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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.
That’s amazing.. it makes me wonder if it’s a 3D generalization of the famous V-E+F=2 relation ..
TL;DR: the alternating sum of numbers of k-dim 'cells' (points - edges + faces - solid 3D cells + ...) is a topological invariant χ.
The Euler characteristic is very cool, and has lots of nice properties and generalisations: https://en.wikipedia.org/wiki/Euler_characteristic … But I don’t think this result is connected to it.
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