Tessellating the Hyperbolic Plane

The main goal of this paper will be to determine which hyperbolic polygons can be used to tessellate the hyperbolic plane. Sections 1-4 will be devoted to providing the context of the hyperbolic plane and developing the basic tools needed to prove the key theorems of this paper. In these sections I will cover two common models of the plane and the isometries of these spaces. Section 4 will be a brief digression into the nature of triangles which will provide a good background for more general theorems about tessellation. Section 5 will cover the basics of tessellation. Poincaré’s theorem for compact polygons which is the central element of the argument of this paper will be introduced at the end of section 5. Section 6 will be devoted to the mechanical tools needed to prove this theorem. In the final two sections,Poincaré’s theorem is proved and the implications of the proof are discussed. Significant information supplementary to the highlighted theorem is discovered in the course of its