Cube 20-Compound

Author

Eric W. Weisstein
September 22, 2023

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Polyhedra/Cube20-Compound.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/Cube20-Compound.html.

©2023 Wolfram Research, Inc. except for portions noted otherwise

Variants

variants=PolyhedronData["CubeTwentyCompound*"]

Polyhedra

GraphicsGrid[Transpose[{
Graphics3D[{PolyhedronData[#,"Polyhedron"]},Boxed->False],
With[{poly=PolyhedronData[#,"Polyhedra"]},Graphics3D[Table[{Hue[(i/(Length[poly]))],poly[[i]]},{i,Length[poly]}],Boxed->False]]
}&/@variants],Dividers->All,ImageSize->500]

Solids and duals

GraphicsGrid[
Table[
With[
{
s={Yellow,PolyhedronData[pname,"Midsphere"]},
dname=PolyhedronData[pname,"Dual","Name"],
style=Directive[Black,Italic,FontFamily->"Times",14]
},
{
Graphics3D[{s,Green,Opacity[.5],PolyhedronData[pname,"Polyhedron"]},Boxed->False,PlotLabel->TextCell[Style[PolyhedronData[pname,"Name"],style],PageWidth->200,TextAlignment->Center]],

Graphics3D[{s,Red,Opacity[.5],PolyhedronData[pname,"Dual","Polyhedron"]},Boxed->False,PlotLabel->Style[PolyhedronData[dname,"Name"],style]],

Graphics3D[{
{Green,Opacity[.5],PolyhedronData[pname,"Polyhedron"]},
{Red,Opacity[.5],PolyhedronData[pname,"Dual","Polyhedron"]},
s
},Boxed->False,PlotLabel->Style["compound with midsphere",style]],

Graphics3D[{
PolyhedronData[pname,"Polyhedron"],
PolyhedronData[pname,"Dual","Polyhedron"]
},Boxed->False,PlotLabel->Style["compound",style]]
}],
{pname,variants}
],
Dividers->All,ImageSize->800,Alignment->Top
]

Interior and convex hull

GraphicsGrid[
With[{style=Directive[Black,Italic,FontFamily->"Times",16]},
Table[{
Graphics3D[PolyhedronData[pname,"Polyhedron"],PlotLabel->TextCell[Style[PolyhedronData[pname,"Name"],style],PageWidth->200,TextAlignment->Center],Boxed->False],
Graphics3D[{AbsoluteThickness[.2],PolyhedronData[pname,"Lines"],Opacity[.8],PolyhedronData[pname,"CompoundInterior","Polyhedron"]},PlotLabel->Style["interior",style],Boxed->False],
Graphics3D[{
{EdgeForm[None],PolyhedronData[pname,"Polyhedron"]},
{Opacity[.7],EdgeForm[Thickness[.005]],PolyhedronData[pname,"ConvexHull","Polyhedron"]}},PlotLabel->Style["convex hull",style],Boxed->False]
},
{pname,variants}]
],
Dividers->All,ImageSize->700]

Polyhedron: {"CubeTwentyCompound", 1}

PolyhedronData[{"CubeTwentyCompound",1},"Orientations"]

Graphics3D[PolyhedronData[{"CubeTwentyCompound",1},"Polyhedron"],Boxed->False]

Show[PolyhedronData[{{"CubeTwentyCompound",1},#}],Boxed->False]&/@PolyhedronData[{"CubeTwentyCompound",1},"Orientations"]

Polyhedron

Graphics3D[{FaceForm[Green,Red],PolyhedronData[{"CubeTwentyCompound",1},"Polyhedron"]}]

Region

PolyhedronData[{"CubeTwentyCompound",1},"Region"]

Solid and Dual

PolyhedronData[{"CubeTwentyCompound",1},#]&/@{"Graphics3D","Dual","DualCompound"}

Show[Most@%]

Polyhedron: {"CubeTwentyCompound", 2}

PolyhedronData[{"CubeTwentyCompound",2},"Orientations"]

Graphics3D[PolyhedronData[{"CubeTwentyCompound",2},"Polyhedron"],Boxed->False]

Show[PolyhedronData[{{"CubeTwentyCompound",2},#}],Boxed->False]&/@PolyhedronData[{"CubeTwentyCompound",1},"Orientations"]

Polyhedron

Graphics3D[{FaceForm[Green,Red],PolyhedronData[{"CubeTwentyCompound",2},"Polyhedron"]}]

Region

PolyhedronData[{"CubeTwentyCompound",2},"Region"]

Solid and Dual

PolyhedronData[{"CubeTwentyCompound",2},#]&/@{"Graphics3D","Dual","DualCompound"}

Show[Most@%]

Properties : {"CubeTwentyCompound", 1}

Initialization

<<MathWorld`Polyhedra`

p=PolyhedronData[pname={"CubeTwentyCompound",1},"Polyhedron"]

AdjacentFaceIndices

adj1=PolyhedronData[pname,"AdjacentFaceIndices"]

adj2=AdjacentFaceIndices[p]

adj1===adj2

Centroid

PolyhedronData[pname,"Centroid"]

Circumsphere

sphere=PolyhedronData[pname,"Circumsphere"]

sphere=Circumsphere[p]

(sphere=MyCircumsphere[p])//Timing

Graphics3D[{{Opacity[.2],Yellow,sphere},p},Boxed->False]

Classes

PolyhedronData[pname,"Classes"]

Compound

PolyhedronData[pname,"Compound"]

CompoundInterior

intname=PolyhedronData[pname,"CompoundInterior","Name"]

PolyhedronData[pname,"CompoundInterior","Scale"]

PolyhedronData[pname,"CompoundInterior","RotationMatrix"]

PolyhedronData[pname,"CompoundInterior"]

Graphics3D[{
AbsoluteThickness[.4],PolyhedronData[pname,"Lines"],
Opacity[.8],interior=PolyhedronData[pname,"CompoundInterior","Polyhedron"]},Boxed->False]

MyCircumsphere[interior]//Quiet

Insphere[interior]//RootReduce//Quiet

Midsphere[interior]

Construction

int=RegionIntersection[Sequence@@PolyhedronData[pname,"Polyhedra"],PerformanceGoal->"Quality"]

Graphics3D[int]

Graphics3D[{{Opacity[0],p},Red,int},Boxed->False]

Union[Chop@int[[1]]]//Length

Exactify

vexact=DeleteDuplicates[RootApproximant[Chop[int[[1]]]]]

Length[vexact]

inthull=ConvexHullRegion[Union@vexact]

v=inthull[[1]]

faces=inthull[[2]]

gname=RecognizeGraph@PolyhedronGraph[inthull]

GraphData[gname,"PolyhedralEmbeddings"]

SimplifyNumericallyDistinct[PolyhedronEdgeLengths[inthull]]//Counts//Quiet

((Keys[%])/(Min[Keys[%]]))//RootReduce

SimplifyNumericallyDistinct[PolyhedronEdgeLengths[PolyhedronData["DisdyakisTriacontahedron","Polyhedron"]]]//Counts

Numerically group

Union[Chop@int[[1]]]//Length

Length[grouped=SplitBy[SortBy[{#,Round[1*^12#]}&/@Chop[int[[1]]],Last],Last][[All,All,1]]]

rules=Flatten[Function[l,Rule[#,Mean[l]]&/@l]/@grouped,1];

newi=ConvexHullRegion[DeleteDuplicates[Chop[int[[1]]]/.rules]]

Midsphere[newi]

RecognizeGraph@PolyhedronGraph@newi

lengths=PolyhedronEdgeLengths[newi]

Divide@@MinMax[%lengths]

Round[10^12lengths]/10^12.//Counts

PolyhedronData["PentakisDodecahedron","EdgeLengths"]

Divide@@%//N

RootReduce[PolyhedronEdgeLengths[PolyhedronData["PentakisDodecahedron","Polyhedron"]]]//Counts

newi[[1]]

newi[[2]]

RootApproximant[((newi[[1]])/(Max[PolyhedronEdgeLengths[newi]])),4]

Graphics3D[{
AbsoluteThickness[1],PolyhedronData[pname,"Lines"],
Opacity[.8],newi},Boxed->False]

Find scale

intname=XXX;

PolyhedronData/@{pname,intname}

xyratio=((Max[Norm/@vexact[[All,;;2]]])/(Max[Norm/@PolyhedronData[intname,"VertexCoordinates"][[All,;;2]]]))//RootReduce//FullSimplify//Quiet

zratio=((Max[vexact[[All,3]]])/(Max[PolyhedronData[intname,"VertexCoordinates"][[All,3]]]))//RootReduce//FullSimplify

