Fred, can you provide some background and justification for this? Such as, why the solid angles at a point (if you have a whole lots of planes meeting there) add up to 4 Pi? Jim Propp On Fri, May 29, 2015 at 3:40 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:

A + B + C - pi , where A,B,C denote dihedral angles between (normals to) three planes meeting at the common point.

WFL

On 5/29/15, Henry Baker <hbaker1@pipeline.com> wrote:

...

BTW, what exactly IS a "solid angle" ?

A 2D angle is acos(A.B), where A,B are unit vectors, and A.B is the dot product.

What is the analogous quaternion formula for a solid angle ?

_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

Actually there *are* higher-dimensional versions of this kind of thing, greatly generalized. One of the earliest version is a 1943 article by Carl Allendoerfer and André Weil, "The Gauss-Bonnet Theorem for Riemannian Polyhedra". The Gauss-Bonnet was eventually taken to the max by Shiing-Shen Chern in 1945 ("The Curvatura Integra for Riemannian Manifolds"). There is a more elementary paper — that I can't locate right now — by Branko Grunbaum and a coauthor, that counts all the angles of all dimensions of a polytope and relates some kind of sum of them to the Euler characteristic. This is probably the closest thing to a direct higher-dimensional generalization of what Fred stated. ——Dan

On May 29, 2015, at 6:24 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:

This is classical spherical trigonometry --- a beautiful topic which once played an important part in navigation and astronomy, but is nowadays almost entirely neglected.

The measure of each "solid angle" is defined by the area of the corresponding"spherical triangle" bounded by great circles where your planes intersect a unit sphere, centred at their intersection.

Hence the sum of all possible solid angles is simply the area of the unit sphere, 4 pi . [If planes at a point are replaced by a convex polyhedron, dilating the sphere to infinity yields a trivial discrete version of Gauss-Bonnet.]

See https://en.wikipedia.org/wiki/Spherical_trigonometry ; note that many of the formulae carry over to the hyperbolic plane, simply by substituting cosh, sinh, ... for cos, sin, ...

The elegant theorem I quoted equates the area of a triangle to the excess of the sum of its vertex angles over that (pi) for a plane triangle. But there appears to be no analogue of this result in higher dimensions: the only formula available for the volume of a spherical tetrahedron involves a grisly multiple integral.