Hyperbolic plane geometry revisited

On the asymptotic geometry of the hyperbolic plane

Asymptotic subcone of an unbounded metric space is another metric space, capturing the structure of the original space at infinity. In this paper we define a functional metric space S which is an asymptotic subcone of the hyperbolic plane. This space is a real tree branching at every its point. Moreover, it is a homogeneous metric space such that any real tree with countably many vertices can b...

Hyperbolic Geometry

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...

Hyperbolic Geometry: Isometry Groups of Hyperbolic Space

The goal of this paper is twofold. First, it consists of an introduction to the basic features of hyperbolic geometry, and the geometry of an important class of functions of the hyperbolic plane, isometries. Second, it identifies a group structure in the set of isometries, specifically those that preserve orientation, and deals with the topological properties of their discrete subgroups. In the...

Emergent Hyperbolic Network Geometry

A large variety of interacting complex systems are characterized by interactions occurring between more than two nodes. These systems are described by simplicial complexes. Simplicial complexes are formed by simplices (nodes, links, triangles, tetrahedra etc.) that have a natural geometric interpretation. As such simplicial complexes are widely used in quantum gravity approaches that involve a ...