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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.

vert={{-(1/2),0,0},{1/2,0,0},{-0.0678\[Ellipsis],0.552\[Ellipsis],-0.126\[Ellipsis]},{0.0678\[Ellipsis],0.552\[Ellipsis],0.126\[Ellipsis]}};
Graphics3D[Tube/@Subsets[vert,{2}]]

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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