Elongated Square Pyramid

Author

Eric W. Weisstein
October 16, 2023

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

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

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

Solid

PolyhedronData["ElongatedSquarePyramid","Orientations"]

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

Net

PolyhedronData["ElongatedSquarePyramid","Net"]

PolyhedronData["ElongatedSquarePyramid","Net","Colored"]

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

Properties

Initialization

<<MathWorld`Polyhedra`

p=PolyhedronData[pname="ElongatedSquarePyramid","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[%]

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

Integrate over polyhedron [unevaluated integral]

$Version

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

Divergence theorem

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

N[rxy]

ResourceFunction["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

N[rxy,20]

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

Boole Integrate doing z integral 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,zmin,zmax}]//FullSimplify)//Timing

NIntegrate[(i1/vol),{x,xmin,xmax},{y,ymin,ymax}]

(i2=Integrate[(i1/vol),{x,xmin,xmax},{y,ymin,ymax}])//Timing

i2=(11875021455846485564750162160115070160069893230404710222721326967666891168+8396908198164797892648526841044955516672582915367884731624404467872865712 (Sqrt[2])+10703273057317141466407365013208627220640794211459038896182046859738587328 ArcCoth[(Sqrt[2])]+7568356959720221490261700993435224217472320646371152805018239368282844960 (Sqrt[2]) ArcCoth[(Sqrt[2])]-2306364859152343573758838172163671703968211296091154164557642391865721616 ArcSinh[1]-1630846231796978707886619913377689137437374031168797693657575884449399376 (Sqrt[2]) ArcSinh[1]+1484377681980810695593770270014383770008736653800588777840165870958361396 Log[2]+1049613524770599736581065855130619439584072864420985591453050558484108214 (Sqrt[2]) Log[2]-297313140481031707400204583700239645017799839207195524893945746103849648 Log[8]-210232137770006152507269472039867339374231129065865355694951093563412360 (Sqrt[2]) Log[8]-98739710089619262232192753152277472492556189363167033859721438774468742 Log[64]-69819518576763546509876239835169570243563246203898254061366212965645189 (Sqrt[2]) Log[64])/(192 (6+(Sqrt[2])) (323+198 (Sqrt[2])) (342303527881312182650212836624027+242045145788926466842611529586950 (Sqrt[2])) (90023344979359906888468753123073054+63656117700001004799713347423226241 (Sqrt[2])));

RootReduce[i2]//FullSimplify

TrigToExp@%

N[(1/136) (-2+23 (Sqrt[2])-3 (-6+(Sqrt[2])) (-Log[1-(1/(Sqrt[2]))]+Log[1+(1/(Sqrt[2]))])+(-13+5 (Sqrt[2])) Log[1+(Sqrt[2])])]

N[(1/136) (-2+23 (Sqrt[2])-3 (-6+(Sqrt[2])) (-Log[(Sqrt[2])-1]+Log[(Sqrt[2])+1])+(-13+5 (Sqrt[2])) Log[1+(Sqrt[2])])]

FullSimplify[(1/136) (-2+23 (Sqrt[2])-3 (-6+(Sqrt[2])) (-Log[(Sqrt[2])-1]+Log[(Sqrt[2])+1])+(-13+5 (Sqrt[2])) Log[1+(Sqrt[2])])]

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)

Integrate [unevaluated]

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

N[rxyz]

CAD [unevaluated]

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

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

N[rxyz,20]

ResourceFunction["LogSimplify"][rxyz]//FullSimplify

CAD over  1/8 [unevaluated]

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

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

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

(rxy=8((Integrate[(Sqrt[(x^2)+(y^2)+(z^2)])Boole[cad],{x,y,z}∈FullRegion[3]])/vol))//Timing

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

Divergence theorem [unevaluated integral]

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

N[rxyz]

ResourceFunction["LogSimplify"][rxyz]//FullSimplify

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

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"],]