Take any tetrahedron, and for each face compute its perimeter P, and the solid angle Ω it subtends as seen from the opposite vertex.
Daniil Rudenko and @robinhouston have found a linear relationship between 1/P and cot(Ω/2).
https://mathoverflow.net/questions/336464/a-curious-relation-between-angles-and-lengths-of-edges-of-a-tetrahedron/336777 …pic.twitter.com/6a8q1uzCrX
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The slope of each line is: 12 times the volume of the tetrahedron, divided by the product of its four face perimeters! The intercept is a bit more complicated. I give a formula for it at the linked Math Overflow page, but maybe it can be expressed in simpler geometric terms.
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Replying to @gregeganSF @robinhouston
That’s amazing.. it makes me wonder if it’s a 3D generalization of the famous V-E+F=2 relation ..
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But is this that generalization? (no, I didn’t click on the link..)
12:56 AM - 23 Jul 2019
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