Truncated Dodecahedron

Author

Eric W. Weisstein
March 14, 2024

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

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

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

Figure

With[{pname="TruncatedDodecahedron"},
GraphicsRow[{
Graphics3D[{Opacity[.8],PolyhedronData[pname,"Polyhedron"]},Boxed->False],
Graphics3D[{PolyhedronData[pname,"Lines"]},Boxed->False],
PolyhedronData[pname,"Net"]
},ImageSize->700]]

Solid

PolyhedronData["TruncatedDodecahedron","Orientations"]

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

Solid and Dual

GraphicsRow[PolyhedronData["TruncatedDodecahedron",#]&/@{"Graphics3D","Dual","DualCompound"}]

Net

PolyhedronData["TruncatedDodecahedron","Net"]

PolyhedronData["TruncatedDodecahedron","Net","Colored"]

NetPrintout[PolyhedronData["TruncatedDodecahedron","Polyhedron"],"TruncatedDodecahedron.pdf"]

Properties

Initialization

<<MathWorld`Polyhedra`

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

vol=PolyhedronData[pname,"Volume"]

Centroid

PolyhedronData[pname,"Centroid"]

RegionCentroid[p]//FullSimplify//Timing

PolyhedronCentroid[p]//FullSimplify//Quiet//Timing

Circumradius

PolyhedronData[pname,"Circumradius"]

RootReduce[%]

Circumsphere[p]//RootReduce//Timing

MyCircumsphere[p]//RootReduce//Timing

ConvexHullOf

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

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

DihedralAngles

Show[Graphics3D[{poly,
MapIndexed[Text[#2[[1]],#1]&,Centroid/@poly//N]
}],Boxed->False]

{3,10}

Show[Graphics3D[p=poly[[{1,29}]]],Boxed->False]

π-RootReduce/@(DihedralAngle@@p)

InputForm[%]

N[%]180/Pi

{10,10}

Show[Graphics3D[p=poly[[{1,11}]]],Boxed->False]

RootReduce[π-DihedralAngle@@p]

InputForm[%]

N[%]180/Pi

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]

EdgeLengths

PolyhedronData[pname,"EdgeLengths"]

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

Faces

PolyhedronFaceList[poly]

Show[GraphicsArray[partition[PolyhedronFaces[poly//N],15]]]

Triangular Faces

Together//@ToRadicals[RootReduce[norm[Centroid[Select[poly,Length[#[[1]]]==3&][[1,1]]]]]]

Dodecagonal Faces

Together//@ToRadicals[RootReduce[norm[Centroid[Select[poly,Length[#[[1]]]==10&][[1,1]]]]]]

Inradius

PolyhedronData[pname,"Inradius"]

Insphere[p]

MyInsphere[p]//RootReduce//Timing

InertiaTensor

PolyhedronData[pname,"InertiaTensor"]

(i=((MomentOfInertia[p2,{0,0,0}])/(Volume[p2])))//Timing

i2=Map[RootApproximant[N[#,1000],32]&,i,{2}]//Quiet//FullSimplify

Equal@@Tr[i2,List]

i2[[1,1]]

PolyhedronInertiaTensor[p]//Timing

MeanCylindricalRadius

PolyhedronData[pname,"MeanCylindricalRadius"]

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

ClosestMeanCylindricalRadius[%]

Integrate over polyhedron

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

N[rxy]

Divergence theorem [integral returns unevaluated]

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

N[rxy]

LogSimplify[rxy]//FullSimplify

CAD

...needs 43+ GB...

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

MaxMemoryUsed[]/10^9.

