Optimal ball and horoball packings generated by $3$-dimensional simply truncated Coxeter orthoschemes with parallel faces

Abstract

In this paper we consider the ball and horoball packings belonging to 3dimensional Coxeter tilings that are derived by simply truncated orthoschemes with parallel faces.The goal of this paper to determine the optimal ball and horoball packing arrangements and their densities for all above Coxeter tilings in hyperbolic 3-space H 3 . The centers of horoballs are required to lie at ideal vertices of the polyhedral cells constituting the tiling, and we allow horoballs of different types at the various vertices.

Visualization of sphere and horosphere packings related to Coxeter tilings generated by simply truncated orthoschemes with parallel faces

“…3. The packing density of these two configurations are the same, ≈ 0.8413392, see [46]. In our opinion, non-Euclidean tilings and packings and their investigations will play an important role in the research of material structure in the near future.…”

“…There are 5 complete combinations of bisectors for constructing the candidate of inball center. The complete packings denisties of inball packing (and their optimum density) can be found in [46], that gave the optimum packing density ≈ 0.2623649, attained by sphere packing in (∞, 3, 3, ∞).…”

“…In the former papers, we investigated that Coxeter simplex tilings whose generating simplices do not have parallel faces. In [46] we extended our study to the 3 dimensional Coxeter tilings generated by simple frustum orthoschemes with parallel faces. But first, we recall some results related to the above topic.…”

“…lying on the sphere ∂H n . In [46], we considered the Coxeter tilings in 3-dimensional hyperbolic space H 3 where the generating orthoscheme is a simple truncated Coxeter orthoscheme with parallel faces i.e. their dihedral angle is zero.…”

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In this paper we describe and visualize the densest ball and horoball packing configurations belonging to the simply truncated 3-dimensional hyperbolic Coxeter orthoschemes with parallel faces using the results of [46]. These beautiful packing arrangements describe and show the very interesting structure of the mentioned orthoschemes and the corresponding Coxeter groups.

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