New horoball packing density lower bound in hyperbolic 5-space

Abstract

We determine the optimal horoball packings of the asymptotic or Koszul-type Coxeter simplex tilings of hyperbolic 5-space, where the symmetries of the packings are derived from Coxeter groups. The packing density $\varTheta = \frac{5}{7 \zeta (3)} \approx 0.5942196502\ldots$ is optimal and realized in eleven cases in a commensurability class of arithmetic Coxeter tilings. For the optimal packing arrangements, horoballs are centered at each ideal vertex of the tiling, and horoballs of different types are used. The packings constructed give an effective proof for a new lower bound for the packing density in $\overline{{\mathbb {H}}}^5$.

Appendix: Volume of \({\widehat{AU}}_5\)

It was recently shown that the nonarithmetic Coxeter simplex ${\widehat{AU}}_5$ does not arise from a Gromov–Piatetski-Sharpiro construction [7, Ex. 6.10]; its volume was found in [9, p. 338], or as we verify, to a much lesser degree of accuracy on [9, p. 350]. To avoid ambiguity, we numerically find the volume and provide error bounds, refer to [9] for the general procedure.

The second and third order Lobachevsky functions in terms of the dilogarithm and the trilogarithm are respectively

$$\begin{aligned} \varLambda _2(z )&= \tfrac{1}{2} \text {Im} \left( \text {Li}_2\left( e^{2 i z }\right) \right) = - \int _0^{z}\ln |2 \sin (t)|dt,\\ \varLambda _3(z)&=\tfrac{1}{4} \text {Re}\left( \text {Li}_3\left( e^{2 i z }\right) \right) . \end{aligned}$$

The Coxeter diagram of ${\widehat{AU}}_5$ is with apex figure $\sum (F(t)):$

where $\tan (x_1(t)) = \csc t \sqrt{3 \sin ^2t-1} , ~~~~~ \tan (x_2(t)) = \tan t \sqrt{1-8 \cos (2 t)}.$ Introduce the auxiliary functions

$$\begin{aligned} \omega (t)&= \cos ^{-1}\left( \tfrac{\frac{\cos \left( x_2(t)\right) }{\sqrt{2}}-\cos ^2\left( x_1(t)\right) }{\sqrt{\cos ^2\left( x_1(t)\right) +\frac{1}{2}}}\right) , h(t)=\tfrac{\sqrt{\cos ^2\left( x_1(t)\right) +\cos ^2\left( x_2(t)\right) -\cos ^2(\omega (t))}}{\sin (\omega (t))},\\ \gamma _1(t)&=\cot ^{-1}\left( \sqrt{2} h(t) \cos \left( x_1(t)\right) \right) , \gamma _2(t)=\tan ^{-1}\left( \tfrac{1}{\sqrt{2} \cos \left( x_1(t)\right) }\right) ,\\ \gamma _3(t)&=\cot ^{-1}\left( \tfrac{h(t)}{\sqrt{2} \cos \left( x_1(t)\right) }\right) , \beta _1(t)=\pi - \gamma _1(t)-\omega (t), \beta _2(t)=\omega (t)-\gamma _3(t). \end{aligned}$$

To simplify, notice $\cos \left( x_1(t)\right) =\frac{\sin (t)}{\sqrt{4 \sin ^2(t)-1}}$ and $\cos \left( x_2(t)\right) =\frac{\cos (t)}{3-4 \cos ^2(t)}$. Then

$$\begin{aligned} vol_5({\widehat{AU}}_5) = \frac{1}{4}\int _{\pi /4}^{\pi /3} vol_3(F(t))dt + \varLambda _3\left( \tfrac{\pi }{3}\right) -\varLambda _3\left( \tfrac{\pi }{6}\right) + \tfrac{7}{32} \zeta (3) \end{aligned}$$

(23)

where F(t) can be decomposed into four orthoschemes [9, Eq. (3)–(4)],

$$\begin{aligned} vol_3(F(t))&= vol_3\left( R\left( \tfrac{\pi }{4}, x_1(t), \beta _1(t)\right) \right) + vol_3\left( R\left( \tfrac{\pi }{4}, \gamma _2(t), \gamma _3(t) \right) ) \right. \\&\left. \quad + vol_3(R\left( \gamma _1(t), \tfrac{\pi }{2} - \gamma _2(t), x_1(t) \right) ) + vol_3(R\left( \beta _1(t), x_2(t), x_1(t) \right) \right) . \end{aligned}$$

Here the volume of a hyperbolic 3-orthoscheme $R(\alpha _1, \alpha _2, \alpha _3)$ with $\sum (R)$ : and $\tan \theta = \frac{\sqrt{\cos ^2\alpha _2-\sin ^2\alpha _1 \sin ^2\alpha _3}}{\cos \alpha _1 \cos \alpha _3} \in [0, \tfrac{\pi }{2}]$ is given by

$$\begin{aligned} vol_3\left( R(\alpha _1, \alpha _2, \alpha _3)\right)= & {} \frac{1}{4} \left( \varLambda _2(\alpha _1 + \theta ) - \varLambda _2(\alpha _1 - \theta ) + \varLambda _2(\tfrac{\pi }{2} + \alpha _2 - \theta )+ \right. \nonumber \\&+ \left. \varLambda _2(\tfrac{\pi }{2} - \alpha _2 - \theta ) + \varLambda _2(\alpha _3 + \theta ) - \varLambda _2(\alpha _3 - \theta ) \right. \nonumber \\&\left. + 2\varLambda _2(\tfrac{\pi }{2} - \theta ) \right) . \end{aligned}$$

(24)

Table 6 Numerical approximations of the volume of ${\widehat{AU}}_5$

In the following table we compare the results of [9] with our own numerical approximation of Eq. (23) using Mathematica V. 10.0.2.0 using with integration method Trapezoidal, optional arguments PrecicionGoal set to 12, and MaxRucursion as shown in Table 6. The final row is the default output of Mathematica with no optional arguments, this agrees with that in [9, p. 338] to the precision given.