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

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

CAD over  1/XX

reg=FullSimplify[PolyhedronData[pname,"RegionFunction"][x,y,z]&&0<y<Tan[Pi/5]x]

Show[Graphics3D[{Opacity[.2],p}],RegionPlot3D[ImplicitRegion[reg,{x,y,z}],PlotPoints->50]]

((Volume[ImplicitRegion[reg,{x,y,z}]])/vol)//FullSimplify

cad=CylindricalDecomposition[reg,{x,y,z},"Function"]

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

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

N[rxy]

ByteCount[rxylog=LogSimplify[rxy,"ZeroTestNumericalPrecision"->10000,WorkingPrecision->10000,Debug->True]]//Timing

ByteCount[rxycollect=Collect[TrigToExp[rxylog],Log[_]]//RootReduce]//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[%]

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

N[rxy2]

(rxy2simp=RootReduce[rxy]//FullSimplify)//Timing

N[rxy2simp]

MeanSphericalRadius

PolyhedronData[pname,"MeanSphericalRadius"]

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

ClosestMeanSphericalRadius[%]

Integrate over polyhedron

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

N[rxyz]

Divergence theorem

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

N[rxyz]

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)

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

N[rxyz,20]

rxysimp1=ResourceFunction["LogSimplify"][rxy]//FullSimplify

CAD over  1/XX

reg=FullSimplify[PolyhedronData[pname,"RegionFunction"][x,y,z]&&0<y<Tan[Pi/5]x]

Show[Graphics3D[{Opacity[.2],p}],RegionPlot3D[ImplicitRegion[reg,{x,y,z}],PlotPoints->50]]

((Volume[ImplicitRegion[reg,{x,y,z}]])/vol)//FullSimplify

cad=CylindricalDecomposition[reg,{x,y,z},"Function"]

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

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

ByteCount[rxyzlog=LogSimplify[rxyz,"ZeroTestNumericalPrecision"->10000,WorkingPrecision->10000,Debug->True]]//Timing

ByteCount[rxyzcollect=Collect[TrigToExp[rxyzlog],Log[_]]//RootReduce]//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"]

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

i//FullSimplify//Timing

i//RootReduce//FullSimplify//Timing

Midradius

PolyhedronData[pname,"Midradius"]

(mid=Midsphere[p])//Timing

SurfaceArea

PolyhedronData[pname,"SurfaceArea"]

RootReduce[%]

SurfaceArea[p]//RootReduce//Timing

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

Volume

PolyhedronData[pname,"Volume"]

Volume[p]//FullSimplify//Timing

PolyhedronVolume[p]//FullSimplify//Timing

Face diagram

polygon[n_,a_,th_:0]:=Module[{R=a Csc[Pi/n]/2},Line[R Table[{Cos[2Pi(i+th)/n],Sin[2Pi(i+th)/n]},{i,0,n}]]
]

poly5=polygon[5,1];
poly10=polygon[10,1/Sqrt[5],1/2];

Show[Graphics[{poly5,
{RGBColor[1,0,0],poly10},
Line[{(poly10[[1,1]]+poly10[[1,-2]])/2,{0,0},(poly10[[1,1]]+poly10[[1,2]])/2}],
Line[{{0,0},poly10[[1,1]]}]
}],AspectRatio->Automatic]

Projections

PolyhedronProjectionGraph[PolyhedronData["TruncatedDodecahedron","Polyhedron"],#]&/@{{0,1,0,(1/2) (π-2 ArcTan[(1/2) (1+(Sqrt[5]))])},{(1/4) (7-3 (Sqrt[5])),(NumericalApproximation[Root, (RowDefault["-0.225", FEExpressionsNumericalApproximationElider]), Root[5-100 ((#1)^2)+16 ((#1)^4)&,2], ]),1,-ArcCot[(Sqrt[10])]},{(1/2) (3-(Sqrt[5])),0,-1},{1,(1/41) (Sqrt[365+158 (Sqrt[5])]),(1/41) (17+5 (Sqrt[5]))},{(Sqrt[1-(2/(Sqrt[5]))]),1,0},{1,1,0},{1,0,(1/4) (3-(Sqrt[5]))},{0,0,1}}

GraphData["TruncatedDodecahedralGraph","Graph","Perspective"]

