Gyroelongated Square Pyramid

Author

Eric W. Weisstein
September 14, 2023

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

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

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

Polyhedron

PolyhedronData["GyroelongatedSquarePyramid","Orientations"]

PolyhedronData["GyroelongatedSquarePyramid"]

Net

PolyhedronData["GyroelongatedSquarePyramid","Net"]

PolyhedronData["GyroelongatedSquarePyramid","Net","Colored"]

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

Properties

Initialization

<<MathWorld`Polyhedra`

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

vol=PolyhedronData[pname,"Volume"]

Centroid

PolyhedronData[pname,"Centroid"]

RegionCentroid[p]//FullSimplify

RegionCentroid[p]//RootReduce//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"]

FullSimplify[%]

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

PolyhedronInertiaTensor[p]//Timing

MeanCylindricalRadius

PolyhedronData[pname,"MeanCylindricalRadius"]

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

Integrate over polyhedron [unevaluated]

(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)])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  1/8

reg=FullSimplify[PolyhedronData[pname,"RegionFunction"][x,y,z]&&0<y<Tan[Pi/4]x]

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

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

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

N[rxy]

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])/vol)

Integrate over polyhedron

(rxy=((Integrate[(Sqrt[(x^2)+(y^2)+(z^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)

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

(rxy=2((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])/(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"]

Midsphere[p]

Region

PolyhedronData[pname,"Region"]

RegionFunction

PolyhedronData[pname,"RegionFunction"]

{RegionPlot3D[ImplicitRegion[PolyhedronData[pname,"RegionFunction"][x,y,z],{x,y,z}],PlotPoints->50],
Graphics3D[p]}

Collect[#,{x,y,z},RootReduce]&/@PolyhedronInequalities[p,{x,y,z}]

ineq=FullSimplify[%]

ineq/.{x->#,y->#2,z->#3}

{Region@ImplicitRegion[ineq,{x,y,z}],Graphics3D[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"]

FullSimplify[%]

Volume[p]//FullSimplify

PolyhedronVolume[p]//FullSimplify

Volume[PolyhedronData[pname,"Region"]]//FullSimplify

FullSimplify[Subtract@@{(1/3) ((1/(Sqrt[2]))+(2^(1/4))+(2^(3/4))),(1/6) ((Sqrt[2])+2 (Sqrt[4+3 (Sqrt[2])]))}]

Volume[ImplicitRegion[Evaluate@PolyhedronData[pname,"RegionFunction"][x,y,z],{x,y,z}]]//Timing

Perspective projections

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

XXX

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