Lifelong interests in math, physics, astronomy, music, including performing (amateur). Adult lifetime interests in running, bicycling. I have benefited from some excellent tutelage in mathematics and physics, and would like to pay that forward in whatever small measure possible.
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.1 AnswerMathematics6 years ago
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!]2 AnswersMathematics7 years ago