Elongated Square Gyrobicupola

Author

Eric W. Weisstein
September 12, 2023

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

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

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

Polyhedron

PolyhedronData["ElongatedSquareGyrobicupola","Orientations"]

PolyhedronData["ElongatedSquareGyrobicupola"]

Net

PolyhedronData["ElongatedSquareGyrobicupola","Net"]

PolyhedronData["ElongatedSquareGyrobicupola","Net","Colored"]

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

Properties

Initialization

<<MathWorld`Polyhedra`

p=PolyhedronData[pname="ElongatedSquareGyrobicupola","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]

Circumsphere

PolyhedronData[pname,"Circumsphere"]

Circumsphere[p]//FullSimplify

sphere=MyCircumsphere[p]//FullSimplify

Graphics3D[{{Opacity[.2],Yellow,sphere},p},Boxed->False]

Convex

ConvexPolyhedronQ[p]

DehnInvariant

PolyhedronData[pname,"DehnInvariant"]

(dehn=DehnInvariant[p,Debug->True,"BasisTerms"->{30,20}]//FullSimplify)//Timing

ToConwayBasis[dehn]

Complement[Select[PolyhedronData[All],PolyhedronData[#,"DehnInvariant"]==dehn&&PolyhedronData[#,"Volume"]==vol&],{pname}]

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]))//RootReduce//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

(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  1/16

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

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

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

(rxy=16((Integrate[(Sqrt[(x^2)+(y^2)])Boole[cad],{x,y,z}∈FullRegion[3]])/vol))//Timing

N[rxy]

LogSimplify[rxy]//FullSimplify

-(((220684595423336375202306612465207421413774273729409688346845835626257+156047573927250881499136561007246548941877112155090057271764536816261 (Sqrt[2])) ArcSinh[1-(Sqrt[2])]+(1134154388468966834133663804202689175601211731413169326189911724753+801968258998888349937255242445989647684469099871972925408872451632 (Sqrt[2])) ArcSinh[1+(Sqrt[2])]+(NumericalApproximation[Root, (RowDefault["-1.84", FEExpressionsNumericalApproximationElider, " × ", (Superscript["10", "68"])]), Root[14992047489084758167507806216458298140066617910799910938752-33699481494022509154250638333139760873838103852866073697156722799601747852703177133186665595999725321797883234469531806121881438165330336 ((#1)^2)+((#1)^4)&,1], ])))((237193742732842952066792563994816504487231382195179923379651117837408-167721303941410628324669607611030140225805972100762946934444693336912 (Sqrt[2]))/Expand[(237193742732842952066792563994816504487231382195179923379651117837408+167721303941410628324669607611030140225805972100762946934444693336912 (Sqrt[2]))((237193742732842952066792563994816504487231382195179923379651117837408-167721303941410628324669607611030140225805972100762946934444693336912 (Sqrt[2])))])//FullSimplify

LogSimplify[%]

Boole Integrate doing z integral first

{{xmin,xmax},{ymin,ymax},{zmin,zmax}}==CoordinateBounds[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

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

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

N[rxyz]

Divergence theorem [unevaluated integral]

(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  1/16 [TerminatedEvaluation[RecursionLimit]]

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

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

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

(rxyz=16((Integrate[(Sqrt[(x^2)+(y^2)+(z^2)])Boole[cad],{x,y,z}∈FullRegion[3]])/vol))//Timing

bug(440097): Integrate[Sqrt[x^2 + y^2 + z^2] Boole[cad], {x, y, z} ∈ FullRegion[3]] returns TerminatedEvaluation["RecursionLimit"]

N[rxyz,20]

rxysimp1=LogSimplify[rxy]//FullSimplify

Boole Integrate doing z integral first

{{xmin,xmax},{ymin,ymax},{zmin,zmax}}=CoordinateBounds[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

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"]

sphere=Midsphere[p]//FullSimplify

Graphics3D[{{Opacity[.5],p},{Red,Midsphere[p]}},Boxed->False]

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"],]