Small Rhombicosidodecahedron

Author

Eric W. Weisstein
January 5, 2024

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

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

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

Figure

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

Solid

PolyhedronData

PolyhedronData["SmallRhombicosidodecahedron","DefaultOrientation"]

PolyhedronData["SmallRhombicosidodecahedron"]

All orientations

PolyhedronData["SmallRhombicosidodecahedron","Orientations"]

PolyhedronData["SmallRhombicosidodecahedron","DefaultOrientation"]

PolyhedronData[{"SmallRhombicosidodecahedron",#}]&/@PolyhedronData["SmallRhombicosidodecahedron","Orientations"]

Solid and Dual

Show[GraphicsArray[Graphics3D[#,Boxed->False]&/@{
poly,
dual=Polyhedron[PolyhedronDual[SmallRhombicosidodecahedron]],{dual,poly}
}]]

UniformPolyhedron

uniform=UniformPolyhedron[stdname="Rhombicosidodecahedron"]

PolyhedronFaceTally[uniform]

indices=UniformPolyhedronIndex[stdname]

UniformPolyhedron/@{
stdname,
Lookup[indices,"Wenninger"],
Sequence@@(List@@@Normal[indices]/."Maeder"->"Uniform")
}

SameQ@@%

Graphics3D/@(polys=Polygon[uniform[[1]],#]&/@uniform[[2]])

Graphics3D[#,Boxed->False]&/@{uniform,polys}

Bulatov

BulatovUniformName[UniformPolyhedronIndex[stdname,"Kaleido"]]

BulatovUniformGraphics3D[UniformPolyhedronIndex[stdname,"Kaleido"],Boxed->False]

bulatov=BulatovUniformPolyhedron[UniformPolyhedronIndex[stdname,"Kaleido"]]

Graphics3D/@(polys=Polygon[bulatov[[1]],#]&/@bulatov[[2]])

PolyhedronFaceTally[bulatov]

Graphics3D[#,Boxed->False]&/@{bulatov,polys}

Maeder

MaederUniformName[UniformPolyhedronIndex[stdname,"Maeder"]]

wythoff=MaederWythoffSymbol[UniformPolyhedronIndex[stdname,"Maeder"]]

With[{n=UniformPolyhedronIndex[stdname,"Maeder"]},Import["https://www.mathconsult.ch/static/unipoly/"<>If[n<10,"0",""]<>ToString[n]<>".html","Images"]][[1]]

maeder=MaederUniformPolyhedron[UniformPolyhedronIndex[stdname,"Maeder"]]

Graphics3D/@(polys=Polygon[maeder[[1]],#]&/@maeder[[2]])

PolyhedronFaceTally[maeder]

Graphics3D[#,Boxed->False]&/@{maeder,polys}

Net

PolyhedronData["SmallRhombicosidodecahedron","Net"]

PolyhedronData["SmallRhombicosidodecahedron","Net","Colored"]

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

Properties

Initialization

<<MathWorld`Polyhedra`

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

vol=PolyhedronData[pname,"Volume"]//Factor

Centroid

PolyhedronData[pname,"Centroid"]

RegionCentroid[p]//FullSimplify

PolyhedronCentroid[p]//FullSimplify

Circumsphere

PolyhedronData[pname,"Circumsphere"]

Circumsphere[p]//FullSimplify//Timing

MyCircumsphere[p]//FullSimplify//Timing

Classes

PolyhedronData[pname,"Classes"]

Compound

PolyhedronData[pname,"Compound"]

Concave

PolyhedronData[pname,"Concave"]

Convex

PolyhedronData[pname,"Convex"]

ConvexPolyhedronQ[p]//Timing

ConvexHull

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

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

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

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

Counts[SimplifyNumericallyDistinct[PolyhedronEdgeLengths[convexhull]]]

Keys[%]//N

{
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

PolyhedronData[pname,"DihedralAngles"]//Tally

DihedralAngles[p]//FullSimplify//Tally

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

Insphere

PolyhedronData[pname,"Insphere"]

Insphere[p]

MyInsphere[p]

InertiaTensor

PolyhedronData[pname,"InertiaTensor"]

((MomentOfInertia[p])/(Volume[p]))//RootReduce//Timing

PolyhedronInertiaTensor[p]//Timing

MeanCylindricalRadius

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

(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}∈p])/(PolyhedronData[pname,"Volume"]))//FullSimplify

