Packings in hyperbolic geometry

Shallow Packings in Geometry

We refine the bound on the packing number, originally shown by Haussler, for shallow geometric set systems. Specifically, let V be a finite set system defined over an n-point set X; we view V as a set of indicator vectors over the n-dimensional unit cube. A δ-separated set of V is a subcollection W, s.t. the Hamming distance between each pair u,v ∈ W is greater than δ, where δ > 0 is an integer...

Density of Tube Packings in Hyperbolic Space

Packing problems have long been a topic of interest. Traditionally, efforts had been focused on Euclidean space, but as interest in hyperbolic space has grown, many of the Euclidean problems have been translated into the hyperbolic arena, in which the problems are almost always vastly more complicated. The particular packing problem of interest here is a hyperbolic version of packing congruent ...

Apollonian Circle Packings: Number Theory II. Spherical and Hyperbolic Packings

Apollonian circle packings arise by repeatedly filling the interstices between mutually tangent circles with further tangent circles. In Euclidean space it is possible for every circle in such a packing to have integer radius of curvature, and we call such a packing an integral Apollonian circle packing. There are infinitely many different integral packings; these were studied in the paper [8]....

Hyperbolic Geometry

Problems in Hyperbolic Geometry

The purpose of this document is to list some outstanding questions in the subject of hyperbolic geometry (real, complex, quaternionic, and the Cayley plane), with the primary focus on the latter three. This list was inspired by the problem sessions held at the conference on Complex Hyperbolic Geometry in Luminy (CIRM) in July of 2003. The questions are roughly grouped by similarity. The name(s)...