
Each face in this order-3 apeirogonal tiling (shown in the Poincaré disk model) can be replaced by part of a binary tiling as modified by Radin. [ 4 ]