Binary tiling

In geometry, a binary tiling (sometimes called a Böröczky tiling)^[1] 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 pentagons, or non-convex shapes with four sides, alternatingly line segments and 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, 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 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.

In one version of the tiling, each tile is a subset of the hyperbolic plane that lies between two hyperbolic lines and two horocycles that are all asymptotic to the same ideal point, with the horocycles at distance ${\displaystyle \log 2}$ from each other. The resulting shape has four right angles, like a rectangle, with its sides alternating between segments of hyperbolic lines and arcs of horocycles. The choice of ${\displaystyle \log 2}$ as the distance between the two horocycles causes one of the two arcs of horocycles (the one farther from the asymptotic point) to be twice as long as the other. These tiles may be packed along their line segment sides to fill out the annular region between the two horocycles, and to pack a nested family of congruent annuli between equally spaced horocycles on either side of them. When these annular packings line up so that each half of the inner horocyclic arc of a tile in one annulus matches up with the outer horocyclic arc of a tile in the next annulus, the result is a binary tiling.^[1]

In the Poincaré half-plane model of hyperbolic geometry, with the ideal point chosen to be a point at infinity for the half-plane, hyperbolic lines asymptotic to this point are modeled as vertical rays, and horocycles asymptotic to this point are modeled as horizontal lines.^[2] This gives each tile the overall shape in the model of an axis-parallel square or rectangle.^[3][4] For this model, the hyperbolic distance between points with the same ${\displaystyle y}$-coordinate is their Euclidean distance divided by ${\displaystyle y}$, while the hyperbolic distance between points with the same ${\displaystyle x}$-coordinate is the logarithm of the ratio of their ${\displaystyle y}$-coordinates.^[5] From these facts one can calculate that successive horocycles of a binary tiling, at hyperbolic distance ${\displaystyle \ln 2}$, are modeled by horizontal lines whose Euclidean distance from the ${\displaystyle x}$-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 horocyclic segments, so that each tile becomes a hyperbolic convex pentagon. This makes the tiling a proper pentagonal tiling.^[4][6]

If one considers only adjacencies between tiles of different sizes, omitting the side-to-side adjacencies, this adjacency pattern gives the tiles of a binary tiling the structure of a binary tree. Representative points within each tile, connected according to this adjacency structure, give an embedding of an infinite binary tree as a hyperbolic tree.^[7]

The tiles of a binary tiling are not all symmetric to each other; for instance, for the four tiles two levels below any given tile, no symmetry takes a middle tile to an outer tile. There are uncountably many different tilings of the hyperbolic plane by these tiles, even when they are modified by adding protrusions and indentations to force them to meet edge-to-edge. None of these different tilings are periodic (having a cocompact symmetry group),^[3] although some (such as the one in which there exists a line that is completely covered by tile edges) have a one-dimensional infinite symmetry group.^[1] As a tile all of whose tilings are not fully periodic, the prototile of a binary tiling solves an analogue of the einstein problem in the hyperbolic plane.^[8]

More strongly than having all tiles the same shape, all first coronas of tiles, the set of tiles touching a single central tile, have the same pattern of tiles (up to symmetries of the hyperbolic plane allowing reflections). For tilings of the Euclidean plane, having all first coronas the same implies that the tiling is periodic and isohedral (having all tiles symmetric to each other); binary tilings provide a strong counterexample for the corresponding property in the hyperbolic plane.^[9]

Corresponding to the fact that these tilings are non-periodic but monohedral (having only one tile shape), the dual tilings of these tilings are non-periodic but monocoronal (having the same pattern of tiles surrounding each vertex). These dual tilings are formed by choosing a reference point within each tile of a binary tiling, and connecting pairs of reference points of tiles that share an edge with each other.^[4]

Binary tilings were first studied mathematically in 1974 by Károly Böröczky [hu].^[3][10][11] Böröczky was investigating the density of a discrete planar point set, the average number of points per unit area. This quantity is used, for instance, to study Danzer sets. For points placed one per tile in a monohedral tiling of the Euclidean plane, the density is inverse to the tile area. But for the hyperbolic plane, paradoxical issues ensue.^[3][11] The tiles of a binary tiling can be grouped into three-tile subunits, with each subunit consisting of one tile above two more (as viewed in the Poincaré half-plane model). Points centered within the upper tile of each subunit have one point per subunit, for an apparent density equal to one third of the area of a binary tile. However, the same points and the same binary tiling can be regrouped in a different way, with two points per subunit, centered in the two lower tiles of each subunit, with two times the apparent density. This example shows that it is not possible to determine the density of a hyperbolic point set from tilings in this way.^[11][12]

Adjusting the distance between the two vertical sides of the tiles in a binary tiling causes their area to vary, proportional to this distance. By making this distance arbitrarily small, this tiling can be used to show that the hyperbolic plane has tilings by congruent tiles of arbitrarily small area.^[10] Jarkko Kari has used a system of colorings of tiles from a binary tiling, analogous to Wang tiles, to prove that determining whether a given system of hyperbolic prototiles can tile the hyperbolic plane is an undecidable problem.^[6] 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.^[13] Recursive data structures resembling quadtrees, based on binary tiling, have been used for approximate nearest neighbor queries in the hyperbolic plane.^[7]

Structure of Escher's Regular Division of the Plane VI

A Smith chart

Four sheets from the Cayley graph of the Baumslag–Solitar group ${\displaystyle BS(1,2)}$

A 1957 print by M. C. Escher, Regular Division of the Plane VI, has this tiling as its underlying structure, with each tile of a binary tiling (as seen in its quadtree form) subdivided into three right triangles.^[14] It is one of several Escher prints based on the half-plane model of the hyperbolic plane.^[15] When interpreted as Euclidean shapes rather than hyperbolically, the tiles are squares and the subdivided triangles are isosceles right triangles. The print itself replaces each triangle by a stylized lizard.^[14]

The Smith chart, from 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.^[16] It was first developed in the late 1930s by Tōsaku Mizuhashi,^[17] Phillip Hagar Smith,^[18] and Amiel R. Volpert.^[19]

The Cayley graph of the Baumslag–Solitar group ${\displaystyle BS(1,2)}$ can be decomposed into "sheets", planar structures with a geometry quasi-isometric to the hyperbolic plane. The Cayley graph is embedded onto each sheet as the graph of vertices and edges of a binary tiling. At each level of a binary tiling, there are two choices for how to continue the tiling at the next higher level. Any two sheets will coincide for some number of levels until separating from each other by following different choices at one of these levels, giving the sheets the structure of an infinite binary tree.^[20][21]

The dual graph of a binary tiling has a vertex for each tile, and an edge for each pair of tiles that share an edge. 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.^[22] Ringed trees have also been studied in terms of their hyperbolic metric properties in connection with small-world networks.^[23]

A related tiling of the hyperbolic plane by Roger Penrose can be interpreted as being formed by adjacent pairs of binary tiles, one above the other, whose unions form L-shaped tiles. Like binary tiling, it is weakly aperiodic.^[24]

• Tetrapentagonal tiling, a periodic tiling of the hyperbolic plane by pentagons