{xyratio,xyratio,zratio}

Concave

PolyhedronData[pname,"Concave"]

Convex

PolyhedronData[pname,"Convex"]

ConvexPolyhedronQ[p]

ConvexHull

convexhullname=PolyhedronData[pname,"ConvexHull","Name"]

((Max[Norm/@p[[1]]])/(Max[Norm/@PolyhedronData[convexhullname,"Polyhedron"][[1]]]))//FullSimplify//Quiet

PolyhedronData[pname,"ConvexHull","Scale"]

PolyhedronData[pname,"ConvexHull"]

convexhull=ConvexHullRegion[PolyhedronData[pname,"Vertices"]]

v=PolyhedronCoordinates[convexhull]

Sort[(convexhull[[2]]/.i_Integer:>v[[i]])/.Thread[PolyhedronData[pname,"Vertices"]->Range[PolyhedronData[pname,"V"]]]]

Graphics3D[{FaceForm[Green,Red],convexhull,MapIndexed[Text[#2[[1]],#,Background->White]&,v]}]

{Graphics3D[{Red,Opacity[.5],p}],Graphics3D[{Blue,Opacity[.5],PolyhedronData[pname,"ConvexHull"][[1]]}]}

Show[%]

Graphics3D[{p,MapIndexed[Text[#2[[1]],#,Background->White]&,p[[1]]]}]

DehnInvariant

PolyhedronData[pname,"DehnInvariant"]

(dehn=DehnInvariant[p,Debug->True])//Timing

ToConwayBasis[dehn]

DihedralAngles

(dih1=PolyhedronData[pname,"DihedralAngles"])//Tally

(dih2=SimplifyNumericallyDistinct[DihedralAngles[p]])//Tally

dih1===dih2

Dual

pname

dname=PolyhedronData[pname,"Dual","Name"]

PolyhedronData[pname,"Dual","Scale"]

midrs={PolyhedronData[pname,"Midradius"],PolyhedronData[dname,"Midradius"]}

Divide@@midrs//FullSimplify

GraphicsRow[With[{
s={Yellow,PolyhedronData[pname,"Midsphere"]},
style=Directive[Black,Italic,FontFamily->"Times",14]
},
{
Graphics3D[{s,Green,Opacity[.2],PolyhedronData[pname,"Polyhedron"]},Boxed->False,PlotLabel->TextCell[Style[PolyhedronData[pname,"Name"],style],PageWidth->200,TextAlignment->Center]],
Graphics3D[{s,Red,Opacity[.1],PolyhedronData[pname,"Dual","Polyhedron"]},Boxed->False,PlotLabel->Style[PolyhedronData[dname,"Name"],style]],
Graphics3D[{
{Green,Opacity[.1],PolyhedronData[pname,"Polyhedron"]},
{Red,Opacity[.1],PolyhedronData[pname,"Dual","Polyhedron"]},
s
},Boxed->False,PlotLabel->Style["compound with midsphere",style]],
Graphics3D[{
PolyhedronData[pname,"Polyhedron"],
PolyhedronData[pname,"Dual","Polyhedron"]
},Boxed->False,PlotLabel->Style["compound",style]]
}],Alignment->Top,ImageSize->800]

EdgeCount

PolyhedronData[pname,"EdgeCount"]

EdgeLengths

Tally[PolyhedronEdgeLengths[p]//RootReduce]

PolyhedronData[pname,"EdgeLengths"]

PolyhedronData[pname,"Lines"]/.Line[l_]:>EuclideanDistance@@l//FullSimplify//Union//Quiet

lens=SimplifyNumericallyDistinct[PolyhedronEdgeLengths[p]];//Quiet

(distinctlens=Mean/@Split[Sort[lens],Abs[#1-#2]<1.*^-10&])//N//Quiet

v=p[[1]];

colors=With[{n=Length[distinctlens]},Table[Hue[i/n],{i,0,n-1}]]

Graphics3D[{Thick,Sequence@@Nearest[Thread[distinctlens->colors],EuclideanDistance@@v[[#]]],Line[v[[#]]]}&/@PolyhedronEdges[p],Boxed->False]//Quiet

