Angles and volumes of regular polytopes in geometries of constant curvature

Source abstract

We derive closed-form expressions for the internal and external angles of $d$-dimensional cubes, regular simplices and regular crosspolytopes in geometries of constant sectional curvature $κ\in \mathbb R$. More generally, we determine internal and external angles at arbitrary faces of rectangular boxes, acute orthocentric simplices, rectangular orthocentric simplices and asymmetric crosspolytopes in arbitrary dimension $d$. We also characterize Riemannian tangent and normal cones of these polytopes, up to isometry. Combining internal angle formulas with the Poincaré relation, we derive formulas for the Riemannian volume of these polytopes if the dimension $d$ is even. All formulas are stated in terms of the standard normal distribution function $Φ(x)$ and its imaginary version $Φ({\rm{i}} x)$. For example, if $d\geq 2$ is even, then the hyperbolic volume of the ideal regular simplex in the $d$-dimensional hyperbolic space of curvature $κ= -1$ is $$\frac{π^{d/2}} {\sqrt{2}\, {\rm{i}}^{d}\, Γ\left(\frac{d+1}{2}\right)} \int_{-\infty}^{\infty} \left[ Φ\left(\frac{{\rm i} y}{\sqrt d}\right)^{d+1} + Φ\left(-\frac{{\rm i} y}{\sqrt d}\right)^{d+1} \right] {\rm e}^{-y^2/2} {\rm d} y.$$