HYPERBALL PACKINGS RELATED TO TRUNCATED CUBE AND OCTAHEDRON TILINGS IN HYPERBOLIC SPACE

In this paper, we study congruent and noncongruent hyperball (hypersphere) packings to the truncated regular cube and octahedron tilings. These are derived from the Coxeter truncated orthoscheme tilings {4, 3,p} (6 < p is an element of N) and {3, 4, p} (4 < p is an element of N), respectively, by their Coxeter reflection groups in hyperbolic space H-3. We determine the densest hyperball packing arrangement and its density with congruent and noncongruent hyperballs. We prove that the locally densest (noncongruent half) hyperball configuration belongs to the truncated cube with a density of approximately 0.86145 if we allow 6 < p is an element of R. for the dihedral angle 2 pi/p. This local density is larger than the Boroczky-Florian density upper bound for balls and horoballs. But our locally optimal noncongruent hyperball packing configuration cannot be extended to the entire hyperbolic space H-3. We determine the extendable densest noncongruent hyperball packing arrangement to the truncated cube tiling {4, 3,p = 7} with a density of approximately 0.84931.