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In plane geometry, knowing all the side lengths of a polygon of m sides isn't enough, by itself, to determine the area; unless m=3, in which case there is Heron's formula:

A = √[s(s - a)(s - b)(s - c)], where s (the semiperimeter) = ½(a + b + c)

In solid geometry, a similar situation exists for the tetrahedron -- knowing all 6 edge lengths is sufficient to completely determine its shape, and thus, its volume.

A) What is the Heron-like formula for that?

B) What is the formula for a simplex (hyperpyramid) in n dimensions, given all ½n(n+1) edge lengths?

Unlike my first Y!A question, I don't have prior knowledge of the answer to this.

And elegance/simplicity in the final expression will get extra consideration.

3 decades ago, I was fiddling with simulations of 2-body gravitational orbits on a dec Pro-350 in compiled BASIC, and stumbled across a remarkable feature that all my years of math and physics education (including a graduate-level course in celestial mechanics!) had failed to divulge. At the time, I was able to verify the apparent result mathematically, but I have not been able to reconstruct the proof.

As you well know, these orbital trajectories (in position space) are conic sections. (Is this starting to sound eerily similar to a certain episode in Newton’s development of the Principia?) But what about in velocity space? That is, if you plot each velocity vector as a point in the xy-plane, what curve does it trace, throughout the entire orbit?

All are hereby invited to characterize the velocity-space trajectories that result from an infinitesimal mass in motion near an inverse-square, gravitational source, whether in bound or unbound orbit, and to prove your result. Be sure to put on your crafts(wo)man’s visor, because elegance counts here.

[After almost 4 years on this site, this is my first question here. Hope it’s a good one!]