CITE THIS NOTEBOOK: The compound of ten tetrahedra by Oliver Seipel. Wolfram Community JAN 5 2023.
The compound of ten tetrahedra is one of the five regular polyhedral compounds. This polyhedron can be seen as either a stellation of the icosahedron or a compound of ten tetrahedra. This geometrical object was first described by Edmund Hess in 1876.
I never was really satisfied with the tetrahedron ten compounds I saw. This notebook is the result of my efforts:)

Mathematica code:
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Card-Boards:

Stellation diagram of the tetrahedron ten compound:
In[]:=
With[{cp=CirclePoints[9],pts={9,1,3,4,6,7}},Graphics[DoubleTriangle[cp[[pts]]],ImageSize->300]]
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Stellation diagrams L- and R-form:
In[]:=
Grid[{{With[{cp=CirclePoints[9],pts={9,1,3,4,6,7}},Graphics[{EdgeForm[Black],White,DoubleTriangleR[cp[[pts]]],Black,DoubleTriangleL[cp[[pts]]]},ImageSize->250]],With[{cp=CirclePoints[9],pts=Reverse[{9,1,3,4,6,7}]},Graphics[{EdgeForm[Black],White,DoubleTriangleR[cp[[pts]]],Black,DoubleTriangleL[cp[[pts]]]},ImageSize->250]]}}]
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Tetrahedron Ten Compound:

In[]:=
tetrahedrontencompound=Graphics3D[{EdgeForm[Black],Map[DoubleTriangle[vc[[#]]]&,{{9,17,7,11,3,1},{2,4,10,6,16,12},{18,10,1,3,12,8},{4,2,11,15,5,9},{15,3,6,10,17,19},{14,16,11,7,4,18},{3,16,20,18,9,5},{17,4,8,12,15,13},{7,2,20,14,3,15},{18,4,19,13,1,6},{2,7,13,5,10,18},{6,1,15,11,8,20},{13,15,16,14,10,9},{18,20,12,11,19,17},{20,8,7,19,9,10},{5,13,11,12,14,6},{2,8,16,3,13,19},{5,1,14,20,4,17},{6,14,8,2,17,9},{19,7,12,16,1,5}}]},Boxed->False,Lighting->"Neutral",ViewProjection"Orthographic",ViewPoint->3Mean[vc[[{4,2,7,19,17}]]],ViewVertical{0,0,1}]
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Tetrahedron ten compound without shading:
In[]:=
Show[tetrahedrontencompound,Lighting{{"Ambient",White}}]
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L-Tetrahedron Five Compound inside the Ten Compound

Every tetrahedron of the L-five compound has a unique color:
In[]:=
tencompoundL=Graphics3D[{EdgeForm[Black],Map[{Blue,DoubleTriangleL[vc[[Reverse[#]]]],White,DoubleTriangleR[vc[[Reverse[#]]]]}&,{{2,7,13,5,10,18},{2,4,10,6,16,12},{2,8,16,3,13,19},{13,15,16,14,10,9}}],Map[{Yellow,DoubleTriangleL[vc[[Reverse[#]]]],White,DoubleTriangleR[vc[[Reverse[#]]]]}&,{{4,2,11,15,5,9},{14,16,11,7,4,18},{5,1,14,20,4,17},{5,13,11,12,14,6}}],Map[{Green,DoubleTriangleL[vc[[Reverse[#]]]],White,DoubleTriangleR[vc[[Reverse[#]]]]}&,{{18,10,1,3,12,8},{19,7,12,16,1,5},{18,4,19,13,1,6},{18,20,12,11,19,17}}],Map[{Red,DoubleTriangleL[vc[[Reverse[#]]]],White,DoubleTriangleR[vc[[Reverse[#]]]]}&,{{6,14,8,2,17,9},{15,3,6,10,17,19},{6,1,15,11,8,20},{17,4,8,12,15,13}}],Map[{Orange,DoubleTriangleL[vc[[Reverse[#]]]],White,DoubleTriangleR[vc[[Reverse[#]]]]}&,{{3,16,20,18,9,5},{9,17,7,11,3,1},{20,8,7,19,9,10},{7,2,20,14,3,15}}]},Boxed->False,Lighting->"Neutral",ViewProjection"Orthographic",ViewPoint->3Mean[vc[[{3,1,5,13,15}]]],ViewVertical{0,0,-1}]
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Every tetrahedron of the L-five compound inside the ten compound has the same color:
Out[]=
Reverse color:
L-Tetrahedron ten compound without shading:

R-Tetrahedron Five Compound inside the Ten Compound

Every tetrahedron of the R-five compound inside the ten compound has unique color:
Every tetrahedron of the R-five compound inside the ten compound has the same color:
Reverse color:
R-Tetrahedron ten compound without shading:

Symmetric Faceting:

Symmetric coloring of the tetrahedron ten compound:
Symmetric coloring of the tetrahedron ten compound II:
Reverse color:
Symmetric-Tetrahedron ten compound without shading:

Adam Wyss Ten Compound Hull

Adam Wyss Hull without shading:
References: Paul Adam, Arnold Wyss, “Platonische und Archimedische Körper, ihre Sternformen und polaren Gebilde” 1984. (Platonic and Archimedean Polyhedrons, their Stellations and Duals)
​https://en.wikipedia.org/wiki/Compound_of_ten _tetrahedra​
​https://en.wikipedia.org/wiki/Compound_of_five _tetrahedra​
​https://en.wikipedia.org/wiki/Edmund_Hess