Horosphere Packings of the (3, 3, 6) Coxeter Honeycomb in Three-Dimensional Hyperbolic Space

The horosphere packings of the (3,3,6) Coxeter honeycomb in hyperbolic three-space are shown. The horoball centers are located in the lattice points of the honeycomb, while the radii of the horoballs are varied. Results due to Boroczky give an upper bound of the packing density using congruent horoballs. The purpose of this Demonstration is to show this density can be achieved with a different configuration of the balls.



  • [Snapshot]
  • [Snapshot]
  • [Snapshot]


The outermost sphere is the Cayley-Klein model of hyperbolic three-space; its points represent points at infinity.
The (3, 3, 6) Coxeter honeycomb is a tiling of three-dimensional hyperbolic space with regular asymptotic tetrahedra. A horosphere is a sphere with infinite radius. We give horosphere packings where the centers of the balls are at lattice points of the honeycomb. This Demonstration shows one tetrahedral cell of the tiling, with balls around its vertices, and the corresponding density. Density is defined as what part of the volume of the tetrahedra the horospheres occupy. This density can be generalized for the entire hyperbolic space. Densest packings are achieved when the balls touch. The "S" parameter varies the radius of "ball 4", which determines the radius of the touching balls.
Snapshot 1: Böröczky-Florian case, congruent balls
Snapshot 2: minimum case
Snapshot 3: another optimal case
K. Böröczky and A. Florian, "Über die dichteste Kugelpackung im hyperbolischen Raum," Acta Math. Acad. Sci. Hungar., 15, 1964 pp. 237–245.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.

Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2018 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+