The Inradius of a Hyperbolic Truncated $$n$$ n -Simplex

Abstract

Hyperbolic truncated simplices are polyhedra bounded by at most 2n + 2 hyperplanes in hyperbolic n-space. They provide important models in the context of hyperbolic space forms of small volume. In this work, we derive an explicit formula for their inradius by algebraic means and by using the concept of reduced Gram matrix. As an illustration, we discuss implications for some polyhedra related to small volume arithmetic orientable hyperbolic orbifolds.

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

“…In [15] M. JACQUEMET determined the inradii of truncated simplices in n-dimensional hyperbolic spaces if their inballs do not have common inner points with the corresponding truncating hyperplanes. We first recall some of the statements from that mentioned paper that we will apply to our calculations.…”

Visualization of Sphere and Horosphere Packings Related to Coxeter Tilings by Simply Truncated Orthoschemes with Parallel Faces

“…Hence the group gives regular tessellations. We note here that the Coxeter groups are finite for S n , and infinite for E n or H n [1,5,7,8,9,17,23].…”

“…4. In constructing the insphere, the largest inscribed classical sphere, in the truncated orthoscheme, we followed in [24] the procedure of [8] by bisector hyperplane.…”

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

“…In constructing the inball, the largest inscribed classical ball, in the truncated orthoscheme, we follow the procedure of [15], that constructed the inscribed sphere into a polyhedra in hyperbolic spaces H n . For i, j ∈ {0, 1, 2, 3}, i = j, let S ij (s ij ) be hyperbolic hyperplane given by the following formula where u i are the above unit pols (normal vectors) of forms u i :…”