Vertex subset hulls

p="TruncatedDodecahedron";
v=PolyhedronData[p,"VertexCoordinates"];

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

Construction

C5

(*TruncatedDodecahedron:=MiscPolyhedra`UniformPolyhedron[26]*)TruncatedDodecahedronExact:=TruncatedDodecahedronExact=With[{d=Polyhedron[Platonic`Dodecahedron]},Module[{v=PolyhedronVertices[d],e=PolyhedronEdges[d],l1=(5-Sqrt[5])/10,l2=(5+Sqrt[5])/10,dods,tris},dods=Sqrt[5]Table[Flatten[{#[[1]]l2+#[[2]]l1,#[[1]]l1+#[[2]]l2}&/@Partition[d[[i,1]],2,1,1],1],{i,Length[d]}];
tris=Sqrt[5]Table[Plus[l2 v[[i]],l1#]&/@Select[Flatten[Select[e,MemberQ[#,v[[i]]]&],1],#≠v[[i]]&],{i,Length[v]}];
OrientCentered[Flatten[ToRadicals[RootReduce[{Polygon/@dods,Polygon/@tris}]]]]]]

C2

Graphics3D[{PolyhedronData["TruncatedDodecahedron","GraphicsComplex"],MapIndexed[Text[#2[[1]],#,Background->White]&,PolyhedronData["TruncatedDodecahedron","VertexCoordinates"]]}]

m=Mean[edge=PolyhedronData["TruncatedDodecahedron","VertexCoordinates"][[{57,53}]]]//FullSimplify

mat=RotationMatrix[{m,{0,0,1}}]//FullSimplify

ArcTan@@Subtract@@FullSimplify[Take[mat.#,2]&/@edge]//FullSimplify

v=(RotationMatrix[-π/5,{0,0,1}].(mat.#))&/@PolyhedronData["TruncatedDodecahedron","VertexCoordinates"]//FullSimplify

p=Polyhedron[v,PolyhedronData["TruncatedDodecahedron","FaceIndices"]]

Graphics3D[p]

reg=PolyhedronData[{"TruncatedDodecahedron","C2"},"RegionFunction"]

bounds=RegionBounds[ImplicitRegion[Evaluate[N[reg[x,y,z]]],{x,y,z}]]

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

RegionPlot3D[Evaluate[N[reg[x,y,z]]],Evaluate[Sequence@@ranges],BoxRatios->Automatic,PlotPoints->50]

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

ineq2=FullSimplify[ineq];//Timing

ineq2

C3

Graphics3D[{PolyhedronData["TruncatedDodecahedron","GraphicsComplex"],MapIndexed[Text[#2[[1]],#,Background->White]&,PolyhedronData["TruncatedDodecahedron","VertexCoordinates"]]}]

triangles=Select[PolyhedronData["TruncatedDodecahedron","FaceIndices"],Length[#]==3&]

ms=FullSimplify[Mean[PolyhedronData["TruncatedDodecahedron","VertexCoordinates"][[#]]]]&/@triangles

ByteCount/@(mats=RotationMatrix[{#,{0,0,1}}]&/@ms)

(mat=FullSimplify[mats[[1]]])//Timing

v=({#1,#2,-#3}&@@@((RotationMatrix[0,{0,0,π}].(mat.#))&/@PolyhedronData["TruncatedDodecahedron","VertexCoordinates"]))//FullSimplify

