Archimedean Dual

Author

Eric W. Weisstein
May 27, 2022

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Polyhedra/ArchimedeanDual.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/ArchimedeanDual.html.

©2022 Wolfram Research, Inc. except for portions noted otherwise

Enumeration

Grid[{PolyhedronData[#,"Name"],PolyhedronData[PolyhedronData[#,"DualName"],"Name"]}&/@PolyhedronData["Archimedean"],Dividers->All,Alignment->Left]//TraditionalForm

Equilateral

TextGrid[{#[[1,1]],#[[All,2]]}&/@SplitBy[Sort[{Length[PolyhedronData[#,"EdgeLengths"]],#}&/@PolyhedronData["ArchimedeanDual"]],First],Dividers->All]

Graphics

ArchimedeanDualCompound[poly_,opts___]:=Module[{
dual=Dual[Polyhedron[poly]],
lines=ClosedLine@@@VertexFigures[Polyhedron[poly]]
},
Show[GraphicsArray[
Block[{$DisplayFunction=Identity},
Show[#,Boxed->False]&/@Graphics3D/@{Polyhedron[poly],{dual,lines},{Polyhedron[poly],lines,dual}}]
],opts]
]

V5

ArchimedeanDualCompound[Cuboctahedron]

ArchimedeanDual[poly_]:=Module[{
dual=Dual[Polyhedron[poly]],
},
Block[{$DisplayFunction=Identity},
Show[#,Boxed->False]&/@Graphics3D/@{Polyhedron[poly],dual}
]
]

Show[GraphicsArray[Transpose[ArchimedeanDual/@Take[Archimedeans,7]]]]

Show[GraphicsArray[Transpose[ArchimedeanDual/@Take[Archimedeans,-6]]]]

V6

GraphicsGrid[Transpose[Show[#,Boxed->False]&/@{PolyhedronData[#],PolyhedronData[PolyhedronData[#,"DualName"]],Graphics3D[PolyhedronData[#,"DualCompound"]]}&/@PolyhedronData["Archimedean"]]]

Circumradius

<<Utilities`Typesetting`

Column[R={#,SymbolicUnion[Norm/@PolyhedronData[#,"VertexCoordinates"]]}&/@PolyhedronData["ArchimedeanDual"]]//TraditionalForm

Faces, Sides, and Vertices

Grid[PolyhedronData[#,{"Name","V","E","F"}]&/@PolyhedronData["Archimedean"],Dividers->All,Alignment->{{Left,Right,Right,Right}}]//TraditionalForm

Grid[PolyhedronData[#,{"Name","V","E","F"}]&/@PolyhedronData["ArchimedeanDual"],Dividers->All,Alignment->{{Left,Right,Right,Right}}]//TraditionalForm

Graphics Array

V6

GraphicsGrid[Partition[Show[PolyhedronData[#],Boxed->False,PlotLabel->TextCell[PolyhedronData[#,"Name"],PageWidth->100,TextAlignment->Center],BaseStyle->{Italic,10}]&/@PolyhedronData["ArchimedeanDual"],4,4,{1,1},{}],ImageSize->400,Spacings->{-80,10}]

Nets

GraphicsGrid[Partition[PolyhedronData[#,"NetImage"]&/@PolyhedronData["ArchimedeanDual"],UpTo[4]],ImageSize->400]

Wenninger plots

Show[DualEmbeddedPlot[Polyhedron[TruncatedCube],{{8,-(1+Sqrt[2])/2}},.8]]

Show[DualEmbeddedPlot[Polyhedron[TruncatedOctahedron],{{6,-Sqrt[7]/2}},.65]]

Show[DualEmbeddedPlot[Polyhedron[TruncatedDodecahedron],{{10,-3Sqrt[(25+11Sqrt[5])/10.]}},.8]]