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There is a nice formula for the area of a triangle on the surface of the 2-dimensional sphere; If the triangle is the intersection of three half spheres, and has angles $\alpha$, $\beta$ and $\gamma$, and we normalize the area of the whole sphere to be $4\pi$ then the area of the triangle is $$ \alpha + \beta + \gamma - \pi. $$ The proof is a cute application of inclusion-exclusion of three sets, and involves the fact that the area we want to calculate appears on both sides of the equation, but with opposite signs.

However, when trying to copy the proof to the three dimensional sphere the parity goes the wrong way and you get 0=0.

Is there a simple formula for the volume of the intersection of four half-spheres of $S^3$ in terms of the 6 angles between the four bounding hyperplanes?

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2 Answers 2

up vote 17 down vote accepted

On the volume of a hyperbolic and spherical tetrahedron, by Murakami and Yano. The volume is obtained as a linear combination of dilogarithms and squares of logarithms. The origin of their formula is really interesting: Asymptotics of quantum $6j$ symbols. (These asymptotics have also been studied by many other people: D. Thurston, Roberts, Woodward, Frohman, Kania-Bartoszynska, etc.)

Note that the 3-dimensional formula has to be much more complicated. The 2-dimensional formula comes from Euler characteristic and Gauss-Bonnet, but the Euler characteristic of the 3-sphere, or any odd-dimensional manifold, vanishes. In fact every characteristic class of a 3-sphere vanishes, because the tangent bundle is trivial. There can't be a purely linear treatment of volumes in isotropic spaces in odd dimensions. In even dimensions, there is always a purely linear extension from lower dimensions using generalized Gauss-Bonnet.

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The asymptotics of quantum 6j symbols, oddly enough, is mentioned in the answers to a question from a couple days ago: mathoverflow.net/questions/7258/… . –  Michael Lugo Dec 1 '09 at 19:19
Right, that's one reason that I recognized that Google hit as a good answer. These asymptotics also an important business that I wish I understood better. –  Greg Kuperberg Dec 1 '09 at 19:26
I printed out this thread and posted it on the department bulletin board. So, next we need to form the push-out of this thread's idea and this one's: mathoverflow.net/questions/6160/… –  Ryan Budney Dec 4 '09 at 7:08

Nice answer, Greg. I looked at the linked paper and was sufficiently intimidated. I just want to point out, again, that for those (like me) who have a phobia of differential geometry, and hence don't want to use (generalized)Gauss-Bonnet, it is easy to see, using inclusion-exclusion, that the formula in even dimensions is a neat linear combination of the formulas in lower dimensions.

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