**I. Lambert cube** $\mathfrak L(\alpha_1,\alpha_2,\alpha_3)$

In this paper (p.8), we find the volume $V$ of the hyperbolic *Lambert cube* for the special case $\alpha=\alpha_1 = \alpha_2 = \alpha_3$ as

$$V_\alpha = \int_x^\infty \ln\left(\frac1{t^2}\left(\frac{t^2-\tan^2\alpha}{1+\tan^2\alpha}\right)^3\right)\frac{dt}{t^2+1}$$

where $x$ is the positive root of $x^4-(1+3\tan^2\alpha)x^2-\tan^6\alpha =0$. Hence,

$$\begin{array}{|c|c|c|} \hline \alpha &x& V_{\alpha} & 8V_{\alpha}\\ \hline \pi/3 &\left(\frac{1+\sqrt{13}}2\right)^{3/2}& 0.324423 & \color{brown}{2.59538}\\ \pi/4 &\left(\frac{1+\sqrt{5}}2\right)^{3/2}& 0.538275 & \color{blue}{4.30620}\\ \pi/5 &\left(\frac{5-\sqrt{5}}2\right)^{3/2}& 0.658081 & 5.26465\\ \hline \end{array}$$

**II. Lobell polyhedron $L(n)$**

In this paper (p.33), we find the volume (using a different formula) of $L(5)$ as

$$\begin{array}{|c|c|c|} \hline n & L(n) & V_n\\ \hline 5 & L(5) & \color{blue}{4.30620}\\ 6 & L(6) & 6.02304\\ \hline \end{array}$$

**III. Closed hyperbolic 3-manifolds**

On a hunch, I checked those volumes in the Hodgson & Weeks census and found,

$$\begin{array}{|c|c|c|} \hline \text{Dehn filling}& \text{Symmetry} & \text{Volume}\\ \hline m160(-3, 2) & D6 & \color{brown}{2.59538}\\ \hline s648(-5, 1) & D4 & \color{blue}{4.30620}\\ s921(-3, 1) & D2 & 4.30620\\ m400( 4, 1) & Z/2 & 4.30620\\ \hline \end{array}$$

**IV. Questions**

- Why are the volumes for $\alpha =\pi/3, \pi/4$ and $L(5)$ found in the census?
- Conversely, why for $\alpha =\pi/5$ and $L(6)$ are they NOT present?