Elongated Triangular Cupola

Author

Eric W. Weisstein
September 12, 2023

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

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

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

Solid

PolyhedronData["ElongatedTriangularCupola","Orientations"]

PolyhedronData["ElongatedTriangularCupola"]

Graphics3D[{Opacity[.3],PolyhedronData["ElongatedTriangularCupola","Polyhedron"]}]

Net

PolyhedronData["ElongatedTriangularCupola","Net"]

PolyhedronData["ElongatedTriangularCupola","Net","Colored"]

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

Properties

Initialization

<<MathWorld`Polyhedra`

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

vol=PolyhedronData[pname,"Volume"]

Centroid

Graphics3D[{AbsolutePointSize[10],
Point[{0,0,0}],{Blue,Point[PolyhedronData[pname,"Centroid"]]},Opacity[.2],PolyhedronData[pname,"GraphicsComplex"]}]

PolyhedronData[pname,"Centroid"]

RegionCentroid[p]//FullSimplify

PolyhedronCentroid[p]//FullSimplify

Circumsphere

PolyhedronData[pname,"Circumsphere"]

Circumsphere[p]

sphere=MyCircumsphere[p]

Convex

ConvexPolyhedronQ[p]

DihedralAngles

PolyhedronData[pname,"DihedralAngles"]//Tally

DihedralAngles[p]//FullSimplify//Tally

EdgeLengths

PolyhedronData[pname,"EdgeLengths"]

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

Faces

GeneralizedDiameter

PolyhedronData[pname,"GeneralizedDiameter"]

GeneralizedDiameter[p]//FullSimplify//Quiet

InertiaTensor

PolyhedronData[pname,"InertiaTensor"]

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

PolyhedronInertiaTensor[p]//Timing

Insphere

PolyhedronData[pname,"Insphere"]

Insphere[p]

sphere=MyInsphere[p]

MeanCylindricalRadius

PolyhedronData[pname,"MeanCylindricalRadius"]

N[%]

((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)//FullSimplify)

N[rxy]

Divergence theorem [bad complex integral]

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

N[rxy]

LogSimplify[rxy]//FullSimplify

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  1/2

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)

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

N[rxy]

<<MathWorld`Polyhedra`

rxysimp=LogSimplify[rxy]//Together

rxysimp=(24138082986048+1203106549701384 (Sqrt[2])+1108270912750848 (Sqrt[3])+12375391192728 (Sqrt[6])+102107094273720 (Sqrt[2 (23+4 (Sqrt[6]))])+81481442285840 (Sqrt[3 (23+4 (Sqrt[6]))])-28467446516760 (Sqrt[2 (209+74 (Sqrt[6]))])-24546549252320 (Sqrt[3 (209+74 (Sqrt[6]))])-54310686718608 ArcSinh[(Sqrt[3])]-3093847798182 (Sqrt[2]) ArcSinh[(Sqrt[3])]-2011506915504 (Sqrt[3]) ArcSinh[(Sqrt[3])]-27844630183638 (Sqrt[6]) ArcSinh[(Sqrt[3])]+27155343359304 Log[3]+1027630748102409 (Sqrt[2]) Log[3]+912669214641048 (Sqrt[3]) Log[3]+13922315091819 (Sqrt[6]) Log[3])/(576 (5 (Sqrt[2])+9 (Sqrt[3])) (515641299697+167625576292 (Sqrt[6])));

N[rxysimp]

(24138082986048+1203106549701384 (Sqrt[2])+1108270912750848 (Sqrt[3])+12375391192728 (Sqrt[6])+102107094273720 (Sqrt[2 (23+4 (Sqrt[6]))])+81481442285840 (Sqrt[3 (23+4 (Sqrt[6]))])-28467446516760 (Sqrt[2 (209+74 (Sqrt[6]))])-24546549252320 (Sqrt[3 (209+74 (Sqrt[6]))])-54310686718608 ArcSinh[(Sqrt[3])]-3093847798182 (Sqrt[2]) ArcSinh[(Sqrt[3])]-2011506915504 (Sqrt[3]) ArcSinh[(Sqrt[3])]-27844630183638 (Sqrt[6]) ArcSinh[(Sqrt[3])]+27155343359304 Log[3]+1027630748102409 (Sqrt[2]) Log[3]+912669214641048 (Sqrt[3]) Log[3]+13922315091819 (Sqrt[6]) Log[3])(((576 (5 (Sqrt[2])-9 (Sqrt[3])) (515641299697-167625576292 (Sqrt[6]))))/(Expand[(576 (5 (Sqrt[2])+9 (Sqrt[3])) (515641299697+167625576292 (Sqrt[6])))(576 (5 (Sqrt[2])-9 (Sqrt[3])) (515641299697-167625576292 (Sqrt[6])))]))//Expand

FullSimplify[%]

TrigToExp[%]//FullSimplify

RootReduce[%]

FullSimplify[%]

TrigToExp[%]

N[%]

MeanSquareCylindricalRadius

PolyhedronData[pname,"MeanSquareCylindricalRadius"]

N[%]

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

Integrate over Polyhedron

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

N[rxy2]

MeanSphericalRadius

PolyhedronData[pname,"MeanSphericalRadius"]

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

ClosestMeanSphericalRadius[%]

Integrate over polyhedron [unevaluated]

(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  1/2 [unevaluated]

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

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

2((NIntegrate[(Sqrt[(x^2)+(y^2)+(z^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

N[rxyz,20]

rxysimp1=LogSimplify[rxy]//FullSimplify

CAD over  1/6 [unevaluated]

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

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

N[rxy]

LogSimplify[rxy]//FullSimplify

MeanSquareSphericalRadius

PolyhedronData[pname,"MeanSquareSphericalRadius"]

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

Integrate over polyhedron

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

N[rxy2]

Midsphere

PolyhedronData[pname,"Midsphere"]

Midsphere[p]

SurfaceArea

PolyhedronData[pname,"SurfaceArea"]

SurfaceArea[p]//FullSimplify

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

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"], Null]