Elongated Triangular Pyramid

Author

Eric W. Weisstein
October 5, 2022

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

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

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

Solid

PolyhedronData["ElongatedTriangularPyramid","Orientations"]

PolyhedronData["ElongatedTriangularPyramid"]

Net

PolyhedronData["ElongatedTriangularPyramid","Net"]

PolyhedronData["ElongatedTriangularPyramid","Net","Colored"]

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

Properties

<<MathWorld`Polyhedra`

p=PolyhedronData[pname="ElongatedTriangularPyramid","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]))//RootReduce//Timing

PolyhedronInertiaTensor[p]//Timing

MeanCylindricalRadius

PolyhedronData[pname,"MeanCylindricalRadius"]

N[%]

Integrate over polyhedron

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

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

N[i]

Integrate using Boole

ineq=PolyhedronData[pname,"RegionFunction"][x,y,z]//FullSimplify

(i=((Integrate[(Sqrt[(x^2)+(y^2)])Boole[ineq],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞}])/vol))//Timing

Log[3+2 (Sqrt[3]),45+26 (Sqrt[3])]//N

Log[2+(Sqrt[3]),45+26 (Sqrt[3])]//N

Log[3+(Sqrt[3]),45+26 (Sqrt[3])]//N

((Log[135+78 (Sqrt[3])])/(Log[3+2(Sqrt[3])]))//FullSimplify

(((3-2(Sqrt[3])))^2)//Expand

Cases[Flatten[Table[{a,b,n,(((a+b Sqrt[3]))^n)//Expand},{n,10},{a,-10,10},{b,-10,10}],2],{__,45+26 (Sqrt[3])}]

(1/(1728 ((Sqrt[2])+3 (Sqrt[3]))))(1728+144 (Sqrt[6])-32 (Sqrt[2]) Log[3]-27 (Sqrt[3]) Log[3]+3 (Sqrt[3]) 2Log[2(Sqrt[3])-3]-24 (Sqrt[2]) Log[2+(Sqrt[3])]+246 (Sqrt[3]) Log[2+(Sqrt[3])]+48 (Sqrt[3]) Log[3+2 (Sqrt[3])]+16 (Sqrt[2]) Log[45+26 (Sqrt[3])]+16 (Sqrt[2]) (3Log[3+2(Sqrt[3])]))//FullSimplify

N[%]

Divergence theorem

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

N[rxy]

FullSimplify[rxy]

TrigToExp[%]

Cases[Flatten[Table[{a,b,n,(((a+b Sqrt[3]))^n)//Expand},{n,10},{a,-10,10},{b,-10,10}],2],{__,18817+10864 (Sqrt[3])}]

Cases[Flatten[Table[{a,b,n,(((a+b Sqrt[3]))^n)//Expand},{n,10},{a,-10,10},{b,-10,10}],2],{__,262087+151316 (Sqrt[3])}]

Cases[Flatten[Table[{a,b,n,(((a+b Sqrt[3]))^n)//Expand},{n,10},{a,-10,10},{b,-10,10}],2],{__,7-4 (Sqrt[3])}]

N[(1/(288 (Sqrt[29+6 (Sqrt[6])])))(288+24 (Sqrt[6])-2 (Sqrt[2]) Log[7-4 (Sqrt[3])]+4 (Sqrt[3]) Log[2-(Sqrt[3])]+42 (Sqrt[3]) Log[2+(Sqrt[3])]+(Sqrt[2]) Log[18817+10864 (Sqrt[3])]+(Sqrt[3]) Log[262087+151316 (Sqrt[3])])]

N[(1/(288 (Sqrt[29+6 (Sqrt[6])])))(288+24 (Sqrt[6])-2 (Sqrt[2]) Log[7-4 (Sqrt[3])]+4 (Sqrt[3]) Log[2-(Sqrt[3])]+42 (Sqrt[3]) Log[2+(Sqrt[3])]+(Sqrt[2]) 8Log[2+ (Sqrt[3])]+(Sqrt[3]) 10Log[2+(Sqrt[3])])]

