Cube 5-Compound

Author

Eric W. Weisstein
September 29, 2023

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Polyhedra/Cube5-Compound.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/Cube5-Compound.html.

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

Variants

variants=PolyhedronData["CubeFiveCompound*"]

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[1],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: {"CubeFiveCompound", 1}

Orientations

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

PolyhedronData[{"CubeFiveCompound",1}]

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

Polyhedron

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

Region

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

Solid and dual

PolyhedronData[{"CubeFiveCompound",1},"DualCompoundName"]

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

Graphics3D[{Opacity[.6],
Red,PolyhedronData[{"CubeFiveCompound",1},"Polyhedron"],
Green,PolyhedronData,""Dual","Polyhedron"],
Yellow,PolyhedronData["CubeFiveCompound","Midsphere"]},Boxed->False]

Polyhedron: {"CubeFiveCompound", 2}

Orientations

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

PolyhedronData[{"CubeFiveCompound",2}]

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

Polyhedron

Graphics3D[{Opacity[.3],PolyhedronData[{"CubeFiveCompound",2},"Polyhedron"]}]

Face orientation

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

Region

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

Solid and dual

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

Graphics3D[{Opacity[.6],
Red,PolyhedronData[{"CubeFiveCompound",2},"Polyhedron"],
Green,PolyhedronData[{"CubeFiveCompound",2},"Dual","Polyhedron"],
Yellow,PolyhedronData[{"CubeFiveCompound",2},"Midsphere"]},Boxed->False]

Properties: {"CubeFiveCompound", 1}

Initialization

<<MathWorld`Polyhedra`

p=PolyhedronData[pname={"CubeFiveCompound",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]//RootReduce)//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],PolyhedronData[pname,"CompoundInterior","Polyhedron"]},Boxed->False]

PolyhedronEdgeLengths[PolyhedronData[pname,"CompoundInterior","Polyhedron"]]//FullSimplify//Union

Construction

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

Graphics3D[int]

Graphics3D[{{Opacity[.3],p},Red,int},Boxed->False]

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

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

Length[vexact]

inthull=ConvexHullRegion[Union@vexact]

inthull[[1]]

faces=inthull[[2]]

gname=RecognizeGraph@PolyhedronGraph[inthull]

GraphData[gname,"PolyhedralEmbeddings"]

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

intname="RhombicTriacontahedron";

PolyhedronData/@{pname,intname}

Graphics3D[{{Opacity[.2],p},Red,inthull}]

Concave

PolyhedronData[pname,"Concave"]

Convex

PolyhedronData[pname,"Convex"]

ConvexPolyhedronQ[p]//Timing

ConvexHull

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

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

PolyhedronData[pname,"ConvexHull","RotationMatrix"]

Graphics3D[convexhull=PolyhedronData[pname,"ConvexHull","Polyhedron"]]

RecognizeGraph@PolyhedronGraph[convexhull]

{
Graphics3D[{Red,Opacity[.5],p}],
Graphics3D[{Blue,Opacity[.5],convexhull}]
}

Show[%]

Construct using ConvexHullRegion

Graphics3D[convexhullreg=ConvexHullRegion[PolyhedronData[pname,"Vertices"]]]

v=PolyhedronCoordinates[convexhullreg]

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

Find scale

SimplifyNumericallyDistinct[PolyhedronEdgeLengths[convexhullreg]]//Tally//FullSimplify

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

ConvexHullOf

hullof=DeleteCases[Select[PolyhedronData[All],PolyhedronData[#,"ConvexHullName"]===pname&],pname]

DehnInvariant

PolyhedronData[pname,"DehnInvariant"]

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

ToConwayBasis[dehn]

Complement[Select[PolyhedronData[All],PolyhedronData[#,"DehnInvariant"]==dehn&&PolyhedronData[#,"Volume"]==vol&],{pname}]

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[.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]]
}],Alignment->Top,ImageSize->800]

EdgeCount

PolyhedronData[pname,"EdgeCount"]

EdgeLengths

Tally[lens=SimplifyNumericallyDistinct[PolyhedronEdgeLengths[p],"Simplification"->RootReduce]//FullSimplify]//Timing

PolyhedronData[pname,"EdgeLengths"]

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

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

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

Show[%]

HullOf

Select[PolyhedronData[All],PolyhedronData[#,"HullName"]===pname&]

InertiaTensor

PolyhedronData[pname,"InertiaTensor"]

Insphere

PolyhedronData[pname,"Insphere"]

Insphere[p]

(sphere=MyInsphere[p])//Timing

MeanCylindricalRadius

PolyhedronData[pname,"MeanCylindricalRadius"]

MeanSquareCylindricalRadius

PolyhedronData[pname,"MeanSquareCylindricalRadius"]

MeanSphericalRadius

PolyhedronData[pname,"MeanSphericalRadius"]

MeanSquareSphericalRadius

PolyhedronData[pname,"MeanSquareSphericalRadius"]

Midsphere

sphere=PolyhedronData[pname,"Midsphere"]

Midsphere[p//N]

(sphere=Midsphere[p])//Timing

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

Graphics3D[{{Opacity[.5],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"]

PolyhedronFaceTally[p]

Graphics3D/@PolyhedronData[pname,"Polygons"]

Show[%]

Polyhedra

PolyhedronData[pname,"Polyhedra"]

Graphics3D[%]

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"]

GraphData[gn]

emb=GraphData[gn,"PolyhedralEmbeddings"]

GraphData[gn,"Graph",{"3D","All"}]

g=PolyhedronGraph[p]

SurfaceArea

PolyhedronData[pname,"SurfaceArea"]

SurfaceArea[p]

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

SurfaceArea[PolyhedronData[pname,"Region"]]

Uniform

PolyhedronData[pname,"Uniform"]

VertexCount

PolyhedronData[pname,"VertexCount"]

VertexSubsetHulls

subsethulls=PolyhedronData[pname,"VertexSubsetHulls"]

v=PolyhedronData[pname,"Vertices"];

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

Rule[#,Graphics3D[Function[f,ConvexHullRegion[v[[f]]]]/@#2]]&@@@subsethulls

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

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

Convex hull

Graphics3D[convexhullreg=ConvexHullRegion[v=PolyhedronData[pname,"Vertices"]]]

v=PolyhedronCoordinates[convexhullreg]

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

%===Range@Length@v

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

Platonic

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

Volume

PolyhedronData[pname,"Volume"]

WythoffSymbol

PolyhedronData[pname,"WythoffSymbol"]

Properties: {"CubeFiveCompound", 2}

Initialization

<<MathWorld`Polyhedra`

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

AdjacentFaceIndices

adj1=PolyhedronData[pname,"AdjacentFaceIndices"]

adj2=AdjacentFaceIndices[p]

adj1===adj2

Centroid

PolyhedronData[pname,"Centroid"]

Graphics3D[{AbsolutePointSize[10],
Point[{0,0,0}],{Blue,Point[PolyhedronData[pname,"Centroid"]]},Opacity[.2],PolyhedronData[pname,"GraphicsComplex"]}]

RegionCentroid[p]

PolyhedronCentroid[p]

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[1],PolyhedronData[pname,"Lines"],Opacity[.8],PolyhedronData[pname,"CompoundInterior","Polyhedron"]},Boxed->False]

Construction

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

Graphics3D[int]

Graphics3D[{{Opacity[.3],p},Red,int},Boxed->False]

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

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

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"]

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

PolyhedronData[pname,"ConvexHull","RotationMatrix"]

Graphics3D[convexhull=PolyhedronData[pname,"ConvexHull","Polyhedron"]]

RecognizeGraph@PolyhedronGraph[convexhull]

{
Graphics3D[{Red,Opacity[.5],p}],
Graphics3D[{Blue,Opacity[.5],convexhull}]
}

Show[%]

Construct using ConvexHullRegion

Graphics3D[convexhullreg=ConvexHullRegion[PolyhedronData[pname,"Vertices"]]]

v=PolyhedronCoordinates[convexhullreg]

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

Find scale

SimplifyNumericallyDistinct[PolyhedronEdgeLengths[convexhullreg]]//Tally//FullSimplify

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

Find rotation matrix

Graphics3D[{FaceForm[Green,Red],basesolid=PolyhedronData[convexhullname,"Polyhedron"],MapIndexed[Text[#2[[1]],#,Background->White]&,basev=basesolid[[1]]]}]

rot=RotationMatrix[{Mean[basev[[{11,12}]]],{0,0,1}}]//FullSimplify

{Graphics3D[{Polyhedron[newv=rot.#&/@basesolid[[1]],basesolid[[2]]],MapIndexed[Text[#2[[1]],#,Background->White]&,basesolid[[1]]]}],
Graphics3D[ConvexHullRegion[PolyhedronData[pname,"Vertices"]]]}

ArcTan@@(Most@Mean[newv[[{7,29}]]])//FullSimplify

rot2=RotationMatrix[Pi/2+2Pi/5,{0,0,1}].rot//FullSimplify

{Graphics3D[{Polyhedron[(rot2.#)&/@basesolid[[1]],basesolid[[2]]],MapIndexed[Text[#2[[1]],#,Background->White]&,basesolid[[1]]]}],
Graphics3D[ConvexHullRegion[PolyhedronData[pname,"Vertices"]]]}

ConvexHullOf

hullof=DeleteCases[Select[PolyhedronData[All],PolyhedronData[#,"ConvexHullName"]===pname&],pname]

GraphicsGrid[Partition[Graphics3D[{
{Green,Opacity[.7],PolyhedronData[#,"Polyhedron"]},
{Blue,Opacity[.2],PolyhedronData[#,"ConvexHull","Polyhedron"]}},Boxed->False,PlotLabel->Style[PolyhedronData[#,"Name"],Directive[14,Black,Italic,FontFamily->"Times"]]]&/@hullof,UpTo[3]],Dividers->All,ImageSize->800]

DehnInvariant

PolyhedronData[pname,"DehnInvariant"]

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

ToConwayBasis[dehn]

Complement[Select[PolyhedronData[All],PolyhedronData[#,"DehnInvariant"]==dehn&&PolyhedronData[#,"Volume"]==vol&],{pname}]

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[.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]]
}],Alignment->Top,ImageSize->800]

EdgeCount

PolyhedronData[pname,"EdgeCount"]

EdgeLengths

Tally[lens=SimplifyNumericallyDistinct[PolyhedronEdgeLengths[p],"Simplification"->RootReduce]//FullSimplify]//Timing

PolyhedronData[pname,"EdgeLengths"]

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

Find rotation matrix

Graphics3D[{FaceForm[Green,Red],basesolid=PolyhedronData[hullname,"Polyhedron"],MapIndexed[Text[#2[[1]],#,Background->White]&,basev=basesolid[[1]]]}]

rot=RotationMatrix[{Mean[basev[[{11,12}]]],{0,0,1}}]//FullSimplify

{Graphics3D[{Polyhedron[newv=rot.#&/@basesolid[[1]],basesolid[[2]]],MapIndexed[Text[#2[[1]],#,Background->White]&,basesolid[[1]]]}],
Graphics3D[ConvexHullRegion[PolyhedronData[pname,"Vertices"]]]}

ArcTan@@(Most@Mean[newv[[{7,29}]]])//FullSimplify

rot2=RotationMatrix[Pi/2+2Pi/5,{0,0,1}].rot//FullSimplify

{Graphics3D[{Polyhedron[(rot2.#)&/@basesolid[[1]],basesolid[[2]]],MapIndexed[Text[#2[[1]],#,Background->White]&,basesolid[[1]]]}],
Graphics3D[ConvexHullRegion[PolyhedronData[pname,"Vertices"]]]}

HullOf

Select[PolyhedronData[All],PolyhedronData[#,"HullName"]===pname&]

Select[PolyhedronData[All],PolyhedronData[#,"HullName"]===hullname&]

InertiaTensor

PolyhedronData[pname,"InertiaTensor"]

Insphere

PolyhedronData[pname,"Insphere"]

Insphere[p]

sphere=MyInsphere[p]

Graphics3D[{{Opacity[.7],p},{Red,sphere}},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[.5],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"]

PolyhedronFaceTally[p]

Graphics3D/@PolyhedronData[pname,"Polygons"]

Show[%]

Polyhedra

PolyhedronData[pname,"Polyhedra"]

Graphics3D[%]

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"]

GraphData[gn]

emb=GraphData[gn,"PolyhedralEmbeddings"]

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

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

pname

With[{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

GraphData[gn,"Graph",{"3D","All"}]

g=PolyhedronGraph[p]

SurfaceArea

PolyhedronData[pname,"SurfaceArea"]

SurfaceArea[p]

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

SurfaceArea[PolyhedronData[pname,"Region"]]

Uniform

PolyhedronData[pname,"Uniform"]

VertexCount

PolyhedronData[pname,"VertexCount"]

VertexSubsetHulls

subsethulls=PolyhedronData[pname,"VertexSubsetHulls"]

v=PolyhedronData[pname,"Vertices"];

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

Rule[#,Graphics3D[Function[f,ConvexHullRegion[v[[f]]]]/@#2]]&@@@subsethulls

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

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

Convex hull

Graphics3D[convexhullreg=ConvexHullRegion[v=PolyhedronData[pname,"Vertices"]]]

v=PolyhedronCoordinates[convexhullreg]

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

%===Range@Length@v

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

Platonic

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

Volume

PolyhedronData[pname,"Volume"]

Volume[p]//FullSimplify

PolyhedronVolume[p]//FullSimplify

Volume[PolyhedronData[pname,"Region"]]//FullSimplify

Volume[ImplicitRegion[Evaluate@PolyhedronData[pname,"RegionFunction"][x,y,z],{x,y,z}]]//FullSimplify

WythoffSymbol

PolyhedronData[pname,"WythoffSymbol"]

Interior stellations

Interior stellations by
Michael Trott
February 10, 2006

Polygons

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      Polygon[{{0.726542528005361, -0.23606797749979003, 
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      Polygon[{{0.7739476728124915, -0.381966011250105, 
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         -0.4490279765795845}, {0.7265425280053609, -0.23606797749978992, 
         -0.27751455142577536}}], Polygon[{{0.7739476728124915, 
         0.3819660112501051, -0.20081141588622714}, {0.7265425280053608, 
         0.2360679774997898, -0.44902797657958493}, {0.5257311121191339, 
         0.38196601125010493, -0.3249196962329064}}], 
      Polygon[{{0.7739476728124915, 0.3819660112501051, 
         -0.20081141588622714}, {0.7265425280053608, 0.2360679774997898, 
         -0.44902797657958493}, {0.7265425280053608, 0.2360679774997898, 
         -0.27751455142577586}}], Polygon[{{0.7739476728124917, 
         0.3819660112501054, -0.20081141588622756}, {0.6024342476586819, 
         0.6180339887498949, -0.20081141588622758}, {0.6498393924658127, 
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      Polygon[{{0.7739476728124917, 0.38196601125010543, 
         -0.20081141588622756}, {1.0514622242382674, 0, 
         -0.20081141588622753}, {0.7265425280053608, 0.23606797749978975, 
         -0.27751455142577586}}], Polygon[{{0.8506508083520397, 
         -0.14589803375031574, 0.20081141588622786}, {0.7265425280053608, 
         -0.23606797749978978, 0.4490279765795855}, {0.7265425280053608, 
         -0.23606797749978983, 0.27751455142577575}}], 
      Polygon[{{0.8506508083520398, -0.6180339887498949, 
         0.20081141588622764}, {0.6024342476586816, -0.6180339887498949, 
         -0.20081141588622753}, {0.44902797657958493, -0.6180339887498949, 
         0}}], Polygon[{{0.8506508083520398, -0.6180339887498949, 
         0.20081141588622764}, {0.6024342476586816, -0.6180339887498949, 
         -0.20081141588622753}, {0.6498393924658126, -0.4721359549995792, 
         -0.12410828034667898}}], Polygon[{{0.8506508083520398, 
         -0.6180339887498949, 0.20081141588622764}, {0.8506508083520399, 
         -0.14589803375031585, 0.20081141588622764}, {0.7265425280053609, 
         -0.23606797749978978, 0.27751455142577575}}], 
      Polygon[{{0.8506508083520398, -0.14589803375031563, 
         0.2008114158862276}, {1.0514622242382674, 0, -0.20081141588622753}, 
        {0.803245663544909, 0, 0.12410828034667926}}], 
      Polygon[{{0.8506508083520399, -0.6180339887498949, 
         0.20081141588622764}, {0.4016228317724551, -0.7639320225002101, 
         0.20081141588622764}, {0.44902797657958493, -0.6180339887498949, 
         0}}], Polygon[{{0.8506508083520399, -0.6180339887498949, 
         0.20081141588622764}, {0.6498393924658126, -0.47213595499957933, 
         -0.12410828034667896}, {0.7739476728124917, -0.3819660112501051, 
         -0.2008114158862272}}], Polygon[{{0.8506508083520399, 
         -0.6180339887498949, 0.20081141588622764}, {0.7739476728124915, 
         -0.381966011250105, -0.20081141588622736}, {0.7265425280053607, 
         -0.23606797749978942, 0}}], 
      Polygon[{{0.8506508083520399, -0.14589803375031585, 
         0.20081141588622764}, {0.8506508083520399, 0.14589803375031585, 
         0.20081141588622764}, {0.6498393924658126, 0, 
         0.32491969623290656}}], Polygon[{{0.8506508083520399, 
         -0.14589803375031557, 0.20081141588622764}, {0.8506508083520398, 
         -0.6180339887498949, 0.20081141588622764}, {0.7265425280053607, 
         -0.23606797749978953, 0}}], 
      Polygon[{{0.8506508083520399, 0.1458980337503158, 0.20081141588622764}, 
        {0.8506508083520399, -0.14589803375031557, 0.20081141588622764}, 
        {0.803245663544909, 0, 0.12410828034667923}}], 
      Polygon[{{0.8506508083520399, 0.14589803375031585, 
         0.20081141588622764}, {0.8506508083520399, 0.6180339887498949, 
         0.20081141588622764}, {0.7265425280053609, 0.23606797749978986, 
         0.27751455142577575}}], Polygon[{{0.8506508083520399, 
         0.6180339887498949, 0.20081141588622764}, {0.40162283177245506, 
         0.7639320225002102, 0.20081141588622764}, {0.4490279765795857, 
         0.6180339887498949, 0.27751455142577575}}], 
      Polygon[{{0.8506508083520399, 0.6180339887498949, 0.20081141588622764}, 
        {0.44902797657958576, 0.6180339887498949, 0.44902797657958526}, 
        {0.5257311121191339, 0.3819660112501049, 0.4016228317724545}}], 
      Polygon[{{0.8506508083520399, 0.6180339887498949, 0.20081141588622764}, 
        {0.6024342476586817, 0.6180339887498949, -0.2008114158862273}, 
        {0.4490279765795855, 0.6180339887498949, 0}}], 
      Polygon[{{0.8506508083520399, 0.6180339887498949, 0.20081141588622764}, 
        {0.6498393924658126, 0.47213595499957933, -0.12410828034667902}, 
        {0.6024342476586817, 0.6180339887498949, -0.2008114158862274}}], 
      Polygon[{{0.8506508083520399, 0.6180339887498949, 0.20081141588622764}, 
        {0.7739476728124915, 0.3819660112501051, -0.20081141588622714}, 
        {0.6498393924658127, 0.47213595499957917, -0.12410828034667917}}], 
      Polygon[{{0.8506508083520399, 0.6180339887498949, 0.20081141588622764}, 
        {0.7739476728124915, 0.3819660112501051, -0.20081141588622714}, 
        {0.7265425280053609, 0.23606797749978983, 0}}], 
      Polygon[{{0.8506508083520399, 0.6180339887498949, 0.20081141588622764}, 
        {0.8506508083520399, 0.1458980337503158, 0.20081141588622764}, 
        {0.7265425280053607, 0.23606797749979008, 0}}], 
      Polygon[{{1.0514622242382672, 0, -0.20081141588622764}, 
        {0.6498393924658125, 0, -0.4016228317724549}, {0.7265425280053608, 
         0.23606797749978975, -0.44902797657958554}}], 
      Polygon[{{1.0514622242382672, 0, -0.20081141588622764}, 
        {0.7739476728124915, -0.38196601125010515, -0.20081141588622758}, 
        {0.7265425280053609, -0.23606797749979003, -0.2775145514257756}}], 
      Polygon[{{1.0514622242382672, 0, -0.20081141588622764}, 
        {0.8506508083520397, 0.1458980337503156, 0.20081141588622747}, 
        {0.7265425280053606, 0.23606797749978992, 0}}], 
      Polygon[{{1.0514622242382672, 0, -0.20081141588622764}, 
        {0.8506508083520397, 0.1458980337503156, 0.20081141588622747}, 
        {0.8032456635449089, 0, 0.1241082803466791}}], 
      Polygon[{{1.0514622242382674, 0, -0.20081141588622753}, 
        {0.7265425280053609, -0.23606797749979, -0.4490279765795858}, 
        {0.6498393924658125, 0, -0.4016228317724549}}], 
      Polygon[{{1.0514622242382674, 0, -0.20081141588622753}, 
        {0.7739476728124917, 0.3819660112501054, -0.20081141588622756}, 
        {0.7265425280053607, 0.2360679774997899, 0}}], 
      Polygon[{{1.0514622242382674, 0, -0.20081141588622753}, 
        {0.8506508083520397, -0.14589803375031574, 0.20081141588622786}, 
        {0.7265425280053608, -0.23606797749978992, 0}}]}};

Figures

Show[GraphicsArray[Partition[Graphics3D[#,Boxed->False]&/@fiveCubePolys,3,3,{1,1},{}]],ImageSize->500,GraphicsSpacing->{-.2,-.3}]

Inscriptable In

i=Select[PolyhedronData[All],Head[h=PolyhedronData[#,"VertexSubsetHulls"]]=!=Missing&&Length["Cube"/.h]==5&]

StringJoin@@Riffle["\\eref{"<>PolyhedronData[#,"Name"]<>"}"&/@i,", "]

GraphicsGrid[Partition[Function[poly,Graphics3D[{
{Gray,Opacity[.2],EdgeForm[Thickness[.005]],PolyhedronData[poly,"Faces"]},
ConvexHull3D[N[PolyhedronData[poly,"VertexCoordinates"]][[#]]]&/@("Cube"/.PolyhedronData[poly,"VertexSubsetHulls"])},Boxed->False,PlotLabel->TextCell[PolyhedronData[poly,"Name"],PageWidth->100,TextAlignment->Center],BaseStyle->Italic]]/@i,5,5,{1,1},{}],ImageSize->500,Spacings->{-120,10}]

Interior

First Form

ineqs1=Flatten[List@@(PolyhedronInequalities[#,{x,y,z}]&/@Polyhedron[Cube5Compound])]//FullSimplify

ineqs1={5 y+(Sqrt[50-10 (Sqrt[5])]) z<5+(Sqrt[5 (5+2 (Sqrt[5]))]) x,(Sqrt[5 (5+2 (Sqrt[5]))]) x<5+5 y+(Sqrt[50-10 (Sqrt[5])]) z,2+(Sqrt[10-2 (Sqrt[5])]) x+y+(Sqrt[5]) y>0,(Sqrt[10-2 (Sqrt[5])]) x+y+(Sqrt[5]) y<2,5 (Sqrt[5]) y<5 (2+y)+(Sqrt[10 (5+(Sqrt[5]))]) (x+2 z),2 (Sqrt[10 (5+(Sqrt[5]))]) (x+2 z)<10 (2+(-1+(Sqrt[5])) y),(Sqrt[50-10 (Sqrt[5])]) z<5+(Sqrt[5 (5+2 (Sqrt[5]))]) x+5 y,(Sqrt[5 (5+2 (Sqrt[5]))]) x+5 y<5+(Sqrt[50-10 (Sqrt[5])]) z,y+(Sqrt[5]) y<2+(Sqrt[10-2 (Sqrt[5])]) x,(Sqrt[10-2 (Sqrt[5])]) x<2+y+(Sqrt[5]) y,5 (2+(Sqrt[5]) y)+(Sqrt[10 (5+(Sqrt[5]))]) (x+2 z)>5 y,(Sqrt[5]) (5 y+(Sqrt[2 (5+(Sqrt[5]))]) (x+2 z))<5 (2+y),(Sqrt[5]) y<2+(Sqrt[2 (5+(Sqrt[5]))]) x+y,(Sqrt[2 (5+(Sqrt[5]))]) x+y<2+(Sqrt[5]) y,5 (2+y+(Sqrt[5]) y)+(Sqrt[50-10 (Sqrt[5])]) (x+2 z)>0,5 (1+(Sqrt[5])) y+(Sqrt[50-10 (Sqrt[5])]) (x+2 z)<10,(Sqrt[25-10 (Sqrt[5])]) x+5 y<5+(Sqrt[10 (5+(Sqrt[5]))]) z,(Sqrt[10 (5+(Sqrt[5]))]) z<5+(Sqrt[25-10 (Sqrt[5])]) x+5 y,y<2+(Sqrt[2 (5+(Sqrt[5]))]) x+(Sqrt[5]) y,(Sqrt[2 (5+(Sqrt[5]))]) x+(Sqrt[5]) y<2+y,5 (-2+y+(Sqrt[5]) y)<(Sqrt[50-10 (Sqrt[5])]) (x+2 z),(Sqrt[50-10 (Sqrt[5])]) (x+2 z)<5 (2+y+(Sqrt[5]) y),8 (Sqrt[25-10 (Sqrt[5])]) x+5 (Sqrt[2 (5+(Sqrt[5]))]) z<40+40 y+(Sqrt[4250+1490 (Sqrt[5])]) z,5 y+(Sqrt[10 (5+(Sqrt[5]))]) z<5+(Sqrt[25-10 (Sqrt[5])]) x,5+(Sqrt[10 (5+(Sqrt[5]))]) x+(Sqrt[50-10 (Sqrt[5])]) z>0,(Sqrt[10 (5+(Sqrt[5]))]) x+(Sqrt[50-10 (Sqrt[5])]) z<5,(Sqrt[10 (5+(Sqrt[5]))]) z<5+(Sqrt[50-10 (Sqrt[5])]) x,(Sqrt[50-10 (Sqrt[5])]) x<5+(Sqrt[10 (5+(Sqrt[5]))]) z,y>-(1/2),2 y<1};

Plot

<<Graphics`InequalityGraphics`

InequalityPlot3D[ineqs1//N,{x},{y},{z}]

Volume

v5.2

NIntegrate[Boole[ineqs1],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞}]//Timing

...running out of memory...

v6

NIntegrate[Boole[ineqs],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞}]//Timing

Second Form

ineqs2={-(1/2)+(x/4)-(((Sqrt[5]) x)/4)+(y/2)+(z/4)+(((Sqrt[5]) z)/4)<0,-(1/2)-(x/4)+(((Sqrt[5]) x)/4)-(y/2)-(z/4)-(((Sqrt[5]) z)/4)<0,-(1/2)-(x/2)-(y/4)-(((Sqrt[5]) y)/4)-(z/4)+(((Sqrt[5]) z)/4)<0,-(1/2)+(x/2)+(y/4)+(((Sqrt[5]) y)/4)+(z/4)-(((Sqrt[5]) z)/4)<0,-(1/2)+(x/4)+(((Sqrt[5]) x)/4)+(y/4)-(((Sqrt[5]) y)/4)+(z/2)<0,-(1/2)-(x/4)-(((Sqrt[5]) x)/4)-(y/4)+(((Sqrt[5]) y)/4)-(z/2)<0,-(1/2)-(x/4)+(((Sqrt[5]) x)/4)+(y/2)+(z/4)+(((Sqrt[5]) z)/4)<0,-(1/2)+(x/4)-(((Sqrt[5]) x)/4)-(y/2)-(z/4)-(((Sqrt[5]) z)/4)<0,-(1/2)-(x/4)-(((Sqrt[5]) x)/4)+(y/4)-(((Sqrt[5]) y)/4)+(z/2)<0,-(1/2)+(x/4)+(((Sqrt[5]) x)/4)-(y/4)+(((Sqrt[5]) y)/4)-(z/2)<0,-(1/2)+(x/2)-(y/4)-(((Sqrt[5]) y)/4)-(z/4)+(((Sqrt[5]) z)/4)<0,-(1/2)-(x/2)+(y/4)+(((Sqrt[5]) y)/4)+(z/4)-(((Sqrt[5]) z)/4)<0,-(1/2)-(x/2)+(y/4)+(((Sqrt[5]) y)/4)-(z/4)+(((Sqrt[5]) z)/4)<0,-(1/2)+(x/2)-(y/4)-(((Sqrt[5]) y)/4)+(z/4)-(((Sqrt[5]) z)/4)<0,-(1/2)+(x/4)-(((Sqrt[5]) x)/4)-(y/2)+(z/4)+(((Sqrt[5]) z)/4)<0,-(1/2)-(x/4)+(((Sqrt[5]) x)/4)+(y/2)-(z/4)-(((Sqrt[5]) z)/4)<0,-(1/2)+(x/4)+(((Sqrt[5]) x)/4)-(y/4)+(((Sqrt[5]) y)/4)+(z/2)<0,-(1/2)-(x/4)-(((Sqrt[5]) x)/4)+(y/4)-(((Sqrt[5]) y)/4)-(z/2)<0,-(1/2)-(x/4)+(((Sqrt[5]) x)/4)-(y/2)+(z/4)+(((Sqrt[5]) z)/4)<0,-(1/2)+(x/4)-(((Sqrt[5]) x)/4)+(y/2)-(z/4)-(((Sqrt[5]) z)/4)<0,-(1/2)-(x/4)-(((Sqrt[5]) x)/4)-(y/4)+(((Sqrt[5]) y)/4)+(z/2)<0,-(1/2)+(x/4)+(((Sqrt[5]) x)/4)+(y/4)-(((Sqrt[5]) y)/4)-(z/2)<0,-(1/2)+(x/2)+(y/4)+(((Sqrt[5]) y)/4)-(z/4)+(((Sqrt[5]) z)/4)<0,-(1/2)-(x/2)-(y/4)-(((Sqrt[5]) y)/4)+(z/4)-(((Sqrt[5]) z)/4)<0,-(1/2)+z<0,-(1/2)-z<0,-(1/2)+y<0,-(1/2)-y<0,-(1/2)+x<0,-(1/2)-x<0};

Plot

<<Graphics`InequalityGraphics`

InequalityPlot3D[ineqs2,{x},{y},{z}]

Volume

NIntegrate[Boole[ineqs2],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞}]//Timing

...runs out of memory..

Interior Polyhedron

interior=PolyhedronFromInequalities[ineqs2,{x,y,z}];//Timing

Show[Graphics3D[{
Wireframe[Cube5Compound],
OrientCentered[interior]
}],Boxed->False]

LiveGraphics3D

<<Utilities`LiveGraphics3D`

WriteLiveForm["cube5int.m",
Show[Graphics3D[{
Wireframe[Cube5Compound],
OrientCentered[interior]
}],Boxed->False]
]//Timing

Net construction

<<DiscreteMath`Combinatorica`

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

FaceIntersections[poly_]:=Module[{pairs=KSubsets[Take[#,3]&/@First/@poly,2]},
DeleteCases[Intersection3D@@@Map[Plane,pairs,{2}],{}]
]

Timing[Length[f=FaceIntersections[Flatten[Polyhedron[Cube5Compound]]]]]

Timing[lines=Map[ToExact,f,{3}];]

Length[lines]

Timing[points=Intersection3D@@@KSubsets[lines,2];]

Length[points]

Length[p=First/@Union[Select[DeleteCases[points,{}],.75<=#[[1]].#[[1]]≤2&&#[[1,2]]<0&]]]//Timing

Show[Graphics3D[{Polyhedron[Cube5Compound],
PointSize[.02],
Point/@p,
MapIndexed[Text[#2[[1]],#1+{0,0,.05},Background->White]&,p]
}],Boxed->False,PlotRange->All,ViewPoint->{0,-2,0}]

p1=p[[{1824,244,1456}]];
p2=p[[{1824,1121,1456}]];
p3=p[[{1456,1361,1795}]];
p4=p[[{1456,1121,1795}]];

<<Simplify`

ps=Map[ToExact,Polygon2D/@{p1,p2,p3,p4},{3}]

Show[Graphics[piece1=ClosedLine@@@{ps[[1]],Map[{#[[1]],-#[[2]]}&,ps[[2]],{2}]}],AspectRatio->Automatic]

Show[Graphics[piece2=Map[{-#[[1]],#[[2]]}&,piece1,{3}]],AspectRatio->Automatic]

Show[Graphics[piece3=ClosedLine@@@{ps[[3]],Map[{#[[1]],-#[[2]]}&,ps[[4]],{2}]}],AspectRatio->Automatic]

net={piece1,piece2/.Line[x_]:>Line[#+{3/2,0}&/@x],piece3
/.Line[x_]:>Line[#+{3/2+1/4,0}&/@x]};
Show[Graphics[net],AspectRatio->Automatic]

net2=Map[ToExact,Flatten[net],{3}]

v=Union[Flatten[First/@net2,1]];

Drop[First[#],-1]&/@net2/.Thread[v->Range[Length[v]]]

v//InputForm

Construction

{"CubeFiveCompound", 1} by Rotation

cube=Polyhedron[Cube];
Show[Graphics3D[{
Table[RotateAboutAxis[cube,{1,GoldenRatio,0},2n Pi/5],{n,4}]
}],Boxed->False]

cube=Polyhedron[Cube];
Show[Graphics3D[{
cube,
Table[RotateAboutAxis[cube,{1,GoldenRatio,0},2n Pi/5],{n,4}]
}],Boxed->False]

{"CubeFiveCompound", 1} by Selecting Vertices

<<DiscreteMath`Combinatorica`

v=PolyhedronVertices[Dodecahedron];

Show[Graphics3D[{
Polyhedron[Dodecahedron],
PointSize[.05],Point/@v[[{4,9}]]
}],Boxed->False]

l=RootReduce[Sqrt[#.#]&[Subtract@@v[[{4,9}]]]]

l2=FullSimplify[l^2]

edges[l_]:=#.#&[(#.#-l2)&/@Subtract@@@KSubsets[v[[l]],2]]

Binomial[20,8]

Timing[cubes=Select[KSubsets[Range[20],4],Abs[edges[#]]<10^-8&]]

cubes=;

cube5=Select[KSubsets[cubes,8],Sort[Flatten[#]]==Union[Flatten[#]]&]

Show[Graphics3D[Polygon/@KSubsets[#,4]&/@(v[[#]]&/@tets)],
Boxed->False]

(*
	The 5 cubes used in the cube and octahedron 5-compounds.
	
	This is ugly code.  It searches the dodecahedron vertices for
	cubes, then displays 5 of them.  Better would be to start with a
	single cube, then make the approproate rotations.
*)

FiveCubes:=Module[
	{phi=(1+Sqrt[5])/2,iphi,v,eps=.01,
	e={},i,j,e2,cube8pts={},elist={},dv={},dv2,
	match,thecubes={},len
	},
	iphi=1/phi;
	(* dodecahedron vertices with a=Sqrt[5]-1 *)
	v={{0,iphi,phi},{0,-iphi,phi},{0,iphi,-phi},{0,-iphi,-phi},
		{phi,0,iphi},{-phi,0,iphi},{phi,0,-iphi},{-phi,0,-iphi},
		{iphi,phi,0},{-iphi,phi,0},{iphi,-phi,0},{-iphi,-phi,0},
		{ 1, 1, 1},{-1, 1, 1},{-1,-1, 1},{ 1,-1, 1},
		{-1,-1,-1},{ 1,-1,-1},{ 1, 1,-1},{-1, 1,-1}}/2//N;
	len=1;
	
	v=PolyhedronVertices[Dodecahedron];
	len=(1+Sqrt[5])/2;
	(* Print[Table[norm[v[[i]],v[[j]]],{i,Length[v]},{j,i+1,Length[v]}]]; *)
	(* edges *)
	Do[
		If[Abs[norm[v[[i]],v[[j]]]-len]<eps,AppendTo[e,{v[[i]],v[[j]]}]],
		{i,Length[v]},{j,i+1,Length[v]}
	];
	(*Print[e];*)
	Do[AppendTo[dv,e[[i,2]]-e[[i,1]]],{i,Length[e]}];
	e2=e;
	dv2=dv;
	(* Find edges whose endpoints are the correct distance away *)
	For[j=1,Length[e2]>0,j++,
		match=dv2[[1]];
		elist={};
		For[i=1,i<=Length[e2],i++,
			If[norm[dv2[[i]],match]<eps || norm[dv2[[i]],-match]<eps,
				AppendTo[elist,e2[[i]]];
				dv2=Drop[dv2,{i}];
				e2=Drop[e2,{i}];
				i--,
				Null,
				Message[Cube5Compound::BadVertex]
			];
		];
		elist=Flatten[elist,1];
		(* Print[j," ",elist]; *)
		cube8pts=Append[cube8pts,elist];
	];
	(* Get rid of 3x degeneracy *)
	cube8pts=Union[cube8pts];
	Do[
		cv[0]=cube8pts[[j,1]];
		k=0;
		Do[
			If[Abs[norm[cube8pts[[j,1]],cube8pts[[j,i]]]]-len<eps,
				(* Print[i]; *) 
				k++;
				cv[k]=cube8pts[[j,i]]-cube8pts[[j,1]]
			],
		{i,2,8}
		];
		top={cv[0],cv[0]+cv[1],cv[0]+cv[1]+cv[2],cv[0]+cv[2]};
		bottom=addvec[top,cv[3]];
		left={cv[0],cv[0]+cv[3],cv[0]+cv[1]+cv[3],cv[0]+cv[1]};
		right=addvec[left,cv[2]];
		front={cv[0],cv[0]+cv[2],cv[0]+cv[2]+cv[3],cv[0]+cv[3]};
		back=addvec[front,cv[1]];
	(*	AppendTo[thecubes,OrientCentered[
			Polygon/@{top,bottom,left,right,front,back}]],
	*)
		AppendTo[thecubes,
			Polygon/@{top,bottom,left,right,front,back}],
		{j,Length[cube8pts]}
	];

	(* Hardwire the cubes to display *)
	thecubes[[{1,2,7,8,13}]]
]

FiveCubesExact:=FiveCubesExact=Simplify[FiveCubes/.
	Polygon[x_List]:>Polygon[Map[ToExact[N[Chop[#]],2]&,x,{2}]]]

SplitPolyhedron[Cube5Compound]:=SplitPolyhedron[Cube5Compound]=
	Select[Flatten[First[Chop[NonIntersectingPolygons[Graphics3D[FiveCubes]]]]],
	  Length[#[[1]]]==3&]

{"CubeFiveCompound", 1}

{"CubeFiveCompound", 2} from {"CubeFourCompound", 1}

subsethulls=PolyhedronData[{"CubeFourCompound",1},"VertexSubsetHulls"]

v=PolyhedronData[{"CubeFourCompound",1},"Vertices"];

Rule[#,Graphics3D[Function[f,ConvexHullRegion[v[[f]]]]/@#2]]&@@@subsethulls

Length[v]

cubeindices=Lookup[subsethulls,"Cube"]

Union[Flatten[cubeindices]]==Range[32]

v=PolyhedronData[{"CubeFourCompound",1},"Polyhedron"][[1]];

Graphics3D[{{Opacity[.2],{Green,p}},Blue,ConvexHullRegion[v[[#]]]},Boxed->False]&/@cubeindices

Graphics3D[ConvexHullRegion[v[[#]]]&/@cubeindices,Boxed->False]

originalfaces[v_,orig_]:=Module[{con=ConvexHullRegion[v]},
(con[[2]]/.i_Integer:>con[[1,i]])/.Thread[orig->Range@Length@orig]
]

faces=originalfaces[v[[#]],v]&/@cubeindices

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

Graphics3D[compound]

PolyhedronData

<<MathWorld`Polyhedra`

file=OpenWrite["~/5compound.m"]

PolyhedronDataString[
Polyhedron[Cube5Compound],
"StandardName"->"CubeFiveCompound",
"AlternateStandardNames"->{"Cube5Compound"},
"Name"->"cube 5-compound",
"AlternateNames"->{"cube five-compound"},
"Information"->"Cube5-Compound",
"Classes"->{"Concave","Compound"},
"PolyhedronIndices"->PolyhedronPolyhedra[Cube5Compound],
"Simplification"->FullSimplify,
Debug->True
]

Close[file]