Gyroelongated Pentagonal Pyramid

Author

Eric W. Weisstein
May 3, 2022

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

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

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

Solid

PolyhedronData["GyroelongatedPentagonalPyramid","Orientations"]

PolyhedronData["GyroelongatedPentagonalPyramid"]

Net

PolyhedronData["GyroelongatedPentagonalPyramid","Net"]

PolyhedronData["GyroelongatedPentagonalPyramid","Net","Colored"]

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

Properties

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

Centroid

PolyhedronData[pname,"Centroid"]

RegionCentroid[p]//FullSimplify

PolyhedronCentroid[p]//Timing

Circumsphere

PolyhedronData[pname,"Circumsphere"]

Circumsphere[p]//Timing

sphere=MyCircumsphere[p]

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

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,"Simplification"->FullSimplify]//Timing

InertiaTensor

PolyhedronData[pname,"InertiaTensor"]

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

PolyhedronInertiaTensor[p]//Timing

MeanCylindricalRadius

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

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

N[i]

MeanCylindricalSquareRadius

((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

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

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

N[i]

MeanSphericalSquareRadius

((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"]

sphere=Midsphere[p]

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

Skeleton

PolyhedronData[pname,"Skeleton"]//StyleGraphs

RecognizeGraph[%]

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

Perspective projections

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

XXX

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