Tridiminished Icosahedron

Author

Eric W. Weisstein
September 12, 2023

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

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

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

Polyhedron

PolyhedronData["TridiminishedIcosahedron","Orientations"]

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

Net

PolyhedronData["TridiminishedIcosahedron","Net"]

PolyhedronData["TridiminishedIcosahedron","Net","Colored"]

NetPrintout[PolyhedronData["TridiminishedIcosahedron","Polyhedron"],"SmallTriakisOctahedron.pdf"]

Properties

Initialization

<<MathWorld`Polyhedra`

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

vol=PolyhedronData[pname,"Volume"]

Centroid

PolyhedronData[pname,"Centroid"]

RegionCentroid[p]//FullSimplify

PolyhedronCentroid[p]//FullSimplify

Circumsphere

PolyhedronData[pname,"Circumsphere"]

FullSimplify[%]

Circumsphere[p]//FullSimplify

MyCircumsphere[p]//RootReduce

FullSimplify[%]

DihedralAngles

PolyhedronData[pname,"DihedralAngles"]//Tally

DihedralAngles[p]//RootReduce//Tally

EdgeLengths

PolyhedronData[pname,"EdgeLengths"]

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

Faces

GeneralizedDiameter

PolyhedronData[pname,"GeneralizedDiameter"]

GeneralizedDiameter[p]//FullSimplify

Insphere

PolyhedronData[pname,"Insphere"]

Insphere[p]

MyInsphere[p,"Simplification"->FullSimplify]//Timing

InertiaTensor

PolyhedronData[pname,"InertiaTensor"]

((MomentOfInertia[p,{0,0,0}])/(Volume[p]))//FullSimplify//Timing

PolyhedronInertiaTensor[p]//FullSimplify//Timing

MeanCylindricalRadius

PolyhedronData[pname,"MeanCylindricalRadius"]

N[%]

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

Integrate over polyhedron

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

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)])Boole[cad],{x,y,z}∈FullRegion[3]])/vol))//Timing

N[rxyz,20]

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

CAD over  1/2 [crash]

reg=FullSimplify[PolyhedronData[pname,"RegionFunction"][x,y,z]&&y>0]

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

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

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

bug(439981): Integrate[Sqrt[x^2 + y^2 + z^2] Boole[cad], {x, y, z} ∈ FullRegion[3]] segfaults kernel

N[rxyz,20]

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

CAD over  1/6 [crash]

reg=FullSimplify[PolyhedronData[pname,"RegionFunction"][x,y,z]&&0<y<Tan[Pi/3]x]

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

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

(rxyz=6((Integrate[(Sqrt[(x^2)+(y^2)+(z^2)])Boole[cad],{x,y,z}∈FullRegion[3]])/vol))//Timing

bug(439981): Integrate[Sqrt[x^2 + y^2 + z^2] Boole[cad], {x, y, z} ∈ FullRegion[3]] segfaults kernel

N[rxyz,20]

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

Divergence theorem [unevaluated integral]

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

N[rxy]

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

MeanSquareCylindricalRadius

PolyhedronData[pname,"MeanSquareCylindricalRadius"]

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

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

N[i]

MeanSphericalRadius

PolyhedronData[pname,"MeanSphericalRadius"]

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

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

N[rxyz]

MeanSquareSphericalRadius

PolyhedronData[pname,"MeanSquareSphericalRadius"]

((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"]

(mid=Midsphere[p,"Simplification"->FullSimplify])//Timing

Graphics3D[{{Opacity[.5],p},{Red,mid}},Boxed->False]

SurfaceArea

PolyhedronData[pname,"SurfaceArea"]

RootReduce[%]

SurfaceArea[p]//RootReduce//Timing

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

VertexSubsetHulls

PolyhedronData[pname,"VertexSubsetHulls"]

Platonic

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

Volume

PolyhedronData[pname,"Volume"]

Volume[p]//FullSimplify

PolyhedronVolume[p,Method->"GaussLaw"]//FullSimplify

Perspective projections

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

XXX

PolyhedronProjectionGraph[PolyhedronData["DuererSolid","Polyhedron"],]

Construction

Original

Graphics3D[{{Opacity[.5],p=PolyhedronData["TridiminishedIcosahedron","Polyhedron"]},MapIndexed[Text[#2[[1]],#,Background->White]&,v=p[[1]]]}]

Mean[v]//RootReduce

v[[{2,3,6}]]

v[[{1,4,5}]]

ArcTan@@Most[v[[#]]]&/@{1,6}