Regular Octagons in Hyperbolic Geometry

Abstract

When constructing hyperbolic structures on closed surfaces, one can use hyperbolic geometry (Lobachevsky geometry) on the plane. To do this, the surface must be represented as a 2n-gon on the hyperbolic plane, and a discrete group action, which is a subgroup of the movements of the hyperbolic plane, must be defined, for which the 2n-gon serves as a fundamental domain. If such a surface is a double torus (an oriented surface of genus 2), it can be obtained by gluing opposite sides of an octagon. In fact, the Lobachevsky plane is divided into octagons. The presence of symmetries simplifies calculations. Therefore, a natural problem arises regarding the partitioning into regular octagons. Additionally, it is important to provide examples of such octagons by specifying the coordinates of their vertices in one of the models of hyperbolic geometry.

The models of the upper half-plane and the Poincaré model on the unit disk are used, for which the Riemannian metric is defined (the formula for finding the lengths of arcs of curves). We describe the main properties of hyperbolic lines and the group of movements (the group of isometric transformations) of hyperbolic geometry on the plane using fractional-linear transformations of the complex plane with real coefficients.