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.

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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.
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