https://en.wikipedia.org/wiki/Polytope_compound
A dual compound is composed of a polyhedron and its dual, arranged reciprocally about a common midsphere, such that the edge of one polyhedron intersects the dual edge of the dual polyhedron.
A dual compound is composed of a polyhedron and its dual, arranged reciprocally about a common midsphere, such that the edge of one polyhedron intersects the dual edge of the dual polyhedron.
midpyramid
midpyramid
midpyramid places a pyramid on the midpoints of a given Polygon (and it’s the only routine to construct the dual compounds):
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midpyramid[face_,k_]:=With[{mid=Map[Mean,{face,RotateLeft[face]}]},MapThread[Polygon[{kMean[face],#1,#2}]&,{mid,RotateLeft[mid]}]]
Example:
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Graphics3D[Polygon[Map[Join[#,{1}]&,N[CirclePoints[5]]]],Lighting"Neutral",Boxed->False]/.{Polygon[p_]:>{{Red,Polygon[p]},{White,midpyramid[p,1.5]}}}
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The five Platonic polyhedrons generate only three different Dual Compounds:
The five Platonic polyhedrons generate only three different Dual Compounds:
The Tetrahedron is self dual. The dual compound is Stella Octangula:
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With[{poly=Normal[PolyhedronData["Tetrahedron","GraphicsComplex"]]},Graphics3D[{Red,poly,White,poly/.{Polygon[p_]:>midpyramid[p,3]}},Lighting"Neutral",Boxed->False,ViewProjection"Orthographic"]]
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The dual of a Hexahedron is an Octahedron:
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With[{poly=Normal[PolyhedronData["Cube","GraphicsComplex"]]},Graphics3D[{Red,poly,White,poly/.{Polygon[p_]:>midpyramid[p,2]}},Lighting"Neutral",Boxed->False,ViewProjection"Orthographic"]]
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The dual of an Octahedron is a Hexahedron:
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With[{poly=Normal[PolyhedronData["Octahedron","GraphicsComplex"]]},Graphics3D[{Red,poly,White,poly/.{Polygon[p_]:>midpyramid[p,3/2]}},Lighting"Neutral",Boxed->False,ViewProjection"Orthographic"]]
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The dual of a Dodecahedron is an Icosahedron:
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With[{poly=Normal[PolyhedronData["Dodecahedron","GraphicsComplex"]]},Graphics3D[Red,poly,White,poly/.Polygon[p_]:>midpyramid[p,
2
],Lighting"Neutral",Boxed->False,ViewProjection"Orthographic"]]Out[]=
The dual of an Icosahedron is a Dodecahedron:
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With{poly=Normal[PolyhedronData["Icosahedron","GraphicsComplex"]]},Graphics3DRed,poly,White,poly/.Polygon[p_]:>midpyramidp,3-
3
2
5
,Lighting"Neutral",Boxed->False,ViewProjection"Orthographic"The 13 Archimedean give 13 Dual Compounds:
The 13 Archimedean give 13 Dual Compounds:
The dual of a Cuboctahedron is a Rhombic Dodecahedron:
The dual of a GreatRhombicosidodecahedron is a Disdyakis Triacontahedron:
The dual of a GreatRhombicuboctahedron ia a Disdyakis Dodecahedron
The dual of an Icosidodecahedron is a Rhombic Triacontahedron:
The dual of a SmallRhombicosidodecahedron is a Deltoidal Hexecontahedron
The dual of a SmallRhombicuboctahedron is a Deltoidal Icositetrahedron
The dual of a Snub Cube (Cubus Simus) is a Pentagonal Icositetrahedron:
The dual of a SnubDodecahedron is a Pentagonal Hexecontahedron
The dual of a TruncatedCube is a Small Triakis Octahedron
The dual of a Truncated Dodecahedron is a Triakis Icosahedron
The dual of a Truncated Icosahedron is a Pentakis Dodecahedron
The dual of a Truncated Octahedron is a Tetrakis Hexahedron:
The dual of a Truncated Tetrahedron is a Triakis Tetrahedron: