Three dimensional spherical excess formula

Question

We all know the spherical excess formula: in a unit sphere, the area of a geodesic triangle is equal to the exceeding from $\pi$ of the sum of the three angles of the triangle.

Is there a similar formula for a geodesic tetrahedron in a 3-sphere? I'm sure there must be, but it seems to be tricky. Thanks in advance.

Answer 1

There are indeed generalization of the angle sums relations to spherical $d$-polytopes.
However, the generalization I know doesn't involve the volume of the polytope for odd $d$.

• Let $P$ any spherical/euclidean/hyperbolic $d$-polytope.
• Let $F_j$ be the collection of $j$-faces of $P$ for $0 \le j \le d$.
• For each face $f$ of $P$, take a interior point $p$ of $f$ and define $$\alpha_f(P) = \lim_{\epsilon \to 0}\frac{\verb/Vol/(P \cap B_p(\epsilon) )}{\verb/Vol/(B_p(\epsilon))}$$
• Define a bunch of coefficients $\alpha_{-1}(P), \alpha_{0}(P), \ldots, \alpha_{d}(P)$ by: $$\alpha_{-1}(P) = \frac{\verb/Vol/(P)}{\verb/Vol/(B(1))} ,\quad \alpha_{j}(P) = \sum_{f\in F_j} \alpha_{f}(P)\quad\text{ for } 0 \le j \le d $$ where $B(1)$ is the whole sphere when $P$ is spherical and any unit $d$-ball (I believe) when $P$ is hyperbolic.

We have

Generalized Gram relations (Grünbaum, Sommerville, Heckman) $$ \sum_{j = 0}^d (-1)^j \alpha_j(P) = \varepsilon^{d/2}(1 + (-1)^d)\alpha_{-1}(P) \quad\text{ where }\quad \varepsilon = \begin{cases} \;1, & P \quad\text{ spherical }\\ \;0, & P \quad\text{ euclidean }\\ -1, & P \quad\text{ hyperbolic } \end{cases}$$

For the special case of tetrahedron at $d = 3$, independent of whether it is spherical/euclidean/hyperbolic, above formula reduces to

$$\sum_{v \in V} \frac{\Omega_v}{4\pi} - \sum_{e \in E} \frac{\theta_e}{2\pi} +\frac{|F|}{2} - 1 = 0$$ where $V, E, F$ are the set of vertices/edges/facets of tetrahedron, $\Omega_v$ is the solid angle at vertex $v$ and $\theta_e$ is the dihedral angle at edge $e$.

As one can see, this formula doesn't involve the volume for odd number $d = 3$.

I don't really know this stuff, above formula is extracted from the paper Angle sums on polytopes and polytopal complexes by Kristin A. Camenga. Look at $\S 4.2$ there for a more accurate description of the relations and references there for more details.

Answer 2

As usual, it is interesting to consult the great masters, at a condition, that they are readable. John Milnor is one of these. He is always clear.

I advise you to read "How to Compute Volume in Hyperbolic space" in "John Milnor. Collected papers. Volume1. Geometry. pages 189-212. Publish or Perish Editor 1994.