Augmented Truncated Dodecahedron

Author

Eric W. Weisstein
October 23, 2024

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

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

©2024 Wolfram Research, Inc. except for portions noted otherwise

Graphics

Solid, net, and skeleton

With[{pname="AugmentedTruncatedDodecahedron"},
GraphicsRow[Show[#,ImageSize->{Automatic,250}]&/@{
Graphics3D[{Opacity[.8],PolyhedronData[pname,"Polyhedron"]},Method->{"ShrinkWrap"->True},Boxed->False],
GraphData[pname<>"Graph"]//StyleGraphs,
PolyhedronData[pname,"Net"]
}]
]

PolyhedronData

PolyhedronData["AugmentedTruncatedDodecahedron","Orientations"]

PolyhedronData["AugmentedTruncatedDodecahedron"]

PolyhedronData["AugmentedTruncatedDodecahedron","VertexCoordinates"]

PolyhedronData (V12.1)

PolyhedronData["AugmentedTruncatedDodecahedron"]

PolyhedronData["AugmentedTruncatedDodecahedron","VertexCoordinates"]

Netlib

PolyhedronName[JohnsonSolid[68]]

Show[Graphics3D[Polyhedron[JohnsonSolid[68]]],Boxed->False]

Net

PolyhedronData

PolyhedronData["AugmentedTruncatedDodecahedron","Net"]

PolyhedronData["AugmentedTruncatedDodecahedron","Net","Colored"]

Netlib

Show[Graphics[Net[JohnsonSolid[68]]],AspectRatio->Automatic]

Skeleton graph

PolyhedronData["AugmentedTruncatedDodecahedron","SkeletonGraph"]

PolyhedronData["AugmentedTruncatedDodecahedron","Skeleton","Name"]

GraphData[%,"Graph","All"]

Construction

PolyhedronData/@{"TruncatedDodecahedron","AugmentedTruncatedDodecahedron"}

Construct by adding a pentagonal cupola onto "TruncatedDodecahedron"

Graphics3D[MyPolyhedron[Cupola[5]]]

MyPolyhedron[Cupola[5]]//PolyhedronCoordinates

h=PolyhedronData["TruncatedDodecahedron","VertexCoordinates"][[All,-1]]//Max

MyPolyhedron[Cupola[5]]//InputForm

cup=MyPolyhedron[Cupola[5]]

cuptop=Polyhedron[({0,0,h}+#)&/@PolyhedronCoordinates[cup],Select[cup[[2]],Length[#]≠10&]]

Not allowed because adjacent triangles are coplanar

Graphics3D[{PolyhedronData["TruncatedDodecahedron","Polyhedron"],cuptop}]

PolyhedronData["AugmentedTruncatedDodecahedron","FaceCountRules"]

Select[PolyhedronData[All],PolyhedronData[#,"FaceCountRules"]==={3->25,4->5,5->1,10->11}&]

vcup=({0,0,h}+RotationMatrix[π/5,{0,0,1}].#)&/@PolyhedronCoordinates[cup]//FullSimplify

cuptop2=Polyhedron[({0,0,h}+RotationMatrix[π/5,{0,0,1}].#)&/@PolyhedronCoordinates[cup],Select[cup[[2]],Length[#]≠10&]]

vtop=vcup[[SelectFirst[cuptop[[2]],Length[#]==5&]]]

vnew=Join[PolyhedronData["TruncatedDodecahedron","VertexCoordinates"],vtop]//FullSimplify

Length[vnew]

top=Select[PolyhedronData["TruncatedDodecahedron","FaceIndices"],Union[(PolyhedronData["TruncatedDodecahedron","VertexCoordinates"][[#]])[[All,-1]]]=={Max@PolyhedronData["TruncatedDodecahedron","VertexCoordinates"][[All,-1]]}&]

facesnew=Complement[Join[PolyhedronData["TruncatedDodecahedron","FaceIndices"],{Range[61,65]},{{1,34,64,63},{34,48,64},{64,48,50,65},{65,50,36},{65,36,3,61},{61,3,42},{61,42,46,62},{46,44,62},{62,44,40,63},{63,40,1}}],top]

Graphics3D[{FaceForm[Green,Red],MapIndexed[Text[#2[[1]],#,Background->White]&,vnew],p=Polyhedron[vnew,facesnew]}]

SmallestRotation[l_]:=Sort[NestList[RotateRight,l,Length[l]-1]][[1]]

facesnew2=Sort[SmallestRotation/@facesnew]

Max[Flatten[facesnew2]]

Graphics3D[{FaceForm[Green,Red],MapIndexed[Text[#2[[1]],#,Background->White]&,vnew],p=Polyhedron[vnew,facesnew2]}]

SimplePolyhedronQ[p//N]

ConvexPolyhedronQ[p]

Properties

<<MathWorld`PolyhedronOperations`

p=PolyhedronData[pname="AugmentedTruncatedDodecahedron","Polyhedron"]

p2=Polyhedron[RootReduce[PolyhedronData[pname,"VertexCoordinates"]],PolyhedronData[pname,"FaceIndices"]]

AdjacentFaceIndices

AdjacentFaceIndices[p]

Centroid

PolyhedronData["AugmentedTruncatedDodecahedron","Centroid"]

N[%]

N[xbar=RegionCentroid[p]]

bug(391999): Volume[PolyhedronData["TruncatedDodecahedron", "Polyhedron"]] hangs uninterruptedly

RegionCentroid[p//N]

numeric does too

N[xbar=PolyhedronCentroid[p,vol,Identity,Identity]]//Timing

ByteCount[xbar]

xbar

ByteCount/@xbar

ByteCount/@(simp=Simplify/@xbar)//Timing

FullSimplify[xbar[[1]]]//Timing

N[xbar[[1]],100]

FullSimplify[xbar[[2]]]//Timing

Since the centroid of the "TruncatedDodecahedron" is at the origin, the z-component of the "AugmentedTruncatedDodecahedron" is just the component added by the cupola

cup=MyPolyhedron[Cupola[5]]

h=PolyhedronData["TruncatedDodecahedron","VertexCoordinates"][[All,-1]]//Max

Graphics3D[offsetcupola=Polyhedron[({0,0,h}+RotationMatrix[π/5,{0,0,1}].#)&/@PolyhedronCoordinates[cup],cup[[2]]]]

(centroidtop=RegionCentroid[offsetcupola][[-1]]Volume[offsetcupola]/vol)//N

ByteCount[centroidtop]

FullSimplify[centroidtop]//Timing

N[((Sqrt[1752673-(947638/(Sqrt[5]))])/16088)-xbar[[-1]],100]

Convex

ConvexPolyhedronQ[p]

DihedralAngles

PolyhedronData["AugmentedTruncatedDodecahedron","DihedralAngles"]

dih=DihedralAngles[p]

N[dih]

(dih2=SimplifyNumericallyDistinct[dih,Debug->True])//Timing

N[dih-dih2]//Chop

GeneralizedDiameter

PolyhedronData["AugmentedTruncatedDodecahedron","GeneralizedDiameter"]

GeneralizedDiameter[p]//FullSimplify

N[%]

InertiaTensor

vol=505/12+(81 Sqrt[5])/4;

PolyhedronInertiaTensor[p//N,vol]//Chop

(i1=MomentOfInertia[p//N,{0,0,0}])//Chop

...hangs...

(i=PolyhedronInertiaTensor[p,vol,Identity,Identity]);//Timing

ByteCount[i]

(i2=PolyhedronInertiaTensor[p,vol]);//Timing

ByteCount[i2]

i2

N[%]

MeanCylindricalRadius

bug(407468): RegionCentroid[PolyhedronData["TruncatedDodecahedron", "Polyhedron"]], Integrate[1, {x, y, z} ∈ PolyhedronData["TruncatedDodecahedron", "Polyhedron"]], ... are uninterruptible

bug(409814): Integrate[x^2 + y^2, {x, y, z} ∈ PolyhedronData["TruncatedDodecahedron", "Polyhedron"]] is uninterruptible

((NIntegrate[(Sqrt[(x^2)+(y^2)]),{x,y,z}∈p2])/(PolyhedronData[pname,"Volume"]))//Timing

(i=((Integrate[(Sqrt[(x^2)+(y^2)]),{x,y,z}∈p])/(PolyhedronData[pname,"Volume"])))//Timing

N[i]

MeanCylindricalSquareRadius

((NIntegrate[(x^2)+(y^2),{x,y,z}∈p2])/(PolyhedronData[pname,"Volume"]))//Timing

(i=((Integrate[(x^2)+(y^2),{x,y,z}∈p2])/(PolyhedronData[pname,"Volume"])))//Timing

N[i]

r=RootApproximant[N[i,1000],32]//FullSimplify

N[r-i,10^4]//Quiet

MeanSphericalRadius

((NIntegrate[(Sqrt[(x^2)+(y^2)+(z^2)]),{x,y,z}∈p2])/(PolyhedronData[pname,"Volume"]))//Timing

(i=((Integrate[(Sqrt[(x^2)+(y^2)+(z^2)]),{x,y,z}∈p])/(PolyhedronData[pname,"Volume"])))//Timing

N[i]

MeanSphericalSquareRadius

((NIntegrate[(x^2)+(y^2)+(z^2),{x,y,z}∈p2])/(PolyhedronData[pname,"Volume"]))//Timing

(i=((Integrate[(x^2)+(y^2)+(z^2),{x,y,z}∈p2])/(PolyhedronData[pname,"Volume"])))//Timing

N[i]

r=RootApproximant[N[i,1000],32]//FullSimplify

N[r-i,10^4]//Quiet

RegionFunction

PolyhedronData["AugmentedTruncatedDodecahedron","RegionFunction"]

bounds=RegionBounds[ImplicitRegion[Evaluate[N[PolyhedronData["AugmentedTruncatedDodecahedron","RegionFunction"][x,y,z]]],{x,y,z}]]

ranges=Flatten/@Transpose[{{x,y,z},.1{-1,1}+#&/@bounds}]

RegionPlot3D[Evaluate[N[PolyhedronData["AugmentedTruncatedDodecahedron","RegionFunction"][x,y,z]]],Evaluate[Sequence@@ranges],BoxRatios->Automatic,PlotPoints->40]

ByteCount[ineq=PolyhedronInequalities[p,{#1,#2,#3}]]//Timing

ByteCount[ineq2=FullSimplify[ineq]]//Timing

ineq2

SkeletonCoordinates

vc=PolyhedronData["AugmentedTruncatedDodecahedron","V"]

g=Graph[Range[vc],UndirectedEdge@@@PolyhedronData["AugmentedTruncatedDodecahedron","EdgeIndices"]]

r=RecognizeGraph[g]

GraphData[r,"Graph","All"]

ToCommonEdges[GraphData[r],g]

Volume

vol=PolyhedronData["AugmentedTruncatedDodecahedron","Volume"]

N[%]

Volume[p//N]

(v1=Volume[p])//N//Timing

bug(391999): Volume[PolyhedronData["TruncatedDodecahedron", "Polyhedron"]] hangs uninterriptibly

(v2=PolyhedronVolume[p])//N//Timing

ByteCount[v2]

v2

(v3=PolyhedronVolume[p,Method->"GaussLaw"])//N//Timing

ByteCount[v3]

vol=FullSimplify[v3]

N[%]