N[(1/(288 (Sqrt[29+6 (Sqrt[6])])))(288+24 (Sqrt[6])-2 (Sqrt[2]) 2Log[2-(Sqrt[3])]+4 (Sqrt[3]) Log[2-(Sqrt[3])]+42 (Sqrt[3]) Log[2+(Sqrt[3])]+(Sqrt[2]) 8Log[2+ (Sqrt[3])]+(Sqrt[3]) 10Log[2+(Sqrt[3])])]

FullSimplify[(1/(288 (Sqrt[29+6 (Sqrt[6])])))(288+24 (Sqrt[6])-2 (Sqrt[2]) 2Log[2-(Sqrt[3])]+4 (Sqrt[3]) Log[2-(Sqrt[3])]+42 (Sqrt[3]) Log[2+(Sqrt[3])]+(Sqrt[2]) 8Log[2+ (Sqrt[3])]+(Sqrt[3]) 10Log[2+(Sqrt[3])])]

ArcCosh[2]//TrigToExp

N[((72+6 (Sqrt[6])+(2 (Sqrt[2])+13 (Sqrt[3]))Log[2+(Sqrt[3])]+((Sqrt[3])-(Sqrt[2])) Log[2-(Sqrt[3])])/(72 (Sqrt[29+6 (Sqrt[6])])))]

N[Subtract@@{(1/(1728 (Sqrt[29+6 (Sqrt[6])])))(-((32 (Sqrt[2])+27 (Sqrt[3])) Log[3])+6 (-4 (Sqrt[2])+41 (Sqrt[3])) Log[2+(Sqrt[3])]+6 (Sqrt[3]) Log[-3+2 (Sqrt[3])]+16 (108+9 (Sqrt[6])+3 ((Sqrt[2])+(Sqrt[3])) Log[3+2 (Sqrt[3])]+(Sqrt[2]) Log[45+26 (Sqrt[3])])),((72+6 (Sqrt[6])+(2 (Sqrt[2])+13 (Sqrt[3]) )Log[2+(Sqrt[3])]+((Sqrt[3])-(Sqrt[2])) Log[2-(Sqrt[3])])/(72 (Sqrt[29+6 (Sqrt[6])])))},1000]

FullSimplify[(Sqrt[29+6 (Sqrt[6])])(Sqrt[29-6 (Sqrt[6])])]

N[(((72+6 (Sqrt[6])+(2 (Sqrt[2])+13 (Sqrt[3]))Log[2+(Sqrt[3])]+((Sqrt[3])-(Sqrt[2])) Log[2-(Sqrt[3])])(Sqrt[29-6 (Sqrt[6])]))/(25 72))]

(((72+6 (Sqrt[6])+(2 (Sqrt[2])+13 (Sqrt[3]))Log[2+(Sqrt[3])]+((Sqrt[3])-(Sqrt[2])) Log[2-(Sqrt[3])])(Sqrt[29-6 (Sqrt[6])]))/(25 72))

FullSimplify[(((72+6 (Sqrt[6])+(2 (Sqrt[2])+13 (Sqrt[3]))Log[2+(Sqrt[3])]+((Sqrt[3])-(Sqrt[2])) Log[2-(Sqrt[3])])(Sqrt[29-6 (Sqrt[6])]))/(25 72))]

N[%]

MeanSphericalRadius

PolyhedronData[pname,"MeanSphericalRadius"]

Integrate over polyhedron

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

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

Divergence theorem

(rxy=MeanSphericalRadius[p,Debug->True])//Timing

Integrate over z first

{{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^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

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]

MeanSquareCylindricalRadius[p]//FullSimplify//Timing

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])/vol)//FullSimplify)//Timing

N[i]

MeanSquareSphericalRadius[p]//FullSimplify//Timing

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["ElongatedTriangularPyramid","Polyhedron"],{a,b,c}],{a,-1,1},{b,-1,1},{c,-1,1}]

XXX

PolyhedronProjectionGraph[PolyhedronData["ElongatedTriangularPyramid","Polyhedron"],]