A Characterization of Hyperbolic Geometry among Hilbert Geometry

Hyperbolic Geometry

Topics in Hyperbolic Geometry

There are two types of parallel lines in Hyperbolic Geometry. There are those who diverge from each other in both directions (type 1) and those that diverge in one direction but come arbitrarily close to each other in the other direction (type 2). The first type have a minimum positive distance at points on a common perpendicular. The second type have no minimum distance: the distance tends to ...

Introduction to Hyperbolic Geometry

What Is Hyperbolic Geometry?

(1) Each pair of points can be joined by one and only one straight line segment. (2) Any straight line segment can be indefinitely extended in either direction. (3) There is exactly one circle of any given radius with any given center. (4) All right angles are congruent to one another. (5) If a straight line falling on two straight lines makes the interior angles on the same side less than two ...

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 ...