Octahedron 4-Compound

Author

Eric W. Weisstein
September 26, 2023

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Polyhedra/Octahedron4-Compound.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/Octahedron4-Compound.html.

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

Compounds

variants=PolyhedronData["OctahedronFourCompound*"]

PolyhedronData[#,"FaceCount"]&/@variants

Length[PolyhedronData[#,"Polyhedra"]]&/@variants

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->700]

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[.5],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]

TextGrid[
Prepend[
Table[{
pname,
RecognizeGraph@PolyhedronGraph@PolyhedronData[pname,"CompoundInterior","Polyhedron"],
RecognizeGraph@PolyhedronGraph@PolyhedronData[pname,"ConvexHull","Polyhedron"]
},
{pname,variants}],
Thread[Style[{"compound","interior","convex hull"},Bold]]],
Dividers->All]

Polyhedron: {"OctahedronFourCompound", 1}

Orientations

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

PolyhedronData[{"OctahedronFourCompound",1}]

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

Polyhedron

Graphics3D[{Opacity[.7],PolyhedronData[{"OctahedronFourCompound",1},"Polyhedron"]}]

Face orientation

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

Region

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

Solid and dual

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

Graphics3D[{{Opacity[.3],
Red,PolyhedronData[{"OctahedronFourCompound",1},"Polyhedron"],
Green,PolyhedronData[{"OctahedronFourCompound",1},"Dual","Polyhedron"]},
{Yellow,PolyhedronData[{"OctahedronFourCompound",1},"Midsphere"]}},Boxed->False]

Polyhedron: {"OctahedronFourCompound", 2}

Orientations

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

PolyhedronData[{"OctahedronFourCompound",2}]

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

Polyhedron

Graphics3D[{Opacity[.7],PolyhedronData[{"OctahedronFourCompound",2},"Polyhedron"]}]

Face orientation

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

Region

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

Solid and dual

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

Graphics3D[{{Opacity[.3],
Red,PolyhedronData[{"OctahedronFourCompound",2},"Polyhedron"],
Green,PolyhedronData[{"OctahedronFourCompound",2},"Dual","Polyhedron"]},
{Yellow,PolyhedronData[{"OctahedronFourCompound",2},"Midsphere"]}},Boxed->False]

Polyhedron: {"OctahedronFourCompound", 3}

Orientations

PolyhedronData[{"OctahedronFourCompound",3},"Orientations"]

PolyhedronData[{"OctahedronFourCompound",3}]

Show[PolyhedronData[{{"OctahedronFourCompound",3},#}],Boxed->False]&/@PolyhedronData[{"OctahedronFourCompound",3},"Orientations"]

Polyhedron

Graphics3D[{Opacity[.7],PolyhedronData[{"OctahedronFourCompound",3},"Polyhedron"]}]

Face orientation

Graphics3D[{FaceForm[Green,Red],PolyhedronData[{"OctahedronFourCompound",3},"Polyhedron"]}]

Region

PolyhedronData[{"OctahedronFourCompound",3},"Region"]

Solid and dual

PolyhedronData[{"OctahedronFourCompound",3},#]&/@{"Graphics3D","Dual","DualCompound"}

Graphics3D[{{Opacity[.3],
Red,PolyhedronData[{"OctahedronFourCompound",3},"Polyhedron"],
Green,PolyhedronData[{"OctahedronFourCompound",3},"Dual","Polyhedron"]},
{Yellow,PolyhedronData[{"OctahedronFourCompound",3},"Midsphere"]}},Boxed->False]

Polyhedron: {"OctahedronFourCompound", 4}

Orientations

PolyhedronData[{"OctahedronFourCompound",4},"Orientations"]

PolyhedronData[{"OctahedronFourCompound",4}]

Show[PolyhedronData[{{"OctahedronFourCompound",4},#}],Boxed->False]&/@PolyhedronData[{"OctahedronFourCompound",4},"Orientations"]

Polyhedron

Graphics3D[{Opacity[.7],PolyhedronData[{"OctahedronFourCompound",4},"Polyhedron"]}]

Face orientation

Graphics3D[{FaceForm[Green,Red],PolyhedronData[{"OctahedronFourCompound",4},"Polyhedron"]}]

Region

PolyhedronData[{"OctahedronFourCompound",4},"Region"]

Solid and dual

PolyhedronData[{"OctahedronFourCompound",4},#]&/@{"Graphics3D","Dual","DualCompound"}

Graphics3D[{{Opacity[.3],
Red,PolyhedronData[{"OctahedronFourCompound",4},"Polyhedron"],
Green,PolyhedronData[{"OctahedronFourCompound",4},"Dual","Polyhedron"]},
{Yellow,PolyhedronData[{"OctahedronFourCompound",4},"Midsphere"]}},Boxed->False]

Polyhedron: {"OctahedronFourCompound", 5}

Orientations

PolyhedronData[{"OctahedronFourCompound",5},"Orientations"]

PolyhedronData[{"OctahedronFourCompound",5}]

Show[PolyhedronData[{{"OctahedronFourCompound",5},#}],Boxed->False]&/@PolyhedronData[{"OctahedronFourCompound",5},"Orientations"]

Polyhedron

Graphics3D[{Opacity[.7],PolyhedronData[{"OctahedronFourCompound",5},"Polyhedron"]}]

Face orientation

Graphics3D[{FaceForm[Green,Red],PolyhedronData[{"OctahedronFourCompound",5},"Polyhedron"]}]

Region

PolyhedronData[{"OctahedronFourCompound",5},"Region"]

Solid and dual

PolyhedronData[{"OctahedronFourCompound",5},#]&/@{"Graphics3D","Dual","DualCompound"}

Graphics3D[{{Opacity[.3],
Red,PolyhedronData[{"OctahedronFourCompound",5},"Polyhedron"],
Green,PolyhedronData[{"OctahedronFourCompound",5},"Dual","Polyhedron"]},
{Yellow,PolyhedronData[{"OctahedronFourCompound",5},"Midsphere"]}},Boxed->False]

Properties: {"OctahedronFourCompound", 1}

Initialization

<<MathWorld`Polyhedra`

p=PolyhedronData[pname={"OctahedronFourCompound",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[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[.2],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

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"]

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

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

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

{
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]

pname

Graphics3D[{{Opacity[.2],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"]

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

g=PolyhedronGraph[p]

SurfaceArea

PolyhedronData[pname,"SurfaceArea"]

SurfaceArea[p]

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

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"]

Properties: {"OctahedronFourCompound", 2}

Initialization

<<MathWorld`Polyhedra`

p=PolyhedronData[pname={"OctahedronFourCompound",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])//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[.3],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]]

pname

gname=RecognizeGraph@PolyhedronGraph[inthull]

GraphData[gname,"PolyhedralEmbeddings"]

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

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"]

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

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

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

{
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[.5],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

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"]

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"]]

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"]

Properties: {"OctahedronFourCompound", 3}

Initialization

<<MathWorld`Polyhedra`

p=PolyhedronData[pname={"OctahedronFourCompound",3},"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[1],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[.2],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]]

pname

gname=RecognizeGraph@PolyhedronGraph[inthull]

GraphData[gname,"PolyhedralEmbeddings"]

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

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"]

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

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

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

{
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[.2],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

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"]

GraphData[gn]

emb=GraphData[gn,"PolyhedralEmbeddings"]

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

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

pname

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

g=PolyhedronGraph[p]

SurfaceArea

PolyhedronData[pname,"SurfaceArea"]

SurfaceArea[p]

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

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"]

Properties: {"OctahedronFourCompound", 4}

Initialization

<<MathWorld`Polyhedra`

p=PolyhedronData[pname={"OctahedronFourCompound",4},"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[1],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[.2],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]]

pname

gname=RecognizeGraph@PolyhedronGraph[inthull]

GraphData[gname,"PolyhedralEmbeddings"]

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

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"]

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

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

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

{
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[.5],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

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"]

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"]]

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"]

Properties: {"OctahedronFourCompound", 5}

Initialization

<<MathWorld`Polyhedra`

p=PolyhedronData[pname={"OctahedronFourCompound",5},"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[1],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[.2],p},Red,int},Boxed->False]

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

Exactify

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

vexact=Map[ToAlgebraicRoot[SetPrecision[#,14],"MaxDegree"->4]&,Chop[int[[1]]],{2}]//ToRadicals

Length[vexact]

inthull=ConvexHullRegion[Union@vexact]

v=inthull[[1]]

faces=inthull[[2]]

pname

gname=RecognizeGraph@PolyhedronGraph[inthull]

GraphData[gname,"PolyhedralEmbeddings"]

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

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"]

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

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

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

{
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[.5],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

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"]

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"]]

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"]

Construction

<<MathWorld`Polyhedra`

PolyhedronData["Octahedron","Orientations"]

{"OctahedronFourCompound", 1}: PolyhedronRotationCompound[PolyhedronData["Octahedron", "Polyhedron"], "Face"

Graphics3D[PolyhedronRotationCompound[PolyhedronData["Octahedron","Polyhedron"],"Face",Debug->True]]

{"OctahedronFourCompound", 1} pieces

<<DiscreteMath`Combinatorica`;
<<SolidGeometry`;
<<Simplify`;
<<Platonic`;

poly=Flatten[Polyhedron[Octahedron4Compound]];

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

Timing[Length[lines=FaceIntersections[poly]]]

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

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

Timing[Length[p2=Union[Map[ToExact,p,{2}]]]]

p3=Select[p2,#[[2]]<0&];

Show[Graphics3D[{Polyhedron[Octahedron4Compound],
PointSize[.02],
Point/@p3,
MapIndexed[Text[#2[[1]],#1+{0,0,.05},Background->White]&,p3]
}],Boxed->False]

{Sqrt[#.#]&[Subtract@@p3[[{114,135}]]],Sqrt[#.#]&[Subtract@@p3[[{144,160}]]]}//N

Sqrt[#.#]&/@(Subtract@@@Partition[p3[[{114,121,144}]],2,1,1])

poly1=Polygon2D[p3[[{114,135,143}]]];
poly2=Polygon2D[p3[[{114,115,143}]]];

Show[Graphics[piece1=ClosedLine/@{poly1[[1]],{{1,0},{0,-1}}.#&/@poly2[[1]]}],AspectRatio->Automatic]

poly41=({{-1,0},{0,1}}.#&/@First[Polygon2D[p3[[{114,144,160,135}]]]]);

ang=ArcTan@@(poly41[[2]]-poly41[[1]])-Pi/3;

poly3=First@Polygon2D[p3[[{114,121,144}]]];
poly4=RotationMatrix[ang].#&/@poly41;
poly42={{-1,0},{0,1}}.#+{Sqrt[2]/3,0}&/@poly4;
poly5=(#+{Sqrt[2]/6,1/Sqrt[6]}&/@First[Polygon2D[p3[[{144,162,166,160}]]]]);

Sqrt[2/9-2/36]

Show[Graphics[{piece2=ClosedLine/@{poly3,poly4,poly42,poly5}},AspectRatio->Automatic]]

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

<<Platonic`

Show[Graphics[NetDiagram[Octahedron4Compound]],AspectRatio->Automatic]

s=FullSimplify/@NetEdgeLengths[Octahedron4Compound]

sides=Sort[Union[s],Less]

sides//N

NetSurfaceArea[Octahedron4Compound]//FullSimplify

N[%]

PolyhedronEdgeLengths[Polyhedron[Cube4Compound]]//FullSimplify

{"OctahedronFourCompound", 2}: VertexSubsetCompound["DeltoidalIcositetrahedron","Octahedron"]]

Graphics3D[VertexSubsetCompound["DeltoidalIcositetrahedron","Octahedron"]]

{"OctahedronFourCompound", 3}: PolyhedronRotationCompound[PolyhedronData[{"Octahedron", "C2"}, "Polyhedron"], "C2", 4]

Graphics3D[compound=PolyhedronRotationCompound[PolyhedronData[{"Octahedron","C2"},"Polyhedron"],"C2",4,Debug->True]]

{"OctahedronFourCompound", 4}: PolyhedronRotationCompound[PolyhedronData[{"Octahedron", "C3"}, "Polyhedron"], "C3", 4]

Graphics3D[compound=PolyhedronRotationCompound[PolyhedronData[{"Octahedron","C3"},"Polyhedron"],"C3",4,Debug->True]]

{"OctahedronFourCompound", 5}: PolyhedronRotationCompound[PolyhedronData[{"Octahedron", "C4"}, "Polyhedron"], "C4", 4]

Graphics3D[compound=PolyhedronRotationCompound[PolyhedronData[{"Octahedron","C4"},"Polyhedron"],"C4",4,Debug->True]]

PolyhedronData

PolyhedronDataString

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

WriteString[file,With[
{
base="Octahedron",
n=4,
facecnt=PolyhedronFaceCount[compound],
ind=5
},
PolyhedronDataString[compound,
"StandardName"->{base<>StringJoin@@(Capitalize/@StringSplit[IntegerName[n],"\[Hyphen]"])<>"Compound",ind},
"AlternateStandardNames"->{},
"Name"->IntegerName[ind,"Ordinal"]<>" "<>ToLowerCase[base]<>" "<>ToString[n]<>"\[Hyphen]compound",
"Information"->base<>ToString[n]<>"-Compound",
"PolyhedronIndices"->"Partition[Range["<>ToString[facecnt]<>"], "<>ToString[facecnt/n]<>"]",
"Classes"->{"Compound"}
]]]

Close[file]