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Rational tetrahedra: edge lengths from given angles

Posted 3 months ago
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At stackexchange I posted a question:

In the paper Space Vectors Forming Rational Angles a special set of tetrahedra is mentioned.

"The remaining three are in the R-orbit of the tetrahedron with dihedral angles (π/7, 3π/7, π/3, π/3, 4π/7, 4π/7)."

What is a set of edge lengths or vertices for this tetrahedron? I've written a function that converts edges to angles, but I need the reverse.

I found the tetrahedron. First, the angles need to be arranged so that a particular determinant is zero. Here's one arrangement that works.

{a, b, c, d, e, f} = {\[Pi]/7, \[Pi]/3, (4 \[Pi])/7, (4 \[Pi])/7, \[Pi]/3, (3 \[Pi])/7};
RootReduce[Det[{{-1, Cos[a], Cos[b], Cos[c]},
   {Cos[a], -1, Cos[d], Cos[e]},
   {Cos[b], Cos[d], -1, Cos[f]},
   {Cos[c], Cos[e], Cos[f], -1}}]]

Then I used TetrahedronEdgeAngles on two set points and two random points, their distances, millions of random points, and annealing to boil things down to exact values.


seventh tetrahedron

I'm not sure how to do the R-orbit, but that's the first one. I'd like to get all of the rational tetrahedra and put them into something like Solid and Dihedral Angles of a Tetrahedron.

A notebook with this tetrahedron is below

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