Triangular Hebesphenorotunda

Author

Eric W. Weisstein
May 15, 2021

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

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

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

Solid

PolyhedronData (V11.1+)

PolyhedronData["TriangularHebesphenorotunda","Orientations"]

PolyhedronData["TriangularHebesphenorotunda"]

PolyhedronData (pre V11.1)

PolyhedronData["TriangularHebesphenorotunda"]

Netlib

PolyhedronName[JohnsonSolid[92]]

Show[Graphics3D[Polyhedron[JohnsonSolid[92]]],Boxed->False]

Net

PolyhedronData

PolyhedronData["TriangularHebesphenorotunda","Net"]

PolyhedronData["TriangularHebesphenorotunda","Net","Colored"]

Netlib

Show[Graphics[Net[JohnsonSolid[92]]],AspectRatio->Automatic]

Properties

<<MathWorld`PolyhedronOperations`

p=PolyhedronData[pname="TriangularHebesphenorotunda","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

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"]))//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

Oriented

gc=PolyhedronData[pname,"GraphicsComplex"]

Graphics3D[{FaceForm[Green,Red],gc}]

OrientedQ[Normal[gc]]

And@@First[%]

Graphics3D[{{AbsolutePointSize[10],Black,Point[{0,0,0}]},Opacity[.5],FaceForm[{Blue,Red}],Normal[gc]}]

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

Construction

Embedding by
Bill Gosper
14 Oct 2016

gc=GraphicsComplex[{{-1,0,1/32 (Sqrt[3]+5 Sqrt[15])},{-(1/2),-(1/(2 Sqrt[3])),-((45+Sqrt[5])/(32 Sqrt[3]))},{-(1/2),-(Sqrt[3]/2),1/32 (Sqrt[3]+5 Sqrt[15])},{-(1/2),Sqrt[3]/2,1/32 (Sqrt[3]+5 Sqrt[15])},{-(1/2),(2+Sqrt[5])/(2 Sqrt[3]),-((13+Sqrt[5])/(32 Sqrt[3]))},{0,1/Sqrt[3],-((45+Sqrt[5])/(32 Sqrt[3]))},{0,-((3+Sqrt[5])/(2 Sqrt[3])),(-29+15 Sqrt[5])/(32 Sqrt[3])},{1/2,-(1/(2 Sqrt[3])),-((45+Sqrt[5])/(32 Sqrt[3]))},{1/2,-(Sqrt[3]/2),1/32 (Sqrt[3]+5 Sqrt[15])},{1/2,Sqrt[3]/2,1/32 (Sqrt[3]+5 Sqrt[15])},{1/2,(2+Sqrt[5])/(2 Sqrt[3]),-((13+Sqrt[5])/(32 Sqrt[3]))},{1,0,1/32 (Sqrt[3]+5 Sqrt[15])},{1/4 (-3-Sqrt[5]),1/12 (Sqrt[3]-Sqrt[15]),-((13+Sqrt[5])/(32 Sqrt[3]))},{1/4 (-3-Sqrt[5]),(3+Sqrt[5])/(4 Sqrt[3]),(-29+15 Sqrt[5])/(32 Sqrt[3])},{1/4 (-1-Sqrt[5]),-((5+Sqrt[5])/(4 Sqrt[3])),-((13+Sqrt[5])/(32 Sqrt[3]))},{1/4 (1+Sqrt[5]),-((5+Sqrt[5])/(4 Sqrt[3])),-((13+Sqrt[5])/(32 Sqrt[3]))},{1/4 (3+Sqrt[5]),1/12 (Sqrt[3]-Sqrt[15]),-((13+Sqrt[5])/(32 Sqrt[3]))},{1/4 (3+Sqrt[5]),(3+Sqrt[5])/(4 Sqrt[3]),(-29+15 Sqrt[5])/(32 Sqrt[3])}},Polygon[{{14,5,6,2,13},{5,14,4},{14,1,4},{1,14,13},{1,13,15,3},{15,13,2},{7,15,2,8,16},{3,15,7},{9,3,7},{9,7,16},{17,16,8},{9,16,17,12},{6,8,2},{6,11,18,17,8},{18,12,17},{18,10,12},{10,18,11},{5,4,10,11},{5,11,6},{10,4,1,3,9,12}}]];

PolyhedronDataString

{skel=SkeletonGraph[gc],GraphData[{"JohnsonSkeleton",92}]}

PolyhedronVertices[gc]

PolyhedronFaces[gc]

PolyhedronEdgeLengths[gc]//FullSimplify

PolyhedronEdges[gc]

PolyhedronDataString[gc,
"StandardName"->"TriangularHebesphenorotunda",
"Name"->"triangular hebesphenorotunda",
Debug->True
]