Triaugmented Triangular Prism

Author

Eric W. Weisstein
October 3, 2022

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

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

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

Solid

PolyhedronData["TriaugmentedTriangularPrism","Orientations"]

PolyhedronData["TriaugmentedTriangularPrism"]

Net

PolyhedronData["TriaugmentedTriangularPrism","Net"]

PolyhedronData["TriaugmentedTriangularPrism","Net","Colored"]

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

Properties

<<MathWorld`Polyhedra`

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

vol=PolyhedronData[pname,"Volume"]

Centroid

PolyhedronData[pname,"Centroid"]

RegionCentroid[p]//RootReduce

PolyhedronCentroid[p]//RootReduce

Circumsphere

PolyhedronData[pname,"Circumsphere"]

Circumsphere[p]

MyCircumsphere[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

Insphere

PolyhedronData[pname,"Insphere"]

Insphere[p]

MyInsphere[p]

InertiaTensor

PolyhedronData[pname,"InertiaTensor"]

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

PolyhedronInertiaTensor[p]

MeanCylindricalRadius

PolyhedronData[pname,"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]

MeanSquareCylindricalRadius

PolyhedronData[pname,"MeanSquareCylindricalRadius"]

N[%]

((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

PolyhedronData[pname,"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[pname,"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)//Timing

N[i]

Midsphere

PolyhedronData[pname,"Midsphere"]

Midsphere[p,Debug->True,"Simplification"->FullSimplify]

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}]

TriaugmentedTriangularPrism

PolyhedronProjectionGraph[PolyhedronData["DuererSolid","Polyhedron"],]

Construction

Canonical

Graphics3D[poly=PolyhedronData["TriaugmentedTriangularPrism","Polyhedron"]]

(poly=FindCanonicalPolyhedron[poly,WorkingPrecision->1000,PrecisionGoal->100])//Timing

Graphics3D[{Opacity[.5],Red,poly,Yellow,Sphere[{0,0,0},1]},Boxed->False]

v=N[poly[[1]],90]

N[Sqrt[3],20]

Map[ToAlgebraicRoot,v,{2}]

dipyr7=Polyhedron[%25,PolyhedronData[{"Dipyramid",3},"FaceIndices"]]