Gyroelongated Pentagonal Cupola

Author

Eric W. Weisstein
May 13, 2021

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

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

©2021 Wolfram Research, Inc. except for portions noted otherwise

Solid

PolyhedronData["GyroelongatedPentagonalCupola","Orientations"]

PolyhedronData["GyroelongatedPentagonalCupola"]

Net

PolyhedronData["GyroelongatedPentagonalCupola","Net"]

PolyhedronData["GyroelongatedPentagonalCupola","Net","Colored"]

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

Properties

<<MathWorld`PolyhedronOperations`

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

Centroid

Graphics3D[{AbsolutePointSize[10],
Point[{0,0,0}],{Blue,Point[PolyhedronData[pname,"Centroid"]]},Opacity[.2],PolyhedronData[pname,"GraphicsComplex"]}]

PolyhedronData[pname,"Centroid"]

RegionCentroid[p]//RootReduce

PolyhedronCentroid[p]//RootReduce

Circumsphere

PolyhedronData[pname,"Circumsphere"]

Circumsphere[p]

(sphere=MyCircumsphere[p])//Timing

Convex

ConvexPolyhedronQ[p]

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

((NIntegrate[(Sqrt[(x^2)+(y^2)]),{x,y,z}∈p])/(PolyhedronData[pname,"Volume"]))//Timing

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

N[i]

MeanSquareCylindricalRadius

((NIntegrate[(x^2)+(y^2),{x,y,z}∈p])/(PolyhedronData[pname,"Volume"]))//Timing

(i=((Integrate[(x^2)+(y^2),{x,y,z}∈p])/(PolyhedronData[pname,"Volume"]))//RootReduce)//Timing

N[i]

MeanSphericalRadius

((NIntegrate[(Sqrt[(x^2)+(y^2)+(z^2)]),{x,y,z}∈p])/(PolyhedronData[pname,"Volume"]))//Timing

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

N[i]

MeanSquareSphericalRadius

((NIntegrate[(x^2)+(y^2)+(z^2),{x,y,z}∈p])/(PolyhedronData[pname,"Volume"]))//Timing

(i=((Integrate[(x^2)+(y^2)+(z^2),{x,y,z}∈p])/(PolyhedronData[pname,"Volume"]))//RootReduce)//Timing

N[i]

Midsphere

sphere=PolyhedronData[pname,"Midsphere"]

(sphere=Midsphere[p])//Timing

SurfaceArea

PolyhedronData[pname,"SurfaceArea"]

RootReduce[%]

SurfaceArea[p]//RootReduce//Timing

Total[Area/@PolyhedronData[pname,"Polygons"]]//RootReduce//Timing

VertexSubsetHulls

PolyhedronData[pname,"VertexSubsetHulls"]

Platonic

(h=VertexSubsetHulls[p,"Platonic","Skip"->{}])//Timing

Volume

PolyhedronData[pname,"Volume"]

Volume[p]//FullSimplify

PolyhedronVolume[p,Method->"GaussLaw"]//FullSimplify

RootReduce[%]

FullSimplify[%]

exprs={(1/6) (5+4 (Sqrt[5])+5 (Sqrt[2 (-1-(Sqrt[5])+(Sqrt[650+290 (Sqrt[5])]))])),(NumericalApproximation[Root, (RowDefault["9.07", FEExpressionsNumericalApproximationElider]), Root[5340175625+13859430000 #1+3828402000 ((#1)^2)-10714248000 ((#1)^3)-4456749600 ((#1)^4)+35769600 ((#1)^5)+27060480 ((#1)^6)-11197440 ((#1)^7)+1679616 ((#1)^8)&,4], ])};

LeafCount/@exprs

ByteCount/@exprs

Perspective projections

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

XXX

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