Triangular Orthobicupola

Author

Eric W. Weisstein
September 6, 2023

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

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

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

Polyhedron

PolyhedronData["TriangularOrthobicupola","Orientations"]

PolyhedronData["TriangularOrthobicupola"]

Net

PolyhedronData["TriangularOrthobicupola","Net"]

PolyhedronData["TriangularOrthobicupola","Net","Colored"]

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

Properties

Initialization

<<MathWorld`Polyhedra`

p=PolyhedronData[pname="TriangularOrthobicupola","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]//FullSimplify

Circumsphere

PolyhedronData[pname,"Circumsphere"]

Circumsphere[p]

(sphere=MyCircumsphere[p])//Timing

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

Convex

ConvexPolyhedronQ[p]

DehnInvariant

PolyhedronData[pname,"DehnInvariant"]

(dehn=DehnInvariant[p,Debug->True]//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[%]

Integrate over polyhedron

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

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

N[rxy]

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  1/4

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

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

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

(rxy=4((Integrate[(Sqrt[(x^2)+(y^2)])Boole[cad],{x,y,z}∈FullRegion[3]])/vol))//Timing

N[rxy,20]

ResourceFunction["LogSimplify"][rxy]//FullSimplify

N[%]

N[(1/960) ((27+(Sqrt[3]))(8+9 Log[3])-2 (1+9 (Sqrt[3])) Log[2+(Sqrt[3])])]

Divergence theorem [wrong integral]

(rxy=MeanCylindricalRadius[p,"Simplification"->FullSimplify,Debug->True])//Timing

N[rxy]

ResourceFunction["LogSimplify"][rxy]//FullSimplify

MeanSquareCylindricalRadius

PolyhedronData[pname,"MeanSquareCylindricalRadius"]

((NIntegrate[(x^2)+(y^2),{x,y,z}∈p])/vol)//Timing

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

N[i]

MeanSphericalRadius

PolyhedronData[pname,"MeanSphericalRadius"]

Integrate over polyhedron

rxy=((NIntegrate[(Sqrt[(x^2)+(y^2)+(z^2)]),{x,y,z}∈p])/vol)

ClosestMeanSphericalRadius[rxy]

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

N[rxy]

Divergence theorem [bad integral(s)]

(rxy=MeanCylindricalRadius[p,"Simplification"->FullSimplify,Debug->True])//Timing

N[rxy]

ResourceFunction["LogSimplify"][rxy]//FullSimplify

CAD [crash]

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)

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

...crash...

CAD over  1/4 [unevaluated]

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

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

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

(rxy=4((Integrate[(Sqrt[(x^2)+(y^2)+(z^2)])Boole[cad],{x,y,z}∈FullRegion[3]])/vol))//Timing

N[rxy,20]

ResourceFunction["LogSimplify"][rxy]//FullSimplify

N[%]

MeanSquareSphericalRadius

PolyhedronData[pname,"MeanSquareSphericalRadius"]

((NIntegrate[(x^2)+(y^2)+(z^2),{x,y,z}∈p])/vol)//Timing

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

N[i]

Midsphere

sphere=PolyhedronData[pname,"Midsphere"]

(sphere=Midsphere[p])//Timing

Graphics3D[{{Opacity[.5],p},{Red,sphere}},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"],]