Packings with horo- and hyperballs generated by simple frustum orthoschemes

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In this paper we deal with the packings derived by horo- and hyperballs (briefly hyp-hor packings) in the n-dimensional hyperbolic spaces H n (n = 2,3) which form a new class of the classical packing problems. We construct in the 2− and 3−dimensional hyperbolic spaces hyphor packings that are generated by complete Coxeter tilings of degree 1 i.e. the fundamental domains of these tilings are simple frustum orthoschemes and we determine their densest packing configurations and their densities. We prove that in the hyperbolic plane (n = 2) the density of the above hyp-hor packings arbitrarily approximate the universal upper bound of the hypercycle or horocycle packing density 3 � and in H 3 the optimal configuration belongs to the [7,3,6] Coxeter tiling with density ≈ 0.83267. Moreover, we study the hyp-hor packings in truncated orthoschemes [p,3,6] (6 < p < 7, p ∈ R) whose density function is attained its maximum for a parameter which lies in the interval [6.05,6.06] and the densities for parameters lying in this interval are larger that ≈ 0.85397. That means that these locally optimal hyp-hor configurations provide =   