FaceCount

PolyhedronData[pname,"FaceCount"]

GeneralizedDiameter

PolyhedronData[pname,"GeneralizedDiameter"]

GeneralizedDiameter[p]//FullSimplify//Quiet

Hull

hullname=PolyhedronData[pname,"Hull","Name"]

PolyhedronData[pname,"Hull","Scale"]

((Max[Norm/@p[[1]]])/(Max[Norm/@PolyhedronData[hullname,"Polyhedron"][[1]]]))//FullSimplify//Quiet

PolyhedronData[pname,"Hull"]

Max[Norm/@PolyhedronData[pname,"Hull","Polyhedron"][[1]]]

Max[Norm/@p[[1]]]

{Graphics3D[{Opacity[.5],Red,p}],Graphics3D[{Opacity[.5],Blue,PolyhedronData[pname,"Hull"][[1]]}]}

Show[%]

InertiaTensor

PolyhedronData[pname,"InertiaTensor"]

Insphere

PolyhedronData[pname,"Insphere"]

Insphere[p]

sphere=MyInsphere[p]

Graphics3D[{{Red,sphere}}]

Graphics3D[{{Red,sphere},Opacity[.05],p},Boxed->False]

MeanCylindricalRadius

PolyhedronData[pname,"MeanCylindricalRadius"]

MeanSquareCylindricalRadius

PolyhedronData[pname,"MeanSquareCylindricalRadius"]

MeanSphericalRadius

PolyhedronData[pname,"MeanSphericalRadius"]

MeanSquareSphericalRadius

PolyhedronData[pname,"MeanSquareSphericalRadius"]

Midsphere

sphere=PolyhedronData[pname,"Midsphere"]

(sphere=Midsphere[p])//Timing

(sphere=Midsphere[p,Method->"PointLineDistance"])//Timing

Graphics3D[{{Opacity[.1],Yellow,p},{Red,sphere}},Boxed->False]

Net

PolyhedronData[pname,"Net"]

NotationRules

PolyhedronData[pname,"NotationRules"]

Orientations

PolyhedronData[pname,"Orientations"]

PolyhedronData[pname,"DefaultOrientation"]

Show[PolyhedronData[{pname,#}],Boxed->False]&/@PolyhedronData[pname,"Orientations"]

Polygons

PolyhedronData[pname,"FaceCount"]

Graphics3D/@PolyhedronData[pname,"Polygons"]

Show[%]

Polyhedra

polyhedra=PolyhedronData[pname,"Polyhedra"]

Length[polyhedra]

Graphics3D[polyhedra]

Graphics3D[Table[{Hue[(i/(Length[polyhedra]))],polyhedra[[i]]},{i,Length[polyhedra]}]]

PolyhedronData[pname,"PolyhedronIndices"]

Region

PolyhedronData[pname,"Region"]

RegionFunction

PolyhedronData[pname,"RegionFunction"]

SimplePolyhedron

PolyhedronData[pname,"Simple"]

SimplePolyhedronQ[p]

Skeleton

PolyhedronData[pname,"Skeleton"]

gn=PolyhedronData[pname,"Skeleton","Name"]

emb=GraphData[gn,"PolyhedralEmbeddings"]

l={{gn,{0,Union[emb,{pname}]}}}

PropertyReplace[l,"PolyhedralEmbeddings","14.0"]

With[{gn=pname,verts=PolyhedronData[pname,"Vertices"],faces=PolyhedronData[pname,"Faces","List"]},
ToCommonEdges[GraphData[gn],Graph[Range[Length[verts]],UndirectedEdge@@@Union[Sort/@Flatten[Partition[#,2,1,1]&/@faces,1]]]]]//Chop[#,1*^-5]&//N

SurfaceArea

PolyhedronData[pname,"SurfaceArea"]

SurfaceArea[p]

Total[Area/@PolyhedronData[pname,"Polygons"]]//RootReduce

SurfaceArea[PolyhedronData[pname,"Region"]]

VertexCount

PolyhedronData[pname,"VertexCount"]

VertexSubsetHulls

subsethulls=PolyhedronData[pname,"VertexSubsetHulls"]

v=PolyhedronData[pname,"Vertices"];