Rule[#,((Integrate[(x^2)+(y^2),{x,y,z}∈PolyhedronData[{pname,#},"Polyhedron"]])/(PolyhedronData[pname,"Volume"]))]&/@PolyhedronData[pname,"Orientations"]//FullSimplify

MeanSphericalRadius

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

(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}∈p])/(PolyhedronData[pname,"Volume"]))

Rule[#,((Integrate[(x^2)+(y^2)+(z^2),{x,y,z}∈PolyhedronData[{pname,#},"Polyhedron"]])/(PolyhedronData[pname,"Volume"]))]&/@PolyhedronData[pname,"Orientations"]//FullSimplify

N[i]

Midsphere

PolyhedronData[pname,"Midsphere"]

Midsphere[p]//FullSimplify//Timing

SurfaceArea

PolyhedronData[pname,"SurfaceArea"]

RootReduce[%]

SurfaceArea[p]//RootReduce//Timing

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

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,Method->"GaussLaw"]//FullSimplify

Perspective projections

Manipulate[PolyhedronProjectionGraph[PolyhedronData["SmallRhombicosidodecahedron","Polyhedron"],{a,b,c}],{a,-1,1},{b,-1,1},{c,-1,1}]

{1,(1/6) (3-(Sqrt[5])),(1/6) (5-(Sqrt[5])),ArcTan[(Sqrt[(1/2)+(1/(Sqrt[5]))])]}

{.69,.09,.315}/.69

g=PolyhedronProjectionGraph[PolyhedronData["SmallRhombicosidodecahedron","Polyhedron"],{1,(1/6) (3-(Sqrt[5])),(1/6) (5-(Sqrt[5])),ArcTan[(Sqrt[(1/2)+(1/(Sqrt[5]))])]},VertexLabels->"Name"]

v=GraphEmbedding[g];

ArcTan@@(v[[60]]-v[[52]])//FullSimplify

g=PolyhedronProjectionGraph[PolyhedronData["SmallRhombicosidodecahedron","Polyhedron"],{1,.1,.4},VertexLabels->"Name"]

g=PolyhedronProjectionGraph[PolyhedronData["SmallRhombicosidodecahedron","Polyhedron"],{1,x,y},VertexLabels->"Name"]

v=GraphEmbedding[g];

eqn=FullSimplify[v[[21]]-v[[52]],x>0&&y>0]

Solve[Thread[eqn==0],{x,y}]//FullSimplify

N[%]

{1,(1/6) (1+(Sqrt[5])),(1/6) (-1+(Sqrt[5]))}

{.69,.37,.14}/.69

g=PolyhedronProjectionGraph[PolyhedronData["SmallRhombicosidodecahedron","Polyhedron"],{1,(1/6) (1+(Sqrt[5])),(1/6) (-1+(Sqrt[5]))},VertexLabels->"Name"]

v[[27,2]]-v[[49,2]]

v[[27,1]]+v[[52,1]]

g=PolyhedronProjectionGraph[PolyhedronData["SmallRhombicosidodecahedron","Polyhedron"],{1,x,y},VertexLabels->"Name"]

v=GraphEmbedding[g];

eqn1=FullSimplify[v[[27,2]]-v[[49,2]],x>0&&y>0]

eqn2=FullSimplify[v[[27,1]]-v[[52,1]],x>0&&y>0]

Solve[Thread[{eqn1,eqn2}==0],{x,y}]//FullSimplify

N[%]

{(1/22) (-5+7 (Sqrt[5])),(1/22) (1+3 (Sqrt[5])),1,-ArcTan[(Sqrt[(5/2)-(Sqrt[5])])]}

{.245,.175,.5}/.5

g=PolyhedronProjectionGraph[PolyhedronData["SmallRhombicosidodecahedron","Polyhedron"],{(1/22) (-5+7 (Sqrt[5])),(1/22) (1+3 (Sqrt[5])),1,-ArcTan[(Sqrt[(5/2)-(Sqrt[5])])]},VertexLabels->"Name"]

v=GraphEmbedding[g];

ArcTan@@(v[[12]]-v[[44]])//FullSimplify

g=PolyhedronProjectionGraph[PolyhedronData["SmallRhombicosidodecahedron","Polyhedron"],{x,y,1},VertexLabels->"Name"]

v=GraphEmbedding[g];

eqn=FullSimplify[v[[1]]-v[[57]],x>0&&y>0]

Solve[Thread[eqn==0],{x,y}]//FullSimplify

N[%]

{0,0,1}

g=PolyhedronProjectionGraph[PolyhedronData["SmallRhombicosidodecahedron","Polyhedron"],{0,0,1},VertexLabels->"Name"]

{0,(1/2)(1+(Sqrt[5])),1}

{0,.455,.28}/.455

1/GoldenRatio//FunctionExpand

g=PolyhedronProjectionGraph[PolyhedronData["SmallRhombicosidodecahedron","Polyhedron"],{0,(1/2)(1+(Sqrt[5])),1},VertexLabels->"Name"]

{0,1/2,1}

g=PolyhedronProjectionGraph[PolyhedronData["SmallRhombicosidodecahedron","Polyhedron"],{0,1/2,1},VertexLabels->"Name"]

g=PolyhedronProjectionGraph[PolyhedronData["SmallRhombicosidodecahedron","Polyhedron"],{0,z,1},VertexLabels->"Name"]

eqn=FullSimplify[v[[54]]-v[[57]],z>0]

Solve[Thread[eqn==0],z]

{0,(1/6) (5-(Sqrt[5])),1}

g=PolyhedronProjectionGraph[PolyhedronData["SmallRhombicosidodecahedron","Polyhedron"],{0,(1/6) (5-(Sqrt[5])),1},VertexLabels->"Name"]

g=PolyhedronProjectionGraph[PolyhedronData["SmallRhombicosidodecahedron","Polyhedron"],{0,z,1},VertexLabels->"Name"]

v=GraphEmbedding[g];

FullSimplify[CollinearPoints[v[[{1,12,13}]]],z>0]

Solve[%,z]//FullSimplify

N[%]

{0,(1/10) (5+(Sqrt[5])),1}

g=PolyhedronProjectionGraph[PolyhedronData["SmallRhombicosidodecahedron","Polyhedron"],{0,(1/10) (5+(Sqrt[5])),1},VertexLabels->"Name"]

g=PolyhedronProjectionGraph[PolyhedronData["SmallRhombicosidodecahedron","Polyhedron"],{0,z,1},VertexLabels->"Name"]

v=GraphEmbedding[g];

eqn=FullSimplify[v[[9]]-v[[12]],z>0]

Solve[Thread[eqn==0],z]//FullSimplify

N[%111]

{0,1,1}

g=PolyhedronProjectionGraph[PolyhedronData["SmallRhombicosidodecahedron","Polyhedron"],{0,1,1},VertexLabels->"Name"]

{1,1,1,-ArcCot[(Sqrt[3 (9+4 (Sqrt[5]))])]}

g=PolyhedronProjectionGraph[PolyhedronData["SmallRhombicosidodecahedron","Polyhedron"],{1,1,1,-ArcCot[(Sqrt[3 (9+4 (Sqrt[5]))])]},VertexLabels->"Name"]

v=GraphEmbedding[g];

ArcTan@@(v[[20]]-v[[58]])//FullSimplify

Construction

C2: square faces along z and y axes

Show[Graphics3D[poly=ConvexHull3D[#.Unit[{0,1,(5-Sqrt[5])/2}]&/@HalfIcosahedralGroup],Boxed-> False]]

poly2=poly/.Polygon[x_]:>Polygon[FullSimplify[#/(3-(Sqrt[5]))]&/@x];

(v=PolyhedronVertices[poly2])//InputForm
First/@poly2/.Thread[v->Range[Length[v]]]

C3: triangular face on top and bottom

Graphics3D[{
MapIndexed[Text[#2[[1]],#,Background->White]&,PolyhedronData["SmallRhombicosidodecahedron","VertexCoordinates"]],,
MapIndexed[Text[#2[[1]],#,Background->Green]&,Mean/@PolyhedronData["SmallRhombicosidodecahedron","Polygons"][[All,1]]],PolyhedronData["SmallRhombicosidodecahedron","GraphicsComplex"]}]

u=Mean[PolyhedronData["SmallRhombicosidodecahedron","Polygons"][[9,1]]]//Simplify

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

Polyhedron[v1=m.#&/@PolyhedronData["SmallRhombicosidodecahedron","VertexCoordinates"],PolyhedronData["SmallRhombicosidodecahedron","FaceIndices"]]//Graphics3D

ang=ArcTan@@Most[(Subtract@@v1[[{4,6}]]//FullSimplify)]//FullSimplify

Graphics3D[{Polyhedron[v2=RotationMatrix[0,{0,0,1}].#&/@v1,PolyhedronData["SmallRhombicosidodecahedron","FaceIndices"]]},Axes->True,AxesOrigin->{0,0,0},AxesLabel->{"x","y","z"}]

v3=v2//FullSimplify

Graphics3D[p3=Polyhedron[v3,PolyhedronData["SmallRhombicosidodecahedron","FaceIndices"]]]

C5: pentagonal face on top and bottom

Graphics3D[{
MapIndexed[Text[#2[[1]],#,Background->White]&,PolyhedronData["SmallRhombicosidodecahedron","VertexCoordinates"]],,
MapIndexed[Text[#2[[1]],#,Background->Green]&,Mean/@PolyhedronData["SmallRhombicosidodecahedron","Polygons"][[All,1]]],PolyhedronData["SmallRhombicosidodecahedron","GraphicsComplex"]}]

u=Mean[PolyhedronData["SmallRhombicosidodecahedron","Polygons"][[60,1]]]//Simplify

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

Polyhedron[v1=m.#&/@PolyhedronData["SmallRhombicosidodecahedron","VertexCoordinates"],PolyhedronData["SmallRhombicosidodecahedron","FaceIndices"]]//Graphics3D

ArcTan@@Most[(Subtract@@v1[[{16,57}]]//FullSimplify)]//FullSimplify

Graphics3D[{Polyhedron[v2=RotationMatrix[π/10,{0,0,1}].#&/@v1,PolyhedronData["SmallRhombicosidodecahedron","FaceIndices"]]},Axes->True,AxesOrigin->{0,0,0},AxesLabel->{"x","y","z"}]

