Augmented Dodecahedron

Author

Eric W. Weisstein
May 12, 2021

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

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

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

Graphic

PolyhedronData[{"Johnson",58},"Name"]

PolyhedronData[{"Johnson",58},"StandardName"]

Show[PolyhedronData[{"Johnson",58}],Boxed->False,Method->{"ShrinkWrap"->True},ImageSize->Small]

Net

PolyhedronData["AugmentedDodecahedron","Net"]

PolyhedronData["AugmentedDodecahedron","Net","Colored"]

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

Properties

<<MathWorld`PolyhedronOperations`

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

Centroid

PolyhedronData[pname,"Centroid"]

RegionCentroid[p]//FullSimplify

PolyhedronCentroid[p]//FullSimplify

Circumsphere

PolyhedronData[pname,"Circumsphere"]

Circumsphere[p]

sphere=MyCircumsphere[p]

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

Faces

GeneralizedDiameter

PolyhedronData[pname,"GeneralizedDiameter"]

GeneralizedDiameter[p]//FullSimplify//Quiet

Insphere

PolyhedronData[pname,"Insphere"]

Insphere[p]

MyInsphere[p,"Simplification"->FullSimplify]//Timing

InertiaTensor

PolyhedronData[pname,"InertiaTensor"]

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

N[i]

Midsphere

PolyhedronData[pname,"Midsphere"]

Midsphere[p,"Simplification"->FullSimplify]//Timing

SurfaceArea

PolyhedronData[pname,"SurfaceArea"]

RootReduce[%]

(s=SurfaceArea[p])//Timing

r=RootApproximant[N[s,1000],64]

N[s-r,5000]

(s2=Total[Area/@PolyhedronData[pname,"Polygons"]])//Timing

r2=RootApproximant[N[s2,1000],64]

N[s2-r2,5000]

VertexSubsetHulls

PolyhedronData[pname,"VertexSubsetHulls"]

Platonic

(h=VertexSubsetHulls[p,"Platonic","Skip"->{}])//Timing

Volume

PolyhedronData[pname,"Volume"]

Volume[p]//FullSimplify

PolyhedronVolume[p,Method->"GaussLaw"]//FullSimplify