Seventh Rational Tetrahedron
The dihedral angles must lead to a determinant of zero within a particular matrix.
{a,b,c,d,e,f}=
π
7
,
π
3
,
4π
7
,
4π
7
,
π
3
,
3π
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}}]]
0
A set of points that work.
vert=-
1
2
,0,0,
1
2
,0,0,
-0.0678
…
,
0.552
…
,
-0.126
…
,
0.0678
…
,
0.552
…
,
0.126
…
;​​Graphics3D[Tube/@Subsets[vert,{2}]]
dist=RootReduce[EuclideanDistance@@#]&/@Subsets[vert,{2}]
1,
0.712
…
,
0.802
…
,
0.802
…
,
0.712
…
,
0.286
…

21
π
7
,
π
3
,
4π
7
,
4π
7
,
π
3
,
3π
7
Pi
{3,7,12,12,7,9}
N[21ResourceFunction["TetrahedronEdgeAngles"][dist]/Pi]
{3.,7.,12.,12.,7.,9.}
vert=-
1
2
,0,0,
1
2
,0,0,
-0.0678
…
,
0.552
…
,
-0.126
…
,
0.0678
…
,
0.552
…
,
0.126
…
;​​Graphics3D[Table[Tube/@Subsets[vert.RotationMatrix[2kPi/7,{1,0,0}],{2}],{k,0,6}]]