Small Rhombicuboctahedron

Author

Eric W. Weisstein
January 5, 2024

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

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

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

Figure

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

Polyhedron

PolyhedronData["SmallRhombicuboctahedron","Orientations"]

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

Polyhedron

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

Region

PolyhedronData["SmallRhombicuboctahedron","Region"]

Solid and Dual

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

UniformPolyhedron

uniform=UniformPolyhedron[stdname="Rhombicuboctahedron"]

PolyhedronFaceTally[uniform]

UniformPolyhedron/@{
stdname,
UniformPolyhedronIndex[stdname,"Wenninger"],
{"Coxeter",UniformPolyhedronIndex[stdname,"Coxeter"]},
{"Kaleido",UniformPolyhedronIndex[stdname,"Kaleido"]},
{"Uniform",UniformPolyhedronIndex[stdname,"Maeder"]},
{"Wenninger",UniformPolyhedronIndex[stdname,"Wenninger"]}
}

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["SmallRhombicuboctahedron","Net"]

PolyhedronData["SmallRhombicuboctahedron","Net","Colored"]

NetPrintout[PolyhedronData["SmallRhombicuboctahedron","Polyhedron"],"Cuboctahedron.pdf"]

Properties

Initialization

<<MathWorld`Polyhedra`

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

vol=PolyhedronData[pname,"Volume"]

AdjacentFaceIndices

adj1=PolyhedronData[pname,"AdjacentFaceIndices"]

adj2=AdjacentFaceIndices[p]

adj1===adj2

Centroid

PolyhedronData[pname,"Centroid"]

RegionCentroid[p]//FullSimplify

PolyhedronCentroid[p]

Circumsphere

sphere=PolyhedronData[pname,"Circumsphere"]

Circumsphere[p]

MyCircumsphere[N[p,50]]

%/.x_?InexactNumberQ:>FullSimplify@RootApproximant[x]

(sphere=MyCircumsphere[p])//Timing

Graphics3D[{{Opacity[.2],Yellow,sphere},p},Boxed->False]

Classes

PolyhedronData[pname,"Classes"]

Compound

PolyhedronData[pname,"Compound"]

Concave

PolyhedronData[pname,"Concave"]

Convex

PolyhedronData[pname,"Convex"]

ConvexPolyhedronQ[p]//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[4]],Dividers->All,ImageSize->800]

DehnInvariant

PolyhedronData[pname,"DehnInvariant"]

(dehn=DehnInvariant[p,"BasisTerms"->{30,20},Debug->True]//FullSimplify)//Timing

ToConwayBasis[dehn]

Complement[Select[PolyhedronData[All],PolyhedronData[#,"DehnInvariant"]==dehn&&PolyhedronData[#,"Volume"]==vol&],{pname}]

DihedralAngles

PolyhedronData[pname,"DihedralAngles"]//Tally

DihedralAngles[p]//Tally

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

{3,4}

Show[Graphics3D[p=poly[[{11,19}]]],Boxed->False]

FullSimplify[π-DihedralAngle@@p]

InputForm[%]

N[%]180/Pi

{4,4}

Show[Graphics3D[p=poly[[{5,7}]]],Boxed->False]

FullSimplify[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]

PolyhedronData[pname,"EdgeLengths"]

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

Faces

PolyhedronFaceList[poly]

Show[GraphicsArray[partition[PolyhedronFaces[poly],5]]]

f1=v[[{2,10,12,4}]];
f2=v[[{10,22,14}]];
Show[GraphicsArray[{
Graphics3D[{
Polygon[f1],LabelVertices[f1]
},Boxed->False],
Graphics3D[{
Polygon[f2],LabelVertices[f2]
},Boxed->False],
Graphics3D[{
Polygon[f1],LabelVertices[f1],
Polygon[f2],LabelVertices[f2],
PointSize[.1],Point[{0,0,0}],
Line[{Centroid[f1][[1]],{0,0,0},Centroid[f2][[1]]}]
},Boxed->False]
}]]

{a,b}=FullSimplify[{Centroid[f1][[1]],Centroid[f2][[1]]}]

Together//@FullSimplify[Sqrt/@{a.a,b.b}]

Insphere

PolyhedronData[pname,"Insphere"]

Insphere[p]

MyInsphere[p]//RootReduce//Timing

InertiaTensor

PolyhedronData[pname,"InertiaTensor"]

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

PolyhedronInertiaTensor[p]//Timing

MeanCylindricalRadius

PolyhedronData[pname,"MeanCylindricalRadius"]

N[%]

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

Integrate over polyhedron

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

N[i]

CAD

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

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

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

CAD over  1/16

reg=FullSimplify[PolyhedronData[pname,"RegionFunction"][x,y,z]&&0<y<Tan[Pi/4]x&&z>0]

