Cube-Octahedron 20-Compound

Author

Eric W. Weisstein
August 30, 2023

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Polyhedra/Cube-Octahedron20-Compound.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/Cube-Octahedron20-Compound.html.

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

Graphic

variants=PolyhedronData["CubeOctahedronTwentyCompound*"]

GraphicsRow[Table[Graphics3D[
{PolyhedronData[{"OctahedronTwentyCompound",n},"Polyhedron"],
PolyhedronData[{"OctahedronTwentyCompound",n},"Dual","Polyhedron"]},Boxed->False],{n,2}]]

Properties: first

Initialization

<<MathWorld`Polyhedra`

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

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"]

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"]]

SimplifyNumericallyDistinct[PolyhedronEdgeLengths[convexhull]]//Tally//Quiet//Timing

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

EdgeCount

PolyhedronData[pname,"EdgeCount"]

EdgeLengths

Tally[lens=Tally[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[%]

InertiaTensor

PolyhedronData[pname,"InertiaTensor"]

Insphere

PolyhedronData[pname,"Insphere"]

Insphere[p]

sphere=MyInsphere[p]

Graphics3D[{{Opacity[.5],p},{Red,sphere}},Boxed->False]

MeanCylindricalRadius

PolyhedronData[pname,"MeanCylindricalRadius"]

N[%]

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"]

Graphics3D/@PolyhedronData[pname,"Polygons"]

Show[%]

Polyhedra

PolyhedronData[pname,"Polyhedra"]

Graphics3D/@%

PolyhedronData[pname,"PolyhedronIndices"]

Region

PolyhedronData[pname,"Region"]

RegionFunction

PolyhedronData[pname,"RegionFunction"]

Collect[#,{x,y,z},RootReduce]&/@PolyhedronInequalities[p,{x,y,z}]

FullSimplify[%]

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"]]

SurfaceArea[PolyhedronData[pname,"Region"]]

Uniform

PolyhedronData[pname,"Uniform"]

With[{rules=PolyhedronData[pname,"NotationRules"]},{
UniformPolyhedron[Lookup[rules,"WenningerIndex"]],
UniformPolyhedron[{"Coxeter",Lookup[rules,"CoxeterIndex"]}],
UniformPolyhedron[{"Kaleido",Lookup[rules,"KaleidoIndex"]}],
UniformPolyhedron[{"Uniform",Lookup[rules,"MaederIndex"]}],
UniformPolyhedron[{"Wenninger",Lookup[rules,"WenningerIndex"]}]
}]

SameQ@@%

VertexCount

PolyhedronData[pname,"VertexCount"]

VertexSubsetHulls

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]

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

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

Platonic

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

What about cubes?!

Volume

PolyhedronData[pname,"Volume"]

Volume[p]//FullSimplify

PolyhedronVolume[p]//FullSimplify

Volume[PolyhedronData[pname,"Region"]]

WythoffSymbol

PolyhedronData[pname,"WythoffSymbol"]

Properties: 2nd

Initialization

<<MathWorld`Polyhedra`

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

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"]

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"]]

SimplifyNumericallyDistinct[PolyhedronEdgeLengths[convexhull]]//Tally//Quiet//Timing

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

EdgeCount

PolyhedronData[pname,"EdgeCount"]

EdgeLengths

Tally[lens=Tally[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[%]

InertiaTensor

PolyhedronData[pname,"InertiaTensor"]

(i=((MomentOfInertia[p,{0,0,0}])/(Volume[p])))//Timing

Map[ToAlgebraicRoot,i,{2}]

PolyhedronInertiaTensor[p]//Timing

Insphere

PolyhedronData[pname,"Insphere"]

Insphere[p]

sphere=MyInsphere[p]

Graphics3D[{{Opacity[.5],p},{Red,sphere}},Boxed->False]

MeanCylindricalRadius

PolyhedronData[pname,"MeanCylindricalRadius"]

N[%]

Integrate over polyhedron

((NIntegrate[(Sqrt[(x^2)+(y^2)]),{x,y,z}∈p])/vol)

(rxy=((Integrate[(Sqrt[(x^2)+(y^2)]),{x,y,z}∈p])/vol))//Timing

N[rxy]

Divergence theorem

(rxy=MeanCylindricalRadius[p,Debug->True])//Timing

N[rxy]

ResourceFunction["LogSimplify"][rxy]//FullSimplify

CAD

cad=CylindricalDecomposition[PolyhedronData[pname,"RegionFunction"][x,y,z],{x,y,z},"Function"]

((NIntegrate[(Sqrt[(x^2)+(y^2)])Boole[cad],{x,y,z}∈FullRegion[3]])/vol)

((Integrate[(Sqrt[(x^2)+(y^2)])Boole[cad],{x,y,z}∈FullRegion[3]])/vol)//Timing

Boole Integrate doing z integral first

{{xmin,xmax},{ymin,ymax},{zmin,zmax}}==CoordinateBounds[p]

ineq=PolyhedronData[pname,"RegionFunction"][x,y,z]

(i1=Integrate[Boole[ineq](Sqrt[(x^2)+(y^2)]),{z,zmin,zmax}])//Timing

NIntegrate[(i1/vol),{x,xmin,xmax},{y,ymin,ymax}]

(i2=Integrate[(i1/vol),{x,xmin,xmax},{y,ymin,ymax}])//Timing

MeanSquareCylindricalRadius

PolyhedronData[pname,"MeanSquareCylindricalRadius"]

N[%]

Integrate over Polyhedron

((NIntegrate[(x^2)+(y^2),{x,y,z}∈p])/vol)//Timing

(rxy2=((Integrate[(x^2)+(y^2),{x,y,z}∈p])/vol))//Timing

N[rxy2]

Divergence theorem

(rxy2=MeanSquareCylindricalRadius[p,Debug->True])//Timing

MeanSphericalRadius

PolyhedronData[pname,"MeanSphericalRadius"]

Integrate over polyhedron

((NIntegrate[(Sqrt[(x^2)+(y^2)+(z^2)]),{x,y,z}∈p])/vol)//Timing

(rxyz=((Integrate[(Sqrt[(x^2)+(y^2)+(z^2)]),{x,y,z}∈p])/vol))//Timing

N[rxyz]

Divergence theorem

(rxyz=MeanSphericalRadius[p,Debug->True])//Timing

N[rxyz]

ResourceFunction["LogSimplify"][rxyz]//FullSimplify

CAD

cad=CylindricalDecomposition[PolyhedronData[pname,"RegionFunction"][x,y,z],{x,y,z},"Function"]

((NIntegrate[(Sqrt[(x^2)+(y^2)+(z^2)])Boole[cad],{x,y,z}∈FullRegion[3]])/vol)

((Integrate[(Sqrt[(x^2)+(y^2)+(z^2)])Boole[cad],{x,y,z}∈FullRegion[3]])/vol)//Timing

Boole Integrate doing z integral first

{{xmin,xmax},{ymin,ymax},{zmin,zmax}}=CoordinateBounds[p]

ineq=PolyhedronData[pname,"RegionFunction"][x,y,z]

(i1=Integrate[Boole[ineq](Sqrt[(x^2)+(y^2)+(z^2)]),{z,zmin,zmax}])//Timing

NIntegrate[(i1/vol),{x,xmin,xmax},{y,ymin,ymax}]

(i2=Integrate[(i1/vol),{x,xmin,xmax},{y,ymin,ymax}])//Timing

MeanSquareSphericalRadius

PolyhedronData[pname,"MeanSquareSphericalRadius"]

Integrate over polyhedron

((NIntegrate[(x^2)+(y^2)+(z^2),{x,y,z}∈p])/vol)//Timing

(rxy2=((Integrate[(x^2)+(y^2)+(z^2),{x,y,z}∈p])/vol))//Timing

N[rxy2]

Integrate over region

reg=PolyhedronData[pname,"RegionFunction"][x,y,z]

(rxy=((Integrate[(Sqrt[(x^2)+(y^2)])Boole[reg],{x,y,z}∈FullRegion[3]])/vol))//Timing

Divergence theorem

(rxy2=MeanSquareSphericalRadius[p,Debug->True])//Timing

Boole Integrate doing z integral first

{{xmin,xmax},{ymin,ymax},{zmin,zmax}}==CoordinateBounds[p]

ineq=PolyhedronData[pname,"RegionFunction"][x,y,z]

(i1=Integrate[Boole[ineq]((x^2)+(y^2)+(z^2)),{z,zmin,zmax}])//Timing

NIntegrate[(i1/vol),{x,xmin,xmax},{y,ymin,ymax}]

(i2=Integrate[(i1/vol),{x,xmin,xmax},{y,ymin,ymax}])//Timing

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"]

Total[Area/@Flatten[PolyhedronData[pname,"Net","Polygon"]]]//ToAlgebraicRoot//Timing

PolyhedronData[pname,"SurfaceArea"]

poly3d=PolyhedronData[pname,"Net","Polygons"]/.Polygon[v_]:>Polygon[Append[#,0]&/@v]

Total[Area/@Flatten[poly3d]]//RootReduce//FullSimplify

Graphics3D[{FaceForm[Green,Red],poly3d},Axes->True,AxesLabel->{"x","y","z"}]

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

PolyhedronData[pname,"Polyhedra"]

Graphics3D[%]

PolyhedronData[pname,"PolyhedronIndices"]

Region

PolyhedronData[pname,"Region"]

RegionFunction

PolyhedronData[pname,"RegionFunction"]

Collect[#,{x,y,z},RootReduce]&/@PolyhedronInequalities[p,{x,y,z}]

FullSimplify[%]

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"]]

SurfaceArea[PolyhedronData[pname,"Region"]]

Uniform

PolyhedronData[pname,"Uniform"]

With[{rules=PolyhedronData[pname,"NotationRules"]},{
UniformPolyhedron[Lookup[rules,"WenningerIndex"]],
UniformPolyhedron[{"Coxeter",Lookup[rules,"CoxeterIndex"]}],
UniformPolyhedron[{"Kaleido",Lookup[rules,"KaleidoIndex"]}],
UniformPolyhedron[{"Uniform",Lookup[rules,"MaederIndex"]}],
UniformPolyhedron[{"Wenninger",Lookup[rules,"WenningerIndex"]}]
}]

SameQ@@%

VertexCount

PolyhedronData[pname,"VertexCount"]

VertexSubsetHulls

subsethulls=PolyhedronData[pname,"VertexSubsetHulls"]

v=PolyhedronData[pname,"Vertices"];

v=PolyhedronData[pname,"Vertices"];

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

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

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

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]//FullSimplify

Volume[PolyhedronData[pname,"Region"]]

WythoffSymbol

PolyhedronData[pname,"WythoffSymbol"]