Length[Lookup[subsethulls,#]]&/@{"Cube","Dodecahedron","Tetrahedron"}

Graphics3D[ConvexHullRegion[v[[#]]]&/@Lookup[subsethulls,#]]&/@{"Cube","Dodecahedron","Tetrahedron"}

subsethulls2=Flatten[Function[{pn,l},Rule[pn,#]&/@l]@@@subsethulls]

v=PolyhedronData[pname,"Vertices"];

Rule[#,Graphics3D[{{Opacity[.3],{Green,p}},Blue,ConvexHullRegion[v[[#2]]]}]]&@@@subsethulls2

Platonic

(h=VertexSubsetHulls[p,"Platonic","Skip"->{}])//Timing

Volume

PolyhedronData[pname,"Volume"]

Volume[p]//FullSimplify

PolyhedronVolume[p]

Volume[PolyhedronData[pname,"Region"]]

Properties: {"CubeTwentyCompound", 2}

Initialization

<<MathWorld`Polyhedra`

p=PolyhedronData[pname={"CubeTwentyCompound",2},"Polyhedron"]

AdjacentFaceIndices

adj1=PolyhedronData[pname,"AdjacentFaceIndices"]

adj2=AdjacentFaceIndices[p]

adj1===adj2

Centroid

PolyhedronData[pname,"Centroid"]

Circumsphere

sphere=PolyhedronData[pname,"Circumsphere"]

sphere=Circumsphere[p]

(sphere=MyCircumsphere[p]//FullSimplify)//Timing

Graphics3D[{{Opacity[.2],Yellow,sphere},p},Boxed->False]

Classes

PolyhedronData[pname,"Classes"]

Compound

PolyhedronData[pname,"Compound"]

CompoundInterior

intname=PolyhedronData[pname,"CompoundInterior","Name"]

PolyhedronData[pname,"CompoundInterior","Scale"]

PolyhedronData[pname,"CompoundInterior","RotationMatrix"]

PolyhedronData[pname,"CompoundInterior"]

Graphics3D[{
AbsoluteThickness[1],PolyhedronData[pname,"Lines"],
Opacity[.8],interior=PolyhedronData[pname,"CompoundInterior","Polyhedron"]},Boxed->False]

MyCircumsphere[interior]//Quiet

Insphere[interior]//Quiet

Midsphere[interior]

Construction

int=RegionIntersection[Sequence@@PolyhedronData[pname,"Polyhedra"],PerformanceGoal->"Quality"]

Graphics3D[int]

Graphics3D[{{Opacity[0],p},Red,int},Boxed->False]

Union[Chop@int[[1]]]//Length

Exactify

vexact=DeleteDuplicates[RootApproximant[Chop[int[[1]]]]]

Length[vexact]

inthull=ConvexHullRegion[Union@vexact]

v=inthull[[1]]

faces=inthull[[2]]

gname=RecognizeGraph@PolyhedronGraph[inthull]

GraphData[gname,"PolyhedralEmbeddings"]

SimplifyNumericallyDistinct[PolyhedronEdgeLengths[inthull]]//Counts//Quiet

(((1/2) (Sqrt[34+15 (Sqrt[5])-(Sqrt[2258+1010 (Sqrt[5])])]))/((1/22) (Sqrt[102+13 (Sqrt[5])-(Sqrt[3142+4346 (Sqrt[5])])])))//RootReduce

SimplifyNumericallyDistinct[PolyhedronEdgeLengths[PolyhedronData["DeltoidalHexecontahedron","Polyhedron"]]]//Counts

Numerically group

Union[Chop@int[[1]]]//Length

Length[grouped=SplitBy[SortBy[{#,Round[1*^12#]}&/@Chop[int[[1]]],Last],Last][[All,All,1]]]

rules=Flatten[Function[l,Rule[#,Mean[l]]&/@l]/@grouped,1];

newi=ConvexHullRegion[DeleteDuplicates[Chop[int[[1]]]/.rules]]

Midsphere[newi]

RecognizeGraph@PolyhedronGraph@newi

lengths=PolyhedronEdgeLengths[newi]

Divide@@MinMax[%lengths]

Round[10^12lengths]/10^12.//Counts

PolyhedronData["PentakisDodecahedron","EdgeLengths"]

