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CITE THIS NOTEBOOK: Dodecahedron small triambic icosahedron compound by Oliver Seipel. Wolfram Community JAN 8 2022.

Ἀγεωμέτρητος μηδεὶς εἰσίτω.
Ageōmetrētos mēdeis eisitō.
“No one should enter without knowledge of the geometry.”
Inscription above the entrance of Plato’s Academy according to the testimony of the Neoplatonists.

Small Triambic Icosahedron

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Show[PolyhedronData["SmallTriambicIcosahedron"],Lighting->"Neutral",Boxed->False]

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Here is a construction of the Small Triambic Icosahedron from a simpler polyhedron, the Icosahedron:

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icosahedron=Graphics3D[{N[PolyhedronData["Icosahedron","Polygons"]]},Lighting->"Neutral",Boxed->False]

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Here is faceting the triangles for the purpose of building a pyramid:

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PyramidizeFace[p_,r_]:=MapThread[Polygon[{rMean[p],#1,#2}]&,{p,RotateLeft[p]}]

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icosahedron/.{Polygon[p_]:>PyramidizeFace[p,1]}

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Taking a streching factor greater then 1:

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smalltriambicicosahedron=icosahedron/.Polygon[p_]:>PyramidizeFacep,N



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And here the streching factor is to large (but the result may look familiar to you):

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icosahedron/.{Polygon[p_]:>PyramidizeFace[p,N[2]]}

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Dodecahedron Small Triambic Icosahedron Compound

This is a compound of the above Small Triambic Icosahedron and a Dodecahedron:

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DodecahedronSmallTriambicIcosahedronCompound=Show[Normal[PolyhedronData["DodecahedronSmallTriambicIcosahedronCompound"]],Boxed->False]

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Here are the compounds separated:

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With[{polys=First@First@DodecahedronSmallTriambicIcosahedronCompound},GraphicsRow[{Graphics3D[Take[polys,60],Boxed->False,Lighting->"Neutral"],Graphics3D[Take[polys,-12],Boxed->False,Lighting->"Neutral"]},ImageSize->500]]

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Here we construct only the visible part of the dodecahedron (in the right scaling for later use):

Here only the Dodecahedron of the compound is cutted:

Here we construct only the visible parts of the Small Triambic Icosahedron:

This is the faceting of the triangles of the Ixosahedron:

Here we strech them in the plane accordingly:

And here we use both streching parameters with the right values:

And this happened with “wrong” streching values:

We use the dodecut from above to make the compound. All inner parts are cut away:

Postscriptum

A special pyramid is built here:

negative carving:

starting from any archimedean: