Biaugmented Pentagonal Prism

Author

Eric W. Weisstein
September 30, 2022

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

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

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

Solid

PolyhedronData["BiaugmentedPentagonalPrism","Orientations"]

PolyhedronData["BiaugmentedPentagonalPrism"]

Net

PolyhedronData["BiaugmentedPentagonalPrism","Net"]

PolyhedronData["BiaugmentedPentagonalPrism","Net","Colored"]

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

Properties

<<MathWorld`Polyhedra`

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

vol=PolyhedronData[pname,"Volume"]

Centroid

PolyhedronData[pname,"Centroid"]//FullSimplify

RegionCentroid[p]//FullSimplify//Timing

PolyhedronCentroid[p]//FullSimplify//Timing

Circumsphere

PolyhedronData[pname,"Circumsphere"]

Circumsphere[p]

sphere=MyCircumsphere[p]

DihedralAngles

PolyhedronData[pname,"DihedralAngles"]//Tally

SimplifyNumericallyDistinct[DihedralAngles[p]]

EdgeLengths

PolyhedronData[pname,"EdgeLengths"]

PolyhedronData[pname,"Lines"]/.Line[l_]:>EuclideanDistance@@l//FullSimplify//Union

Faces

GeneralizedDiameter

PolyhedronData[pname,"GeneralizedDiameter"]

GeneralizedDiameter[p]//RootReduce

FullSimplify[%]

Insphere

PolyhedronData[pname,"Insphere"]

Insphere[p]

MyInsphere[p]

InertiaTensor

PolyhedronData[pname,"InertiaTensor"]

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

PolyhedronInertiaTensor[p]//FullSimplify//Timing

MeanCylindricalRadius

PolyhedronData["BiaugmentedPentagonalPrism","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

{{xmin,xmax},{ymin,ymax},{zmin,zmax}}=MinMax/@Transpose[PolyhedronCoordinates[p]]

ineq=PolyhedronData[pname,"RegionFunction"][x,y,z]

((NIntegrate[Boole[ineq](Sqrt[(x^2)+(y^2)]),{x,xmin,xmax},{y,ymin,ymax},{z,zmin,zmax}])/(PolyhedronData[pname,"Volume"]))

(i1=Integrate[Boole[ineq](Sqrt[(x^2)+(y^2)]),{z,zmin,zmax}]/.p_Piecewise:>RootReduce/@p)//Timing

NIntegrate[(i1/vol),{x,xmin,xmax},{y,ymin,ymax}]

(i2=Integrate[(i1/vol),{x,xmin,xmax},{y,ymin,ymax}])//Timing

MeanSquareCylindricalRadius

PolyhedronData["BiaugmentedPentagonalPrism","MeanSquareCylindricalRadius"]

((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"]))//RootReduce//FullSimplify)//Timing

N[i]

MeanSphericalRadius

PolyhedronData["BiaugmentedPentagonalPrism","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]

MeanSquareSphericalRadius

PolyhedronData["BiaugmentedPentagonalPrism","MeanSquareSphericalRadius"]

((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"]))//RootReduce//FullSimplify)//Timing

N[i]

Midsphere

PolyhedronData[pname,"Midsphere"]

Midsphere[p]//Timing

SurfaceArea

PolyhedronData[pname,"SurfaceArea"]

RootReduce[%]

SurfaceArea[p]//RootReduce//FullSimplify//Timing

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