Binary tiling: Difference between revisions

Revision as of 06:25, 10 October 2024

Revision as of 21:12, 16 October 2024

Line 12:

==Tiles==

A [[Tessellation|tiling]] of a [[surface]] is a covering of the surface by [[geometric shape]]s, called tiles, with no overlaps and no gaps. An example is the familiar tiling of the [[Euclidean plane]] by [[square]]s, meeting edge-to-edge. When all the tiles have the same shape (they are all [[Congruence (geometry)|congruent]]), the tiling is called a [[monohedral tiling]], and the shape of the tiles is called the [[prototile]] of the tiling.{{r|adams}} 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.{{r|radin}} Two common models for the hyperbolic plane are the [[Poincaré disk model]] and [[Poincaré half-plane model]]. In these, the points of the hyperbolic plane are modeled by the points in an open disk or the half-plane above a horizontal line. Hyperbolic lines are modeled by those Euclidean circles and lines that cross the model's boundary [[perpendicular]]ly. The boundary points of the model are called [[ideal point]]s, and a hyperbolic line through an ideal point is said to be ''asymptotic'' to it. The half-plane model has one more ideal point, the [[point at infinity]], asymptotic to all vertical lines. Another important class of hyperbolic curves, called [[horocycle]]s, are modeled as circles that are tangent to the boundary of the model, or as horizontal lines in the half-plane model. Horocycles are asymptotic to their point of tangency, or to the point at infinity if they have none.{{r|rr|stahl}}

In one version of thebinary tiling, each tile is a subsetshape ofbounded the hyperbolic plane that lies betweenby two hyperbolic lines and two [[horocycle]]shorocycles. thatThese arefour allcurves are asymptotic to the same [[ideal point]], withand the two horocycles athave hyperbolic distance <math>\log 2</math> 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 <math>\log 2</math> as the distance between the two horocycles causes one of the two arcs of horocycles (the one farther from the asymptotic point) to behave twice asthe longhyperbolic aslength of the otheropposite arc. TheseCopies tilesof maythis beshape, packedmeeting edge-to-edge along their line segment sides, tocan fill outtile the annularslab or crescent shaped region between the two horocycles,. and to pack aA nested family of congruentthese annulislabs betweenor equallycrescents spacedcan horocyclesthen ontile eitherthe sideentire ofhyperbolic them.plane, When these annular packings linelined up so that each half of the outer horocycliclong arc of aeach tile in one annulusslab matchesis upcovered withby the innershort horocyclic arcarcs of atwo tiletiles in the next annulus,slab. theThe result is a binary tiling.{{r|df}}

[[File:Binary Tiling.png|thumb|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.]]

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.{{r|rr}} This gives eachEach tile theis overallmodeled shape in the model ofas an axis-parallel square or rectangle.{{r|radin|fg}} For this model, theThe hyperbolic length of an arc of a horizontal horocycle is its Euclidean length divided by its <math>y</math>-coordinate, while the hyperbolic distance between points with the same <math>x</math>-coordinate is the [[logarithm]] of the ratio of their <math>y</math>-coordinates.{{r|stahl}} From these facts one can calculate that successive horocycles of a binary tiling, at hyperbolic distance <math>\ln 2</math>, are modeled by horizontal lines whose Euclidean distance from the <math>x</math>-axis doubles at each step, and that the two bottom half-arcs of a binary tile each equal the top arc.

[[File:Binary tiling straight.svg|thumb|Binary tiling with [[pentagonal tiling|convex pentagon tiles]], in the Poincaré half-plane model.]]