Pentagonal Dipyramid

Author

Eric W. Weisstein
May 14, 2021

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

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

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

Solid

PolyhedronData["PentagonalDipyramid","Orientations"]

PolyhedronData["PentagonalDipyramid"]

Obsolete Package

Show[Graphics3D[Dipyramid[5]],Boxed->False,ViewPoint->{2,2.6,.8},PlotRange->All]

Solid and Dual

PolyhedronData["PentagonalDipyramid",#]&/@{"Graphics3D","Dual","DualCompound"}

Net

PolyhedronData["PentagonalDipyramid","Net"]

PolyhedronData["PentagonalDipyramid","Net","Colored"]

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

Properties

<<MathWorld`PolyhedronOperations`

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

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]//FullSimplify

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

Height

ToRadicals[FullSimplify[Sqrt[1-R^2]]]

h=Together//@%

FullSimplify[R/h]

FunctionExpand[GoldenRatio]

InertiaTensor

PolyhedronData[pname,"InertiaTensor"]

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

PolyhedronInertiaTensor[p]//Timing

Insphere

sphere=PolyhedronData[pname,"Insphere"]

Insphere[p]//FullSimplify

sphere=MyInsphere[p]//FullSimplify

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

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"]))//FullSimplify)//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"]))//FullSimplify)//Timing

N[i]

Midsphere

sphere=PolyhedronData[pname,"Midsphere"]

(sphere=Midsphere[p])//Timing

SurfaceArea

PolyhedronData[pname,"SurfaceArea"]

SurfaceArea[p]//RootReduce

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

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"],]