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

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

(rxy=16((Integrate[(Sqrt[(x^2)+(y^2)])Boole[cad],{x,y,z}∈FullRegion[3]])/vol))//Timing

N[rxy]

rxysimp=LogSimplify[rxy]//FullSimplify

Divide@@(Expand[# 16 (6-5 (Sqrt[2]))]&/@NumeratorDenominator[rxysimp])//FullSimplify

N[%]

TrigToExp[(1/224) (4 (Sqrt[290+127 (Sqrt[2])])-(108+71 (Sqrt[2])) ArcSinh[1-(Sqrt[2])]+(6-5 (Sqrt[2])) ArcSinh[1+(Sqrt[2])])]//LogSimplify//FullSimplify

Divergence theorem [bad integral(s)]

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

N[rxy]

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

MeanCylindricalSquareRadius

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

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

N[i]

MeanSphericalRadius

PolyhedronData[pname,"MeanCylindricalRadius"]

N[%]

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

Integrate over polyhedron

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

N[rxyz]

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  1/16 [ran for 2+ days]

reg=FullSimplify[PolyhedronData[pname,"RegionFunction"][x,y,z]&&0<y<Tan[Pi/4]x&&z>0]

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

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

(rxyz=16((Integrate[(Sqrt[(x^2)+(y^2)+(z^2)])Boole[cad],{x,y,z}∈FullRegion[3]])/vol))//Timing

...ran for 2+ days...

N[rxyz,20]

rxysimp1=LogSimplify[rxy]//FullSimplify

Divergence theorem [unevaluated]

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

MeanSphericalSquareRadius

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

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

N[i]

Midsphere

PolyhedronData[pname,"Midsphere"]

Midsphere[p]//FullSimplify//Timing

SurfaceArea

PolyhedronData[pname,"SurfaceArea"]

SurfaceArea[p]//FullSimplify

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

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

Projections

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

{-5+4 (Sqrt[2]),1,0}

{.365,.54,0}/.54

g=PolyhedronProjectionGraph[PolyhedronData["SmallRhombicuboctahedron","Polyhedron"],{-5+4 (Sqrt[2]),1,0},VertexLabels->"Name"]

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

v=GraphEmbedding[g];

FullSimplify[SquaredEuclideanDistance@@v[[{17,15}]]-SquaredEuclideanDistance@@v[[{5,15}]],z>0]

Solve[(-1-2 (Sqrt[2])+(3+2 (Sqrt[2])) z)==0,z]//FullSimplify

{1,1,(Sqrt[2])}

g=PolyhedronProjectionGraph[PolyhedronData["SmallRhombicuboctahedron","Polyhedron"],{1,1,(Sqrt[2]),ArcCot[(Sqrt[2])]},VertexLabels->"Name"]

v=GraphEmbedding[g];

ArcTan@@(v[[5]]-v[[17]])//FullSimplify

g=PolyhedronProjectionGraph[PolyhedronData["SmallRhombicuboctahedron","Polyhedron"],{.6,.7,1},VertexLabels->"Name"]

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

v=GraphEmbedding[g];

FullSimplify[v[[8]]-v[[19]],z>0]

{(3/2)-(1/(Sqrt[2])),(3/2)-(1/(Sqrt[2])),1,ArcTan[(Sqrt[(2/97) (13+6 (Sqrt[2]))])]}

g=PolyhedronProjectionGraph[PolyhedronData["SmallRhombicuboctahedron","Polyhedron"],{(3/2)-(1/(Sqrt[2])),(3/2)-(1/(Sqrt[2])),1,ArcTan[(Sqrt[(2/97) (13+6 (Sqrt[2]))])]},VertexLabels->"Name"]

v=GraphEmbedding[g];

ArcTan@@(v[[5]]-v[[17]])//FullSimplify

PolyhedronProjectionGraph[PolyhedronData["SmallRhombicuboctahedron","Polyhedron"],{.8,.8,1},VertexLabels->"Name"]

g=PolyhedronProjectionGraph[PolyhedronData["SmallRhombicuboctahedron","Polyhedron"],{z,z,1}];

v=GraphEmbedding[g];

FullSimplify[CollinearPoints[v[[{17,12,5}]]],z>0]

Solve[%,z]//FullSimplify

{1,1,2-(Sqrt[2]),-ArcTan[(NumericalApproximation[Root, (RowDefault["-0.383", FEExpressionsNumericalApproximationElider]), Root[1-8 ((#1)^2)+8 ((#1)^4)&,2], ])]}

g=PolyhedronProjectionGraph[PolyhedronData["SmallRhombicuboctahedron","Polyhedron"],{1,1,2-(Sqrt[2]),-ArcTan[(NumericalApproximation[Root, (RowDefault["-0.383", FEExpressionsNumericalApproximationElider]), Root[1-8 ((#1)^2)+8 ((#1)^4)&,2], ])]},VertexLabels->"Name"]

v=GraphEmbedding[g];

ArcTan@@(v[[16]]-v[[24]])//FullSimplify

PolyhedronProjectionGraph[PolyhedronData["SmallRhombicuboctahedron","Polyhedron"],{1,1,.5},VertexLabels->"Name"]

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

v=GraphEmbedding[g];

FullSimplify[v[[16]]-v[[17]],x>0]

Solve[-1+x+(x/(Sqrt[2]))==0,x]//FullSimplify

{0, 1, 1}

PolyhedronProjectionGraph[PolyhedronData["SmallRhombicuboctahedron","Polyhedron"],{1,0,1}]

{0, 0, 1}

PolyhedronProjectionGraph[PolyhedronData["SmallRhombicuboctahedron","Polyhedron"],{0,0,1}]

{1, 1, 1}

PolyhedronProjectionGraph[PolyhedronData["SmallRhombicuboctahedron","Polyhedron"],{1,1,1}]

Construction

Exact Construction

SmallRhombicuboctahedron:=MiscPolyhedra`UniformPolyhedron[10]

SmallRhombicuboctahedronExact:=SmallRhombicuboctahedronExact=Module[{t=1+Sqrt[2]},OrientCentered[{Polygon[{{-1,1,-t},{1,1,-t},{1,-1,-t},{-1,-1,-t}}/2],(*Bottom*)Polygon[{{-1,1,t},{1,1,t},{1,-1,t},{-1,-1,t}}/2],(*Top*)Polygon[{{t,1,-1},{t,-1,-1},{t,-1,1},{t,1,1}}/2],(*Right*)Polygon[{{-t,1,-1},{-t,-1,-1},{-t,-1,1},{-t,1,1}}/2],(*Left*)Polygon[{{-1,-t,1},{1,-t,1},{1,-t,-1},{-1,-t,-1}}/2],(*Front*)Polygon[{{-1,t,1},{1,t,1},{1,t,-1},{-1,t,-1}}/2],(*Back*)Polygon[{{1,-t,1},{t,-1,1},{t,-1,-1},{1,-t,-1}}/2],(*Left Front*)Polygon[{{1,t,1},{t,1,1},{t,1,-1},{1,t,-1}}/2],(*Left Back*)Polygon[{{-1,-t,1},{-t,-1,1},{-t,-1,-1},{-1,-t,-1}}/2],(*Right Front*)Polygon[{{-1,t,1},{-t,1,1},{-t,1,-1},{-1,t,-1}}/2],(*Right Back*)Polygon[{{-1,-1,t},{1,-1,t},{1,-t,1},{-1,-t,1}}/2],(*Top Front*)Polygon[{{-1,1,t},{1,1,t},{1,t,1},{-1,t,1}}/2],(*Top Back*)Polygon[{{1,-1,t},{1,1,t},{t,1,1},{t,-1,1}}/2],(*Top Right*)Polygon[{{-1,-1,t},{-1,1,t},{-t,1,1},{-t,-1,1}}/2],(*Top Left*)Polygon[{{-1,-1,-t},{1,-1,-t},{1,-t,-1},{-1,-t,-1}}/2],(*Bottom Front*)Polygon[{{-1,1,-t},{1,1,-t},{1,t,-1},{-1,t,-1}}/2],(*Bottom Back*)Polygon[{{1,-1,-t},{1,1,-t},{t,1,-1},{t,-1,-1}}/2],(*Bottom Right*)Polygon[{{-1,-1,-t},{-1,1,-t},{-t,1,-1},{-t,-1,-1}}/2],(*Bottom Left*)Polygon[{{1,-t,1},{1,-1,t},{t,-1,1}}/2],Polygon[{{1,t,1},{1,1,t},{t,1,1}}/2],Polygon[{{-1,-t,1},{-1,-1,t},{-t,-1,1}}/2],Polygon[{{-1,t,1},{-1,1,t},{-t,1,1}}/2],Polygon[{{1,-t,-1},{1,-1,-t},{t,-1,-1}}/2],Polygon[{{1,t,-1},{1,1,-t},{t,1,-1}}/2],Polygon[{{-1,-t,-1},{-1,-1,-t},{-t,-1,-1}}/2],Polygon[{{-1,t,-1},{-1,1,-t},{-t,1,-1}}/2]}]]

Another Construction

Show[Graphics3D[ConvexHull3D[With[{y=(1+(Sqrt[2]))/2},Union[Flatten[Permutations/@Flatten[Outer[List,{-y,y},{-1,1}/2,{-1,1}/2],2],1]]]]],Boxed->False]