Divide@@%//N

RootReduce[PolyhedronEdgeLengths[PolyhedronData["PentakisDodecahedron","Polyhedron"]]]//Counts

newi[[1]]

newi[[2]]

RootApproximant[((newi[[1]])/(Max[PolyhedronEdgeLengths[newi]])),4]

Graphics3D[{
AbsoluteThickness[1],PolyhedronData[pname,"Lines"],
Opacity[.8],newi},Boxed->False]

Find scale

intname=XXX;

PolyhedronData/@{pname,intname}

xyratio=((Max[Norm/@vexact[[All,;;2]]])/(Max[Norm/@PolyhedronData[intname,"VertexCoordinates"][[All,;;2]]]))//RootReduce//FullSimplify//Quiet

zratio=((Max[vexact[[All,3]]])/(Max[PolyhedronData[intname,"VertexCoordinates"][[All,3]]]))//RootReduce//FullSimplify

{xyratio,xyratio,zratio}

Concave

PolyhedronData[pname,"Concave"]

Convex

PolyhedronData[pname,"Convex"]

ConvexPolyhedronQ[p]//Timing

ConvexHull

convexhullname=PolyhedronData[pname,"ConvexHull","Name"]

((Max[Norm/@p[[1]]])/(Max[Norm/@PolyhedronData[convexhullname,"Polyhedron"][[1]]]))//FullSimplify//Quiet

PolyhedronData[pname,"ConvexHull","Scale"]

PolyhedronData[pname,"ConvexHull"]

convexhull=ConvexHullRegion[PolyhedronData[pname,"Vertices"]]

v=PolyhedronCoordinates[convexhull]

Sort[(convexhull[[2]]/.i_Integer:>v[[i]])/.Thread[PolyhedronData[pname,"Vertices"]->Range[PolyhedronData[pname,"V"]]]]

Graphics3D[{FaceForm[Green,Red],convexhull,MapIndexed[Text[#2[[1]],#,Background->White]&,v]}]

{Graphics3D[{Red,Opacity[.5],p}],Graphics3D[{Blue,Opacity[.5],PolyhedronData[pname,"ConvexHull"][[1]]}]}

Show[%]

Graphics3D[{p,MapIndexed[Text[#2[[1]],#,Background->White]&,p[[1]]]}]

DehnInvariant

PolyhedronData[pname,"DehnInvariant"]

(dehn=DehnInvariant[p,Debug->True])//Timing

ToConwayBasis[dehn]

Dual

pname

dname=PolyhedronData[pname,"Dual","Name"]

PolyhedronData[pname,"Dual","Scale"]

midrs={PolyhedronData[pname,"Midradius"],PolyhedronData[dname,"Midradius"]}

Divide@@midrs//FullSimplify

GraphicsRow[With[{
s={Yellow,PolyhedronData[pname,"Midsphere"]},
style=Directive[Black,Italic,FontFamily->"Times",14]
},
{
Graphics3D[{s,Green,Opacity[.2],PolyhedronData[pname,"Polyhedron"]},Boxed->False,PlotLabel->TextCell[Style[PolyhedronData[pname,"Name"],style],PageWidth->200,TextAlignment->Center]],
Graphics3D[{s,Red,Opacity[.1],PolyhedronData[pname,"Dual","Polyhedron"]},Boxed->False,PlotLabel->Style[PolyhedronData[dname,"Name"],style]],
Graphics3D[{
{Green,Opacity[.1],PolyhedronData[pname,"Polyhedron"]},
{Red,Opacity[.1],PolyhedronData[pname,"Dual","Polyhedron"]},
s
},Boxed->False,PlotLabel->Style["compound with midsphere",style]],
Graphics3D[{
PolyhedronData[pname,"Polyhedron"],
PolyhedronData[pname,"Dual","Polyhedron"]
},Boxed->False,PlotLabel->Style["compound",style]]
}],Alignment->Top,ImageSize->800]

DihedralAngles

PolyhedronData[pname,"DihedralAngles"]//Tally

SimplifyNumericallyDistinct[DihedralAngles[p]]//Tally

EdgeCount

PolyhedronData[pname,"EdgeCount"]

EdgeLengths

Tally[PolyhedronEdgeLengths[p]//RootReduce]

PolyhedronData[pname,"EdgeLengths"]

PolyhedronData[pname,"Lines"]/.Line[l_]:>EuclideanDistance@@l//FullSimplify//Union//Quiet

lens=SimplifyNumericallyDistinct[PolyhedronEdgeLengths[p],"Simplification"->RootReduce];//Quiet

(distinctlens=Mean/@Split[Sort[lens],Abs[#1-#2]<1.*^-10&])//N//Quiet

v=p[[1]];

colors=With[{n=Length[distinctlens]},Table[Hue[i/n],{i,0,n-1}]]

Graphics3D[{Thick,Sequence@@Nearest[Thread[distinctlens->colors],EuclideanDistance@@v[[#]]],Line[v[[#]]]}&/@PolyhedronEdges[p],Boxed->False]//Quiet

FaceCount

PolyhedronData[pname,"FaceCount"]

GeneralizedDiameter

PolyhedronData[pname,"GeneralizedDiameter"]

GeneralizedDiameter[p]//FullSimplify//Quiet

Hull

hullname=PolyhedronData[pname,"Hull","Name"]

PolyhedronData[pname,"Hull","Scale"]

((Max[Norm/@p[[1]]])/(Max[Norm/@PolyhedronData[hullname,"Polyhedron"][[1]]]))//FullSimplify//Quiet

PolyhedronData[pname,"Hull"]

Max[Norm/@PolyhedronData[pname,"Hull","Polyhedron"][[1]]]

Max[Norm/@p[[1]]]

{Graphics3D[{Opacity[.5],Red,p}],Graphics3D[{Opacity[.5],Blue,PolyhedronData[pname,"Hull"][[1]]}]}

Show[%]

InertiaTensor

PolyhedronData[pname,"InertiaTensor"]

Insphere

PolyhedronData[pname,"Insphere"]

Insphere[p]//Timing

sphere=MyInsphere[p//N]

(sphere=MyInsphere[p])//Timing

Graphics3D[{{Red,sphere},Opacity[.05],p},Boxed->False]

MeanCylindricalRadius

PolyhedronData[pname,"MeanCylindricalRadius"]

MeanSquareCylindricalRadius

PolyhedronData[pname,"MeanSquareCylindricalRadius"]

MeanSphericalRadius

PolyhedronData[pname,"MeanSphericalRadius"]

MeanSquareSphericalRadius

PolyhedronData[pname,"MeanSquareSphericalRadius"]

Midsphere

sphere=PolyhedronData[pname,"Midsphere"]

(sphere=Midsphere[p//N])//Timing

1/Sqrt[2.]

(sphere=Midsphere[p])//Timing

(sphere=Midsphere[p,Method->"PointLineDistance"])//Timing

Graphics3D[{{Opacity[.1],Yellow,p},{Red,sphere}},Boxed->False]

Net

PolyhedronData[pname,"Net"]

NotationRules

PolyhedronData[pname,"NotationRules"]

Orientations

PolyhedronData[pname,"Orientations"]

PolyhedronData[pname,"DefaultOrientation"]

Show[PolyhedronData[{pname,#}],Boxed->False]&/@PolyhedronData[pname,"Orientations"]

Polygons

PolyhedronData[pname,"FaceCount"]

Graphics3D/@PolyhedronData[pname,"Polygons"]

Show[%]

Polyhedra

polyhedra=PolyhedronData[pname,"Polyhedra"]

Length[polyhedra]

Graphics3D[polyhedra]

Graphics3D[Table[{Hue[(i/(Length[polyhedra]))],polyhedra[[i]]},{i,Length[polyhedra]}]]

PolyhedronData[pname,"PolyhedronIndices"]

Region

PolyhedronData[pname,"Region"]

RegionFunction

PolyhedronData[pname,"RegionFunction"]

SimplePolyhedron

PolyhedronData[pname,"Simple"]

SimplePolyhedronQ[p]//Timing

Skeleton

PolyhedronData[pname,"Skeleton"]

gn=PolyhedronData[pname,"Skeleton","Name"]

emb=GraphData[gn,"PolyhedralEmbeddings"]

l={{gn,{0,Union[emb,{pname}]}}}

PropertyReplace[l,"PolyhedralEmbeddings","14.0"]

