# A half-flipped binary tiling

In this tiling of the hyperbolic plane, all of the tiles are the same shape and size (despite their varied appearance), they are not square, and they are not polygons.

The right side of the illustration depicts a binary tiling, in one of its conventional views using the Poincaré half plane model of hyperbolic geometry. The vertical sides of the tiles lie on hyperbolic lines, but their horizontal sides are arcs of horocycles, special hyperbolic curves intermediate between a circle and a line. They are not straight: the tiles are concave on top and convex on the bottom. The black line at the bottom of the image is the boundary of the half plane model.

One can measure the sides of each tile using standard formulas for hyperbolic geometry in the half-plane model. The hyperbolic length of a vertical line segment is the logarithm of the ratio of its top and bottom \(x\)-coordinates. Therefore, the vertical sides of each tile have length \(\ln 2\approx 0.69315\), as they always do in binary tilings. The widths of tiles in binary tilings can vary arbitrarily; I chose this drawing to make the tiles look square but they could be rectangles of arbitrary aspect ratio. The hyperbolic length of a horizontal line segment is its Euclidean length divided by its height above the \(x\)-axis. Therefore, the top edge of each square has hyperbolic length \(\tfrac12\), while the bottom edge has hyperbolic length \(1\). That’s why it’s possible for two top edges of tiles to match up with a single bottom edge.

The vertical black line in the middle of the illustration passes only through edges of tiles, not their interiors. Not every binary tiling has such a line, but when it exists the tiling has one-dimensional symmetry: it can be translated upwards and downwards along this line, and reflected across the line. (Binary tilings cannot have full two-dimensional symmetry.) The tile edges divide this line into intervals, all the same hyperbolic length, \(\ln 2\).

That leaves the black semicircle to explain. It is a hyperbolic line, perpendicular to the vertical symmetry line of the binary tiling. The left side of the top illustration shows what happens if we reflect the binary tiling across this perpendicular. In Euclidean geometry, this reflection is modeled as an inversion through the semicircle. Its (Euclidean) radius is the geometric mean of the two endpoints of the uppermost visible interval on the vertical black line, \(\sqrt2\) times the height of the inner endpoint, causing these two endpoints to invert to each other. The tiles on the right and the inverted tiles on the left match up along the vertical symmetry line, giving us a tiling of the entire hyperbolic plane. The inversion reverses top and bottom, so the bottom sides of tiles on the right (touching two other tiles) become the sides farthest from the origin of tiles on the left (still touching two other tiles). The vertical black line continues to be a symmetry line for the half-flipped tiling, but with different symmetries: translations and \(180^\circ\) rotations through tile corners or tile edge midpoints.

The horizontal horocycles on the right are inverted on the left into circles that are mutually tangent to each other and to the \(x\)-axis at the origin. We see only semicircles, but a complete horocycle would invert to a complete circle. The top point of each semicircle lies on the vertical symmetry line, at a corner of four tiles. These semicircles are scaled by powers of two, just like the horizontal lines on the right, to make them equally spaced at hyperbolic distance \(\ln 2\) from each other. Each row of equal-size squares on the right, between two consecutive horocycles, continues in inverted form on the left as a horn-shaped region between two of these semicircles.

Within a single horn, the tiles are separated from each other by hyperbolic lines, modeled as semicircles that are mutually tangent to each other and to the \(y\)-axis at the origin. On the right, the vertical sides of tiles in a row are equally spaced in Euclidean distance. On the left, this translates into a harmonic sequence: the radii of the semicircles modeling these lines are proportional to \(1,\tfrac12,\tfrac13,\tfrac14,\dots\). The constant of proportionality, for the horn whose top tile is crossed by the inversion circle, is set by the observation that the inversion circle passes through the bottom right corner of the top right red tile, and therefore must also pass through a corner of the top left blue tile. The first semicircle in this harmonic sequence also passes through this corner (which turns out to be its apex). Once we have found the tiles of one horn in this way, all of the other horns are scaled from it by powers of two. The semicircles with proportions \(1,\tfrac12,\tfrac13,\tfrac14,\dots\) in one horn continue with proportions \(\tfrac12,\tfrac14,\tfrac16,\tfrac18,\dots\) in the next horn out, and so on.

With these calculations and measurements, we have all we need to construct the illustration of the half-flipped binary tiling!