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

TL;DR

This paper visualizes the densest ball and horoball packings related to specific hyperbolic Coxeter orthoschemes, revealing their geometric structure using Python-based visualizations in the Beltrami-Cayley-Klein model.

Contribution

It introduces a visualization of complex hyperbolic packings associated with truncated orthoschemes, highlighting their geometric properties and structure.

Findings

Visualization of dense packings using Python

Insights into the structure of Coxeter orthoschemes

Representation in the Beltrami-Cayley-Klein model

Abstract

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. These beautiful packing arrangements describe and show the very interesting structure of the mentioned orthoschemes and the corresponding Coxeter groups. We use for the visualization the Beltrami-Cayley-Klein ball model of $3$-dimensional hyperbolic space $\boldsymbol{H}^3$ and the pictures were made by the Python software.