The Trigonometry of Hyperbolic Tessellations

Abstract

AbstractFor positive integerspandqwith (p− 2)(q− 2) > 4 there is, in the hyperbolic plane, a group [p, q] generated by reflections in the three sides of a triangleABCwith angles π/p, π/q, π/2. Hyperbolic trigonometry shows that the sideAChas length ψ, where cosh ψ =c/s, c = cos π/q,s= sin π/p. For a conformal drawing inside the unit circle with centreA, we may take the sidesABandACto run straight along radiiwhileBCappears as an arc of a circle orthogonal to the unit circle. The circle containing this arc is found to have radius 1/ sinh ψ =s/z, where z =, while its centre is at distance 1/ tanh ψ = c/z fromA. In the hyperbolic triangleABC, the altitude fromABto the right-angled vertex C is ζ, where sinh ζ = z.