v3=v2//FullSimplify

v3={{-(1/2),-(1/2) (Sqrt[1+(2/(Sqrt[5]))]),(Sqrt[(9/4)+(9/(2 (Sqrt[5])))])},{(1/2) (1+(Sqrt[5])),0,-(1/2) (Sqrt[5+2 (Sqrt[5])])},{(1/4) (-1-(Sqrt[5])),(1/4) (Sqrt[2-(2/(Sqrt[5]))]),(Sqrt[(9/4)+(9/(2 (Sqrt[5])))])},{(1/4) (3+(Sqrt[5])),(1/2) (Sqrt[(1/2) (5+(Sqrt[5]))]),-(1/2) (Sqrt[5+2 (Sqrt[5])])},{(1/4) (1+(Sqrt[5])),-(1/4) (Sqrt[34+(62/(Sqrt[5]))]),(Sqrt[(1/4)+(1/(2 (Sqrt[5])))])},{(1/4) (3+(Sqrt[5])),-(1/4) (Sqrt[26+(58/(Sqrt[5]))]),-(1/2) (Sqrt[1-(2/(Sqrt[5]))])},{-(1/2),(Sqrt[(9/4)+(9/(2 (Sqrt[5])))]),(Sqrt[(1/4)+(1/(2 (Sqrt[5])))])},{0,(Sqrt[(5/2)+(11/(2 (Sqrt[5])))]),-(1/2) (Sqrt[1-(2/(Sqrt[5]))])},{-(1/2),-(1/2) (Sqrt[5+2 (Sqrt[5])]),(1/2) (Sqrt[5+2 (Sqrt[5])])},{(1/4) (3+(Sqrt[5])),-(1/2) (Sqrt[(1/2) (5+(Sqrt[5]))]),-(1/2) (Sqrt[5+2 (Sqrt[5])])},{(1/4) (-3-(Sqrt[5])),(1/2) (Sqrt[(1/2) (5+(Sqrt[5]))]),(1/2) (Sqrt[5+2 (Sqrt[5])])},{(1/2),(1/2) (Sqrt[5+2 (Sqrt[5])]),-(1/2) (Sqrt[5+2 (Sqrt[5])])},{(1/4) (-3-(Sqrt[5])),-(1/2) (Sqrt[(1/2) (5+(Sqrt[5]))]),(1/2) (Sqrt[5+2 (Sqrt[5])])},{(1/4) (1+(Sqrt[5])),-(1/4) (Sqrt[2-(2/(Sqrt[5]))]),-(3/2) (Sqrt[1+(2/(Sqrt[5]))])},{(1/2) (-1-(Sqrt[5])),0,(1/2) (Sqrt[5+2 (Sqrt[5])])},{(1/2),(Sqrt[(1/4)+(1/(2 (Sqrt[5])))]),-(3/2) (Sqrt[1+(2/(Sqrt[5]))])},{0,Root[1-25 ((#1)^2)+5 ((#1)^4)&,1],(1/2) (Sqrt[1-(2/(Sqrt[5]))])},{(1/2),-(3/2) (Sqrt[1+(2/(Sqrt[5]))]),-(1/2) (Sqrt[1+(2/(Sqrt[5]))])},{(1/4) (-3-(Sqrt[5])),(Sqrt[(13/8)+(29/(8 (Sqrt[5])))]),(1/2) (Sqrt[1-(2/(Sqrt[5]))])},{(1/4) (-1-(Sqrt[5])),(Sqrt[(17/8)+(31/(8 (Sqrt[5])))]),-(1/2) (Sqrt[1+(2/(Sqrt[5]))])},{(1/4) (1+(Sqrt[5])),Root[1-20 ((#1)^2)+80 ((#1)^4)&,3],(Sqrt[(9/4)+(9/(2 (Sqrt[5])))])},{(1/2) (2+(Sqrt[5])),(Sqrt[(1/4)+(1/(2 (Sqrt[5])))]),-(1/2) (Sqrt[1-(2/(Sqrt[5]))])},{(1/2),-(1/2) (Sqrt[5+2 (Sqrt[5])]),(1/2) (Sqrt[5+2 (Sqrt[5])])},{(1/4) (5+(Sqrt[5])),Root[1-100 ((#1)^2)+80 ((#1)^4)&,1],-(1/2) (Sqrt[1+(2/(Sqrt[5]))])},{-(1/2),(1/2) (Sqrt[5+2 (Sqrt[5])]),(1/2) (Sqrt[5+2 (Sqrt[5])])},{(1/4) (1+(Sqrt[5])),(Sqrt[(17/8)+(31/(8 (Sqrt[5])))]),-(1/2) (Sqrt[1+(2/(Sqrt[5]))])},{(1/4) (3+(Sqrt[5])),-(1/2) (Sqrt[(1/2) (5+(Sqrt[5]))]),(1/2) (Sqrt[5+2 (Sqrt[5])])},{(1/2) (2+(Sqrt[5])),-(1/2) (Sqrt[1+(2/(Sqrt[5]))]),(1/2) (Sqrt[1-(2/(Sqrt[5]))])},{(1/2),(1/2) (Sqrt[5+2 (Sqrt[5])]),(1/2) (Sqrt[5+2 (Sqrt[5])])},{(1/4) (3+(Sqrt[5])),(Sqrt[(13/8)+(29/(8 (Sqrt[5])))]),(1/2) (Sqrt[1-(2/(Sqrt[5]))])},{(1/2) (1+(Sqrt[5])),0,(1/2) (Sqrt[5+2 (Sqrt[5])])},{(1/2) (2+(Sqrt[5])),(1/2) (Sqrt[1-(2/(Sqrt[5]))]),(Sqrt[(1/4)+(1/(2 (Sqrt[5])))])},{(1/4) (3+(Sqrt[5])),(1/2) (Sqrt[(1/2) (5+(Sqrt[5]))]),(1/2) (Sqrt[5+2 (Sqrt[5])])},{(1/4) (5+(Sqrt[5])),(Sqrt[(5/8)+(11/(8 (Sqrt[5])))]),(Sqrt[(1/4)+(1/(2 (Sqrt[5])))])},{(1/2) (1+(Sqrt[5])),-(Sqrt[1+(2/(Sqrt[5]))]),(Sqrt[(1/4)+(1/(2 (Sqrt[5])))])},{(1/2),-(1/2) (Sqrt[1+(2/(Sqrt[5]))]),(Sqrt[(9/4)+(9/(2 (Sqrt[5])))])},{(1/2) (2+(Sqrt[5])),-(1/2) (Sqrt[1-(2/(Sqrt[5]))]),-(1/2) (Sqrt[1+(2/(Sqrt[5]))])},{0,(Sqrt[(1/10) (5+(Sqrt[5]))]),(Sqrt[(9/4)+(9/(2 (Sqrt[5])))])},{(1/2) (1+(Sqrt[5])),(Sqrt[1+(2/(Sqrt[5]))]),-(1/2) (Sqrt[1+(2/(Sqrt[5]))])},{(1/2),(Sqrt[(9/4)+(9/(2 (Sqrt[5])))]),(Sqrt[(1/4)+(1/(2 (Sqrt[5])))])},{(1/4) (-1-(Sqrt[5])),-(Sqrt[(17/8)+(31/(8 (Sqrt[5])))]),(Sqrt[(1/4)+(1/(2 (Sqrt[5])))])},{(1/2),-(1/2) (Sqrt[5+2 (Sqrt[5])]),-(1/2) (Sqrt[5+2 (Sqrt[5])])},{(1/4) (-5-(Sqrt[5])),(Sqrt[(5/8)+(11/(8 (Sqrt[5])))]),(Sqrt[(1/4)+(1/(2 (Sqrt[5])))])},{-(1/2),(1/2) (Sqrt[5+2 (Sqrt[5])]),-(1/2) (Sqrt[5+2 (Sqrt[5])])},{(1/(-3+(Sqrt[5]))),Root[1-260 ((#1)^2)+80 ((#1)^4)&,1],-(1/2) (Sqrt[1-(2/(Sqrt[5]))])},{-(1/2),-(1/2) (Sqrt[5+2 (Sqrt[5])]),-(1/2) (Sqrt[5+2 (Sqrt[5])])},{-1-((Sqrt[5])/2),(Sqrt[(1/4)+(1/(2 (Sqrt[5])))]),-(1/2) (Sqrt[1-(2/(Sqrt[5]))])},{(1/4) (-3-(Sqrt[5])),(1/2) (Sqrt[(1/2) (5+(Sqrt[5]))]),-(1/2) (Sqrt[5+2 (Sqrt[5])])},{(1/4) (-5-(Sqrt[5])),-(1/4) (Sqrt[10+(22/(Sqrt[5]))]),-(1/2) (Sqrt[1+(2/(Sqrt[5]))])},{(1/4) (-3-(Sqrt[5])),-(1/2) (Sqrt[(1/2) (5+(Sqrt[5]))]),-(1/2) (Sqrt[5+2 (Sqrt[5])])},{-1-((Sqrt[5])/2),-(1/2) (Sqrt[1-(2/(Sqrt[5]))]),-(1/2) (Sqrt[1+(2/(Sqrt[5]))])},{(1/2) (-1-(Sqrt[5])),0,-(1/2) (Sqrt[5+2 (Sqrt[5])])},{-(1/2),-(3/2) (Sqrt[1+(2/(Sqrt[5]))]),-(1/2) (Sqrt[1+(2/(Sqrt[5]))])},{(1/2) (-1-(Sqrt[5])),-(Sqrt[1+(2/(Sqrt[5]))]),(Sqrt[(1/4)+(1/(2 (Sqrt[5])))])},{0,Root[1-5 ((#1)^2)+5 ((#1)^4)&,1],-(3/2) (Sqrt[1+(2/(Sqrt[5]))])},{-1-((Sqrt[5])/2),(1/2) (Sqrt[1-(2/(Sqrt[5]))]),(Sqrt[(1/4)+(1/(2 (Sqrt[5])))])},{-(1/2),(Sqrt[(1/4)+(1/(2 (Sqrt[5])))]),-(3/2) (Sqrt[1+(2/(Sqrt[5]))])},{(1/2) (-1-(Sqrt[5])),(Sqrt[1+(2/(Sqrt[5]))]),-(1/2) (Sqrt[1+(2/(Sqrt[5]))])},{-1-((Sqrt[5])/2),-(1/2) (Sqrt[1+(2/(Sqrt[5]))]),(1/2) (Sqrt[1-(2/(Sqrt[5]))])},{(1/4) (-1-(Sqrt[5])),-(1/4) (Sqrt[2-(2/(Sqrt[5]))]),-(3/2) (Sqrt[1+(2/(Sqrt[5]))])}};

Graphics3D[p5=Polyhedron[v3,PolyhedronData["SmallRhombicosidodecahedron","FaceIndices"]]]

ByteCount[reg=PolyhedronInequalities[p5,{x,y,z}]]

Simplify[reg]