v=RotationMatrix[π,{0,0,1}].#&/@({#1,#2,-#3}&@@@{{(Sqrt[3+((4 (Sqrt[5]))/3)]),(1/2) (-1-(Sqrt[5])),(1/2) (Sqrt[(1/6) (3+(Sqrt[5]))])},{-(Sqrt[3+((4 (Sqrt[5]))/3)]),(1/2) (-1-(Sqrt[5])),-(1/2) (Sqrt[(1/6) (3+(Sqrt[5]))])},{(Sqrt[3+((4 (Sqrt[5]))/3)]),(1/2) (1+(Sqrt[5])),(1/2) (Sqrt[(1/6) (3+(Sqrt[5]))])},{-(Sqrt[3+((4 (Sqrt[5]))/3)]),(1/2) (1+(Sqrt[5])),-(1/2) (Sqrt[(1/6) (3+(Sqrt[5]))])},{(NumericalApproximation[Root, (RowDefault["0.178", FEExpressionsNumericalApproximationElider]), Root[1-36 ((#1)^2)+144 ((#1)^4)&,3], ]),(1/4) (-5-3 (Sqrt[5])),(1/2) (Sqrt[(1/6) (3+(Sqrt[5]))])},{(NumericalApproximation[Root, (RowDefault["0.178", FEExpressionsNumericalApproximationElider]), Root[1-36 ((#1)^2)+144 ((#1)^4)&,3], ]),(1/4) (5+3 (Sqrt[5])),(1/2) (Sqrt[(1/6) (3+(Sqrt[5]))])},{(Sqrt[(7/4)+((2 (Sqrt[5]))/3)]),-1-((Sqrt[5])/2),((5+(Sqrt[5]))/(4 (Sqrt[3])))},{(Sqrt[(7/4)+((2 (Sqrt[5]))/3)]),(1/2) (2+(Sqrt[5])),((5+(Sqrt[5]))/(4 (Sqrt[3])))},{-(1/(Sqrt[3])),0,-((9+5 (Sqrt[5]))/(4 (Sqrt[3])))},{((Sqrt[(5/3)])/2),-1-((Sqrt[5])/2),-(Sqrt[(47/24)+((7 (Sqrt[5]))/8)])},{((Sqrt[(5/3)])/2),(1/2) (2+(Sqrt[5])),-(Sqrt[(47/24)+((7 (Sqrt[5]))/8)])},{0,(1/2) (-3-(Sqrt[5])),-(1/2) (Sqrt[(3/2) (3+(Sqrt[5]))])},{0,(1/2) (3+(Sqrt[5])),-(1/2) (Sqrt[(3/2) (3+(Sqrt[5]))])},{-((4+(Sqrt[5]))/(2 (Sqrt[3]))),-1-((Sqrt[5])/2),-((5+(Sqrt[5]))/(4 (Sqrt[3])))},{-((4+(Sqrt[5]))/(2 (Sqrt[3]))),(1/2) (2+(Sqrt[5])),-((5+(Sqrt[5]))/(4 (Sqrt[3])))},{0,(1/2) (-3-(Sqrt[5])),(1/2) (Sqrt[(3/2) (3+(Sqrt[5]))])},{0,(1/2) (3+(Sqrt[5])),(1/2) (Sqrt[(3/2) (3+(Sqrt[5]))])},{(1/(Sqrt[3])),0,((9+5 (Sqrt[5]))/(4 (Sqrt[3])))},{-(1/4) (Sqrt[3]) (3+(Sqrt[5])),(1/4) (-3-(Sqrt[5])),(1/4) ((Sqrt[3])+(Sqrt[15]))},{-(1/4) (Sqrt[3]) (3+(Sqrt[5])),(1/4) (3+(Sqrt[5])),(1/4) ((Sqrt[3])+(Sqrt[15]))},{-((Sqrt[(5/3)])/2),-1-((Sqrt[5])/2),(Sqrt[(47/24)+((7 (Sqrt[5]))/8)])},{-((Sqrt[(5/3)])/2),(1/2) (2+(Sqrt[5])),(Sqrt[(47/24)+((7 (Sqrt[5]))/8)])},{(Sqrt[(1/6) (7+3 (Sqrt[5]))]),0,(Sqrt[(83/24)+((11 (Sqrt[5]))/8)])},{-((3+(Sqrt[5]))/(2 (Sqrt[3]))),(1/2) (-1-(Sqrt[5])),(Sqrt[(47/24)+((7 (Sqrt[5]))/8)])},{-((3+(Sqrt[5]))/(2 (Sqrt[3]))),(1/2) (1+(Sqrt[5])),(Sqrt[(47/24)+((7 (Sqrt[5]))/8)])},{((3+(Sqrt[5]))/(4 (Sqrt[3]))),(1/4) (-3-(Sqrt[5])),-(Sqrt[(83/24)+((11 (Sqrt[5]))/8)])},{((3+(Sqrt[5]))/(4 (Sqrt[3]))),(1/4) (3+(Sqrt[5])),-(Sqrt[(83/24)+((11 (Sqrt[5]))/8)])},{-((3+(Sqrt[5]))/(4 (Sqrt[3]))),(1/4) (-3-(Sqrt[5])),(Sqrt[(83/24)+((11 (Sqrt[5]))/8)])},{-((3+(Sqrt[5]))/(4 (Sqrt[3]))),(1/4) (3+(Sqrt[5])),(Sqrt[(83/24)+((11 (Sqrt[5]))/8)])},{-(1/(2 (Sqrt[3]))),-(1/2),((9+5 (Sqrt[5]))/(4 (Sqrt[3])))},{-(1/(2 (Sqrt[3]))),(1/2),((9+5 (Sqrt[5]))/(4 (Sqrt[3])))},{(1/(2 (Sqrt[3]))),-(1/2),-(Sqrt[(103/24)+((15 (Sqrt[5]))/8)])},{(1/(2 (Sqrt[3]))),(1/2),-(Sqrt[(103/24)+((15 (Sqrt[5]))/8)])},{(Sqrt[(21/8)+((9 (Sqrt[5]))/8)]),(1/4) (-3-(Sqrt[5])),(1/2) (Sqrt[(3/2) (3+(Sqrt[5]))])},{-(Sqrt[(29/8)+((35 (Sqrt[5]))/24)]),(1/4) (-3-(Sqrt[5])),(1/2) (Sqrt[(1/6) (3+(Sqrt[5]))])},{(Sqrt[(21/8)+((9 (Sqrt[5]))/8)]),(1/4) (3+(Sqrt[5])),(1/2) (Sqrt[(3/2) (3+(Sqrt[5]))])},{-(Sqrt[(29/8)+((35 (Sqrt[5]))/24)]),(1/4) (3+(Sqrt[5])),(1/2) (Sqrt[(1/6) (3+(Sqrt[5]))])},{(Sqrt[(1/6) (3+(Sqrt[5]))]),(1/2) (-3-(Sqrt[5])),(1/2) (Sqrt[(5/6) (3+(Sqrt[5]))])},{(Sqrt[(1/6) (3+(Sqrt[5]))]),(1/2) (3+(Sqrt[5])),(1/2) (Sqrt[(5/6) (3+(Sqrt[5]))])},{(Sqrt[(29/8)+((35 (Sqrt[5]))/24)]),(1/4) (-3-(Sqrt[5])),-(1/2) (Sqrt[(1/6) (3+(Sqrt[5]))])},{(NumericalApproximation[Root, (RowDefault["-2.27", FEExpressionsNumericalApproximationElider]), Root[9-84 ((#1)^2)+16 ((#1)^4)&,1], ]),(1/4) (-3-(Sqrt[5])),-(1/2) (Sqrt[(3/2) (3+(Sqrt[5]))])},{(Sqrt[(29/8)+((35 (Sqrt[5]))/24)]),(1/4) (3+(Sqrt[5])),-(1/2) (Sqrt[(1/6) (3+(Sqrt[5]))])},{(NumericalApproximation[Root, (RowDefault["-2.27", FEExpressionsNumericalApproximationElider]), Root[9-84 ((#1)^2)+16 ((#1)^4)&,1], ]),(1/4) (3+(Sqrt[5])),-(1/2) (Sqrt[(3/2) (3+(Sqrt[5]))])},{(Sqrt[(15/4)+((5 (Sqrt[5]))/3)]),-(1/2),-(1/2) (Sqrt[(5/6) (3+(Sqrt[5]))])},{-((3+2 (Sqrt[5]))/(2 (Sqrt[3]))),-(1/2),-(Sqrt[(47/24)+((7 (Sqrt[5]))/8)])},{(Sqrt[(15/4)+((5 (Sqrt[5]))/3)]),(1/2),-(1/2) (Sqrt[(5/6) (3+(Sqrt[5]))])},{-((3+2 (Sqrt[5]))/(2 (Sqrt[3]))),(1/2),-(Sqrt[(47/24)+((7 (Sqrt[5]))/8)])},{(1/2) (Sqrt[(29/3)+4 (Sqrt[5])]),-(1/2),(Sqrt[(47/24)+((7 (Sqrt[5]))/8)])},{-((5+2 (Sqrt[5]))/(2 (Sqrt[3]))),-(1/2),(1/2) (Sqrt[(5/6) (3+(Sqrt[5]))])},{(1/2) (Sqrt[(29/3)+4 (Sqrt[5])]),(1/2),(Sqrt[(47/24)+((7 (Sqrt[5]))/8)])},{-((5+2 (Sqrt[5]))/(2 (Sqrt[3]))),(1/2),(1/2) (Sqrt[(5/6) (3+(Sqrt[5]))])},{(NumericalApproximation[Root, (RowDefault["-1.51", FEExpressionsNumericalApproximationElider]), Root[1-21 ((#1)^2)+9 ((#1)^4)&,1], ]),0,-((11+3 (Sqrt[5]))/(4 (Sqrt[3])))},{(Sqrt[(1/6) (7+3 (Sqrt[5]))]),(1/2) (-1-(Sqrt[5])),-(Sqrt[(47/24)+((7 (Sqrt[5]))/8)])},{(Sqrt[(1/6) (7+3 (Sqrt[5]))]),(1/2) (1+(Sqrt[5])),-(Sqrt[(47/24)+((7 (Sqrt[5]))/8)])},{-(Sqrt[(1/6) (3+(Sqrt[5]))]),(1/2) (-3-(Sqrt[5])),-(1/2) (Sqrt[(5/6) (3+(Sqrt[5]))])},{-(Sqrt[(1/6) (3+(Sqrt[5]))]),(1/2) (3+(Sqrt[5])),-(1/2) (Sqrt[(5/6) (3+(Sqrt[5]))])},{(Sqrt[(21/8)+((9 (Sqrt[5]))/8)]),(1/4) (-3-(Sqrt[5])),-(1/2) (Sqrt[(3/2) (3+(Sqrt[5]))])},{(Sqrt[(21/8)+((9 (Sqrt[5]))/8)]),(1/4) (3+(Sqrt[5])),-(1/2) (Sqrt[(3/2) (3+(Sqrt[5]))])},{((-5+(Sqrt[5]))/(4 (Sqrt[15]))),(1/4) (-5-3 (Sqrt[5])),-(1/2) (Sqrt[(1/6) (3+(Sqrt[5]))])},{((-5+(Sqrt[5]))/(4 (Sqrt[15]))),(1/4) (5+3 (Sqrt[5])),-(1/2) (Sqrt[(1/6) (3+(Sqrt[5]))])}})//FullSimplify

p=Polyhedron[v,PolyhedronData["TruncatedDodecahedron","FaceIndices"]]

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

Graphics3D[{FaceForm[Green,Red],p}]

reg=PolyhedronData[{"TruncatedDodecahedron","C3"},"RegionFunction"]

bounds=RegionBounds[ImplicitRegion[Evaluate[N[reg[x,y,z]]],{x,y,z}]]

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

RegionPlot3D[Evaluate[N[reg[x,y,z]]],Evaluate[Sequence@@ranges],BoxRatios->Automatic,PlotPoints->50]

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

ineq2=FullSimplify[ineq];//Timing

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

ineq2=FullSimplify[ineq];//Timing

ineq2