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Volumes of ideal hyperbolic drums

Eric Chesebro, Justin Lanier, Emma N. McQuire, James Morgan, Jessica S. Purcell, Henry Segerman

Abstract

Milnor computed the volumes of ideal hyperbolic prisms as part of an effort to construct 3-manifolds whose volumes are finite rational sums of the Lobachevsky function evaluated at rational multiples of pi. Motivated by these results and with an eye to related applications, we prove a volume formula for arbitrary ideal hyperbolic antiprisms, also called drums.

Volumes of ideal hyperbolic drums

Abstract

Milnor computed the volumes of ideal hyperbolic prisms as part of an effort to construct 3-manifolds whose volumes are finite rational sums of the Lobachevsky function evaluated at rational multiples of pi. Motivated by these results and with an eye to related applications, we prove a volume formula for arbitrary ideal hyperbolic antiprisms, also called drums.
Paper Structure (3 sections, 1 theorem, 15 equations, 4 figures, 1 table)

This paper contains 3 sections, 1 theorem, 15 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Methods and result
  3. Computations and open problems

Key Result

Theorem 1

The volume of a hyperbolic $n$-drum ${\mathcal{D}}$ determined by parameters $r$ and $\theta$ is where the volumes $\mathop{\mathrm{Vol}}\nolimits(\tau_2)$ and $\mathop{\mathrm{Vol}}\nolimits(\tau_3)$ may be computed using formulas (eq: 1), (eq: 2), and (eq: 3).

Figures (4)

  • Figure 1: A $6$-drum drawn in the Poincaré ball model of ${\mathbb{H}}^3$. This drum corresponds to the parameters $n=6$, translation length $0.78$, and rotation angle $2\pi/9$.
  • Figure 2: The volume of the drum is the volume of the suspended drum minus twice the volume of an ideal pyramid.
  • Figure 3: Three ideal tetrahedra, drawn in the upper half space model.
  • Figure 4: The bipyramid $\widehat{{\mathcal{P}}}$ and its fundamental domain $\tau_1$.

Theorems & Definitions (1)

  • Theorem 1