With[{gn=pname,verts=PolyhedronData[pname,"Vertices"],faces=PolyhedronData[pname,"Faces","List"]},
ToCommonEdges[GraphData[gn],Graph[Range[Length[verts]],UndirectedEdge@@@Union[Sort/@Flatten[Partition[#,2,1,1]&/@faces,1]]]]]//Chop[#,1*^-5]&//N

SurfaceArea

PolyhedronData[pname,"SurfaceArea"]

SurfaceArea[p]

Total[Area/@PolyhedronData[pname,"Polygons"]]//RootReduce

SurfaceArea[PolyhedronData[pname,"Region"]]

VertexCount

PolyhedronData[pname,"VertexCount"]

VertexSubsetHulls

pname

subsethulls=PolyhedronData[pname,"VertexSubsetHulls"]

v=PolyhedronData[pname,"Vertices"];

Rule[#,Length[#2]]&@@@subsethulls

Rule[#,Graphics3D[ConvexHullRegion[v[[#]]]&/@Lookup[subsethulls,#]]]&/@subsethulls[[All,1]]

subsethulls2=Flatten[Function[{pn,l},Rule[pn,#]&/@l]@@@subsethulls]

Rule[#,Graphics3D[{{Opacity[.3],{Green,p}},Blue,ConvexHullRegion[v[[#2]]]}]]&@@@subsethulls2

Platonic

(h=VertexSubsetHulls[p,"Platonic","Skip"->{}])//Timing

Volume

PolyhedronData[pname,"Volume"]

Volume[p]//FullSimplify

PolyhedronVolume[p]

Volume[PolyhedronData[pname,"Region"]]

Construction

First (from {"OctahedronTwentyCompound", 1})

OctahedronDual[oct_Polyhedron]:=ConvexHullRegion[((1/2) (-1+(Sqrt[5])))Normalize[Mean[#[[1]][[#[[2]]]]]]&/@PolyhedronPolygons[oct]]

octs=PolyhedronData[{"OctahedronTwentyCompound",1},"Polyhedra"];

cubes=OctahedronDual/@octs

Graphics3D[cubes,Boxed->False]

Graphics3D/@Transpose[{cubes,octs}]

Show[%]

vcubes=RootReduce[cubes[[All,1]]]

Length[Flatten[vcubes,1]]

Length[DeleteDuplicates[Flatten[vcubes,1]]]

Length[v=DeleteDuplicates[Flatten[vcubes,1]]]

faces=RootReduce[(Function[f,#[[1]][[f]]]/@#[[2]])&/@cubes]/.Thread[v->Range@Length@v]

Flatten[faces,1]

Graphics3D[compound=Polyhedron[v,Flatten[faces,1]]]

Second (from {"OctahedronTwentyCompound", 2})

<<MathWorld`Polyhedra`

OctahedronDual[oct_Polyhedron]:=ConvexHullRegion[((1/2) (-1+(Sqrt[5])))Normalize[Mean[#[[1]][[#[[2]]]]]]&/@PolyhedronPolygons[oct]]

OctahedronDual[oct_Polyhedron]:=ConvexHullRegion[((1/2) (-1+(Sqrt[5])))(Sqrt[6])Mean[#[[1]][[#[[2]]]]]&/@PolyhedronPolygons[oct]]

octs=PolyhedronData[{"OctahedronTwentyCompound",2},"Polyhedra"]

cubes=OctahedronDual/@octs

Graphics3D[cubes,Boxed->False]

Graphics3D/@Transpose[{cubes,octs}]

Show[%]

vcubes=RootReduce[cubes[[All,1]]]

Length[Flatten[vcubes,1]]

Length[v=DeleteDuplicates[Flatten[vcubes,1]]]

faces=RootReduce[(Function[f,#[[1]][[f]]]/@#[[2]])&/@cubes]/.Thread[v->Range@Length@v]//Quiet

Flatten[faces,1]

Graphics3D[compound=Polyhedron[v,Flatten[faces,1]]]

PolyhedronData

PolyhedronDataString

<<MathWorld`Polyhedra`

file=OpenWrite["~/20compound.m"]

WriteString[file,PolyhedronDataString[compound,
"StandardName"->{"CubeTwentyCompound",2},
"AlternateStandardNames"->{},
"Name"->"second cube 20\[Hyphen]compound",
"Information"->"Cube20-Compound",
"Classes"->{"Compound"}
]]

Close[file]