Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Volumes of Regular Hyperbolic Simplices

Zakhar Kabluchko, Philipp Schange

TL;DR

The paper derives an explicit formula for the hyperbolic volume of a regular $d$-dimensional simplex in spaces of constant curvature $\kappa$, valid for all $d\ge 2$. The volume is expressed via a contour integral involving the standard normal distribution function $\Phi$, extended to complex arguments, and depends on orthocentric parameters $\tau_j$ with $s=\sum_j \tau_j^2$. The authors prove this by analyzing the volume as a function of $\kappa$, establishing analyticity on $\mathbb{C}\setminus(-\infty,\kappa_0]$, deriving a spherical-volume representation for $\kappa\ge s$, and performing an analytic continuation to handle all admissible $\kappa$; they then invoke a uniqueness theorem to finalize the formula. Consequences include explicit hyperbolic and ideal simplex volumes and alignment with known maximality results for ideal regular simplices, linking geometric volumes to probabilistic integrals via contour methods.

Abstract

We derive an explicit formula for the volume of a regular simplex in the hyperbolic space of any dimension.

Volumes of Regular Hyperbolic Simplices

TL;DR

The paper derives an explicit formula for the hyperbolic volume of a regular -dimensional simplex in spaces of constant curvature , valid for all . The volume is expressed via a contour integral involving the standard normal distribution function , extended to complex arguments, and depends on orthocentric parameters with . The authors prove this by analyzing the volume as a function of , establishing analyticity on , deriving a spherical-volume representation for , and performing an analytic continuation to handle all admissible ; they then invoke a uniqueness theorem to finalize the formula. Consequences include explicit hyperbolic and ideal simplex volumes and alignment with known maximality results for ideal regular simplices, linking geometric volumes to probabilistic integrals via contour methods.

Abstract

We derive an explicit formula for the volume of a regular simplex in the hyperbolic space of any dimension.

Paper Structure

This paper contains 19 sections, 18 theorems, 73 equations, 1 figure.

Table of Contents

  1. Introduction and main result
  2. Introduction
  3. Volumes of regular hyperbolic simplices
  4. Notation and facts from hyperbolic and spherical geometry
  5. Basic notation.
  6. Upper half-sphere.
  7. Upper hyperboloid.
  8. Gnomonic projection.
  9. Klein model.
  10. Main result
  11. The setting: Orthocentric simplices
  12. Statement of the main result: Hyperbolic volume of orthocentric simplices
  13. Proof of the main result
  14. Outline of the proof
  15. Analytic continuation of the volume
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $d\geq 2$. In the $d$-dimensional hyperbolic space of constant curvature $\kappa = -1$ consider a regular $d$-dimensional hyperbolic simplex $\Delta_\ell^d$ with hyperbolic side length $\ell>0$. Then, the hyperbolic volume of $\Delta_\ell^d$ is given by The hyperbolic volume of the ideal regular $d$-dimensional hyperbolic simplex $\Delta_\infty^d$ is given by

Figures (1)

  • Figure 2.1: Gnomonic projection: Hyperbolic case (left) and spherical case (right)

Theorems & Definitions (36)

  • Theorem 1.1: Volume of a regular hyperbolic simplex
  • Definition 3.1: Vectors in orthocentric position
  • Example 3.2: Regular simplices
  • Example 3.3: Orthocentric simplices
  • Lemma 3.4
  • proof
  • Proposition 3.5
  • proof
  • Theorem 3.6: Volume of an orthocentric simplex, spherical or hyperbolic
  • Theorem 3.7: Volume of a regular hyperbolic simplex
  • ...and 26 more