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computational geometry - How to find the vertices of a regular tetrahedron? a dodecahedron?


My question is: how to find the coordinates of the vertices of regular tetrahedron and dodecahedron? I tried to find the coordinates of the vertices of a regular tetrahedron as the solutions of a certain polynomial system in 8 variables, notating the vertices of a tetrahedron S(0,0,1), A(0,yA,zA), B(xB,yB,zB), and C(xC,yC,zC):


Reduce[

yA^2 + zA^2 == 1 &&
xB^2 + yB^2 + zB^2 == 1 && 
xC^2 + yC^2 + zC^2 == 1 && 
yA^2 + (zA - 1)^2 == xB^2 + yB^2 + (zB - 1)^2 && 
yA^2 + (zA - 1)^2 == xC^2 + yC^2 + (zC - 1)^2 && 
xB^2 + (yB - yA)^2 + (zB - zA)^2 ==
                 xC^2 + (yC - yA)^2 + (zC - zA)^2 && 
xB^2 + (yB - yA)^2 + (zB - zA)^2 ==
                 (xC - xB)^2 + (yC -yB)^2 + (zC - zB)^2 && 
xB^2 + (yB - yA)^2 + (zB - zA)^2 == 

                 yA^2 + (zA - 1)^2,
        {xB, xC, yA, yB, yC, zA, zB, zC}, Reals]

However, that code is spinning for hours without any output. A new idea is required.


P.S. 12.12.13. The answer done with Maple can be seen at http://mapleprimes.com/questions/200438-Around-Plato-And-Kepler-Again. Because nothing but trigonometry is used, I am pretty sure all that is possible in Mathematica.



Answer



Invariant theory construction


We can use Klein's invariants (Φ′ on page 55, H on page 61, Lectures on the Icosahedron) and project the complex roots onto the Riemann sphere, borrowing ubpdqn's projection code:


tetraPoly = -z1^4 - 2 Sqrt[3] z1^2 z2^2 + z2^4;
dodecaPoly = z1^20 + z2^20 - 228 (z1^15 z2^5 - z1^5 z2^15) + 494 z1^10 z2^10;


 (* project onto the Riemann sphere *)
sph[z_?NumericQ] := 
  Module[{den}, den = 1 + Re[z]^2 + Im[z]^2; {2 Re[z]/den, 2 Im[z]/den, (den - 2)/den}];

vTetra2 = sph[z1] /. Solve[(tetraPoly /. z2 -> 1) == 0, z1];

vDodeca2 = sph[z1] /. Solve[(dodecaPoly /. z2 -> 1) == 0, z1];
nf = Nearest[N@vDodeca2 -> Automatic];
edgeIndices2 = 

  Flatten[Cases[nf[vDodeca2[[#]], 4], n_ /; n > # :> {#, n}] & /@ Range[1, 19], 1];

Tetrahedron:


Graphics3D[GraphicsComplex[vTetra2,
  {Darker@Green, Thick, PointSize[Large],
   Point[Range@4],
   Line[Subsets[Range@4, {2}]]
   }]
 ]


tetrahedron


Dodecahedron:


Graphics3D[GraphicsComplex[vDodeca2,
  {Darker@Green, Thick, PointSize[Large],
   Point[Range@20],
   Line[edgeIndices2]
   }]
 ]

dodecahedron



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