Elongated Triangular Dipyramid

Author

Eric W. Weisstein
September 29, 2022

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

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

©2022 Wolfram Research, Inc. except for portions noted otherwise

Solid

PolyhedronData["ElongatedTriangularDipyramid","Orientations"]

PolyhedronData["ElongatedTriangularDipyramid"]

Net

PolyhedronData["ElongatedTriangularDipyramid","Net"]

PolyhedronData["ElongatedTriangularDipyramid","Net","Colored"]

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

Properties

Initialization

<<MathWorld`Polyhedra`

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

vol=PolyhedronData[pname,"Volume"]

Centroid

PolyhedronData[pname,"Centroid"]

RegionCentroid[p]//FullSimplify

PolyhedronCentroid[p]

Circumsphere

PolyhedronData[pname,"Circumsphere"]

Circumsphere[p]

MyCircumsphere[p]

DihedralAngles

PolyhedronData[pname,"DihedralAngles"]//Tally

DihedralAngles[p]//FullSimplify//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]

InertiaTensor

PolyhedronData[pname,"InertiaTensor"]

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

PolyhedronInertiaTensor[p]//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

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  1/4

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

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

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

(rxy=4((Integrate[(Sqrt[(x^2)+(y^2)])Boole[cad],{x,y,z}∈FullRegion[3]])/vol))//Timing

N[rxy]

LogSimplify[rxy]//FullSimplify//TrigToExp

Inequalities

{{xmin,xmax},{ymin,ymax},{zmin,zmax}}=MinMax/@Transpose[PolyhedronCoordinates[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

RootReduce[i2]//FullSimplify

N[%48]

Log[-3+2Sqrt[3],63687141-36769788 (Sqrt[3])]//FullSimplify

N[(1/87552)(-2 (-812+(Sqrt[6])) ArcCosh[2]+16 (1+4 (Sqrt[6])) ArcCosh[18817]-3 (188+87 (Sqrt[6])) Log[3]+(-4+3 (Sqrt[6])) 10Log[-3+2Sqrt[3]]+5 (-4+3 (Sqrt[6])) Log[21-12 (Sqrt[3])]+(-440+558 (Sqrt[6])) Log[2-(Sqrt[3])]+6 (-92+31 (Sqrt[6])) Log[2+(Sqrt[3])]+6 (-204+77 (Sqrt[6])) Log[-3+2 (Sqrt[3])]-128 (18 (Sqrt[2])-84 (Sqrt[3])-19 Log[3+2 (Sqrt[3])]))]

TrigToExp[ArcCosh[18817]]

Log[2+Sqrt[3],18817+10864 (Sqrt[3])]//FullSimplify

Log[2Sqrt[3]-3,21-12 (Sqrt[3])]//FullSimplify

TrigToExp[ArcCosh[2]]

N[expr=(1/87552)(-2 (-812+(Sqrt[6]))Log[2+(Sqrt[3])]+16 (1+4 (Sqrt[6])) 8Log[2+Sqrt[3]]-3 (188+87 (Sqrt[6])) Log[3]+(-4+3 (Sqrt[6])) 10Log[-3+2Sqrt[3]]+5 (-4+3 (Sqrt[6])) 2Log[2Sqrt[3]-3]+(-440+558 (Sqrt[6])) Log[2-(Sqrt[3])]+6 (-92+31 (Sqrt[6])) Log[2+(Sqrt[3])]+6 (-204+77 (Sqrt[6])) Log[-3+2 (Sqrt[3])]-128 (18 (Sqrt[2])-84 (Sqrt[3])-19 Log[3+2 (Sqrt[3])]))]

FullSimplify[expr]

N[%]

MeanSquareCylindricalRadius

PolyhedronData[pname,"MeanSquareCylindricalRadius"]

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

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

N[i]

MeanSphericalRadius

PolyhedronData[pname,"MeanSphericalRadius"]

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

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

N[rxyz]

Divergence theorem [unevaluated integral]

(rxy=MeanSphericalRadius[p,"Simplification"->FullSimplify,Debug->True])//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)+(z^2)])Boole[cad],{x,y,z}∈FullRegion[3]])/vol))//Timing

N[rxyz]

CAD over  1/4 [unevaluated]

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

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

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

(rxy=4((Integrate[(Sqrt[(x^2)+(y^2)+(z^2)])Boole[cad],{x,y,z}∈FullRegion[3]])/vol))//Timing

N[rxy]

LogSimplify[rxy]//FullSimplify//TrigToExp

MeanSquareSphericalRadius

PolyhedronData[pname,"MeanSquareSphericalRadius"]

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

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

N[i]

Midsphere

PolyhedronData[pname,"Midsphere"]

Midsphere[p]

SurfaceArea

PolyhedronData[pname,"SurfaceArea"]

SurfaceArea[p]

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

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"],]