Archimedean Solid

Author

Eric W. Weisstein
December 13, 2018

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

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

©2018 Wolfram Research, Inc. except for portions noted otherwise

Solids

GraphicsGrid[Partition[Graphics3D[#,Boxed->False,Method->{"ShrinkWrap"->True}]&/@PolyhedronData["Archimedean","GraphicsComplex"],UpTo[5]]]

GraphicsGrid[Partition[Graphics3D[Tooltip[PolyhedronData[#,"GraphicsComplex"],#],Boxed->False,Method->{"ShrinkWrap"->True}]&/@PolyhedronData["Archimedean"],UpTo[5]]]

GraphicsGrid[Partition[Show[PolyhedronData[#],Boxed->False,PlotRange->PolyhedronData[#,"Circumradius"],PlotLabel->TextCell[PolyhedronData[#,"Name"],PageWidth->90,TextAlignment->Center],BaseStyle->{FontFamily->"Times",Italic,10}]&/@PolyhedronData["Archimedean"],UpTo[5]],ImageSize->500,Spacings->{-40,10}]

Nets

Grid[Partition[PolyhedronData["Archimedean","Net"],UpTo[5]]]

Grid[Partition[Tooltip[PolyhedronData[#,"Net"],#]&/@PolyhedronData["Archimedean"],UpTo[5]]]

Tables

Faces, Edges, Vertices

TextGrid[PolyhedronData["Archimedean",{"Name","F","E","V"}],Dividers->All,Alignment->{{Left,{Decimal}}}]

Inradius, Midradius, Circumradius

TextGrid[PolyhedronData["Archimedean",{"Name","Inradius","Midradius","Circumradius"}],Dividers->All,Alignment->{{Left,{Center}}}]//TraditionalForm

Radius relations

<<Utilities`Simplify`;
Off[$MaxExtraPrecision::"meprec"];

ns=Flatten[Position[Table[Inradius[Archimedean[n]],{n,13}],_?NumericQ,{1},Heads->False]]

duals=Map[ToExact,Dual[Flatten[Polyhedron[Archimedean[#]]]],{3}]&/@ns;//Timing

TableForm[dualradii=
RootReduce/@ToExact/@Through[{Inradius,Midradius,Circumradius}[duals[[#]]]]&/@ns,TableHeadings->{{PolyhedronName[Archimedean[n]]<>" dual = "<>PolyhedronName[ArchimedeanDual[n]]}&/@ns,{(Subscript[r, d]),(Subscript[ρ, d]),(Subscript[R, d])}}]//TraditionalForm

ρ=(Subscript[ρ, d])

Equal@@@Transpose[Map[#[[2]]&,{radii,dualradii},{2}]]

Map[#[[2]]&,{radii,dualradii},{2}]//TableForm//TraditionalForm

R (Subscript[r, d])=(ρ^2)

FullSimplify[Times@@Drop[#,-1]-Last[#]^2&/@Transpose[{Last/@radii,First/@dualradii,#[[2]]&/@radii}]]

R=(1/2) ((Sqrt[(a^2)+(Subscript[r, d]^2)])+(Subscript[r, d]))

FullSimplify[#[[1,3]]-(#[[2,1]]+Sqrt[#[[2,1]]^2+1])/2&/@Transpose[{radii,dualradii}]]

R=(Sqrt[((a^2)/4)+(ρ^2)])

FullSimplify[#[[1,3]]-Sqrt[1/4+#[[1,2]]^2]&/@Transpose[{radii,dualradii}]]

(Subscript[r, d])=((ρ^2)/(Sqrt[((a^2)/4)+(ρ^2)]))

FullSimplify[#[[2,1]]-#[[1,2]]^2/Sqrt[1/4+#[[1,2]]^2]&/@Transpose[{radii,dualradii}]]

(Subscript[r, d])=(((R^2)-((a^2)/4))/R)

FullSimplify[#[[2,1]]-(#[[1,3]]^2-1/4)/#[[1,3]]&/@Transpose[{radii,dualradii}]]

ρ=(1/2) (Sqrt[2]) (Sqrt[(Subscript[r, d]^2)+(Sqrt[(a^2)+(Subscript[r, d]^2)]) (Subscript[r, d])])

FullSimplify[#[[1,2]]-Sqrt[#[[2,1]]^2+Sqrt[1+#[[2,1]]^2]#[[2,1]]]Sqrt[2]/2&/@Transpose[{radii,dualradii}]]