Gosper encountered a paper (via Askey?) with a generalized law of cosines for polygons and polyhedra. The paper subsumes "my" observation that the pythagorean theorem works for right-angled tetrahedra (cut from a cube corner). The paper doesn't mention higher dimensions, but presumably it's valid. Another nicety is that cos() is even, so you don't need to worry about assigning signs to the angles. Does the law of sines works for tetrahedra? The final comment about splitting up the sides other than 1 vs N-1 should give a very simple 2-2 formula for quadrilaterals, and could lead to an angle-free formula relating the lengths of the sides and the diagonals. Presumably this would give the 2nd diagonal as a quadratic in the 4 sides and the first diagonal. And there's likely something pretty involving the sides of a pentagon and the "inscribed star". Rich ----- Quoting Bill Gosper <billgosper@gmail.com>:
for the nth side of an n-gon. Easy to remember. Per Askey. Rich: Math fun?
http://www.rowan.edu/open/colleges/las_new/departments/math/facultystaff/osl... --Bill