I’m revisiting a lot of the techniques I’ve implemented for working with hyperbolic geometry as I’m rebuilding the Transversion Automaton in order to make the simpler and clearer. The most basic structure that underlies the topology of the surfaces generated by the Transversion Automaton is the hyperbolic tiling. I recently found a much simpler derivation for the fundamental polygon that seeds the tiling and implemented it.

In a regular hyperbolic tiling, each tile is exactly the same with the tile located at the center of the Poincaré disc called the fundamental polygon. By virtue of the expansive nature of hyperbolic space, there are an infinite number of regular tilings of the hyperbolic plane since the polygons can simply be resized until the rate at which they fill space matches the rate at which the hyperbolic plane expands. Thus, the key value to calculate when generating regular hyperbolic tilings is the radius of the fundamental polygon.

The value that determines how big the radius is is the angle of parallelism of the edges of the fundamental polygon. The angle of parallelism essentially says how close to Euclidean space a particular tiling is. The closer to 90 degrees the angle of parallelism, the closer to Euclidean it is. For example, in the image above, the fundamental polygon on the far left is for the (3, 7) tiling. Notice how the radius of the pink circle is large and the arc that’s visible approaches a straight line. The angle that the line tangent to vertex D makes with the horizontal axis is the angle of parallelism. As D moves toward the top of the Poincaré disc circle, this angle approaches 90 degrees. If we changed the tiling to (3, 6), we would in fact have a 90 degree angle of parallelism and a Euclidean tiling. As the fundamental polygon increases in sides and number of polygons meeting at a vertex, the angle of parallelism decreases toward 0 as both values approach infinity.

The diagrams above show how the angle of parallelism varies with the number if polygons incident to a vertex (left) and the number of sides (right). Calculating the location of the fundamental polygon’s vertices is equivalent to calculating the location of the circles that the polygon’s arcs lie on. Since they’re all equidistant from the origin, we just need to figure out how far away the circle is. To do this, we use right triangles AEF and ABF along with the fact that angles \(\angle{}BAF = \frac{\pi}{p}\) and \(\angle{}ABF = \frac{\pi}{q}\) are known to derive the fact that:

- \(sin(\angle{}EBF) = \frac{\overline{EF}}{\overline{BF}}\) and \(sin(\angle{}BAF) = \frac{\overline{EB}}{\overline{AF}}\)
- \(1+\overline{BF}^2 = \overline{AF}^2\)
- \(sin(\angle{}EBF) = \frac{\pi}{2} – \frac{\pi}{q}\)
- \(sin(\angle{}BAF) = \frac{\pi}{p}\)
- \(\sqrt{\overline{AF}^2-1} \dot{} sin(\frac{\pi}{2} – \frac{\pi}{q}) = \overline{AF} \dot{} sin(\frac{\pi}{2})\)
- \(\overline{AF} = \sqrt{\frac{sin^2(\frac{\pi}{2}- \frac{\pi}{q})}{sin^2(\frac{\pi}{2}- \frac{\pi}{q}) – sin^2(\frac{\pi}{p})}}\)