Ten-of-Diamonds Decahedron

Author

Eric W. Weisstein
August 21, 2023

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Polyhedra/Ten-of-DiamondsDecahedron.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/Ten-of-DiamondsDecahedron.html.

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

Sources

https://en.wikipedia.org/wiki/Ten-of-diamonds_decahedron

Goldberg:

https://upcommons.upc.edu/bitstream/handle/2099/990/st7-06-a3.pdf?sequence=1&isAllowed=y

Solid

PolyhedronData["TenOfDiamondsDecahedron","Orientations"]

Graphics3D[{Opacity[.5],PolyhedronData["TenOfDiamondsDecahedron","Polyhedron"]},Boxed->False]

Show[PolyhedronData[{"TenOfDiamondsDecahedron",#}],Boxed->False]&/@PolyhedronData["TenOfDiamondsDecahedron","Orientations"]

Polyhedron

Graphics3D[{FaceForm[Green,Red],PolyhedronData["TenOfDiamondsDecahedron","Polyhedron"]}]

Solid and Dual

PolyhedronData["TenOfDiamondsDecahedron",#]&/@{"Graphics3D","Dual","DualCompound"}

Net

PolyhedronData["TenOfDiamondsDecahedron","Net"]

PolyhedronData["TenOfDiamondsDecahedron","Net","Colored"]

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

Properties

Initialization

<<MathWorld`Polyhedra`

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

vol=PolyhedronData[pname,"Volume"]

Centroid

PolyhedronData[pname,"Centroid"]

Graphics3D[{AbsolutePointSize[10],
Point[{0,0,0}],{Blue,Point[PolyhedronData[pname,"Centroid"]]},Opacity[.2],PolyhedronData[pname,"GraphicsComplex"]}]

RegionCentroid[p]

PolyhedronCentroid[p]

Circumsphere

sphere=PolyhedronData[pname,"Circumsphere"]

sphere=Circumsphere[p]

(sphere=MyCircumsphere[p])//Timing

Classes

PolyhedronData[pname,"Classes"]

Compound

PolyhedronData[pname,"Compound"]

Convex

PolyhedronData[pname,"Convex"]

ConvexPolyhedronQ[p]

ConvexHull

PolyhedronData[pname,"ConvexHull"]

convexhull=ConvexHullRegion[PolyhedronData[pname,"Vertices"]]

v=PolyhedronCoordinates[convexhull]

Sort[(convexhull[[2]]/.i_Integer:>v[[i]])/.Thread[PolyhedronData[pname,"Vertices"]->Range[PolyhedronData[pname,"V"]]]]

Graphics3D[{FaceForm[Green,Red],convexhull,MapIndexed[Text[#2[[1]],#,Background->White]&,v]}]

{Graphics3D[{Red,Opacity[.5],p}],Graphics3D[{Blue,Opacity[.5],PolyhedronData[pname,"ConvexHull"][[1]]}]}

Show[%]

Graphics3D[{p,MapIndexed[Text[#2[[1]],#,Background->White]&,p[[1]]]}]

DehnInvariant

PolyhedronData[pname,"DehnInvariant"]

DehnInvariant[p,Debug->True]

ToConwayBasis[%]

DihedralAngles

PolyhedronData[pname,"DihedralAngles"]//Tally

N[%[[All,1]]]

SimplifyNumericallyDistinct[DihedralAngles[p]]//Tally

EdgeCount

PolyhedronData[pname,"EdgeCount"]

EdgeLengths

Tally[PolyhedronEdgeLengths[p]]

PolyhedronData[pname,"EdgeLengths"]

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

lens=SimplifyNumericallyDistinct[PolyhedronEdgeLengths[p]];//Quiet

(distinctlens=Mean/@Split[Sort[lens],Abs[#1-#2]<1.*^-10&])//N//Quiet

v=PolyhedronCoordinates[p];

colors=With[{n=Length[distinctlens]},Table[Hue[i/n],{i,0,n-1}]]

Graphics3D[{Thick,Sequence@@Nearest[Thread[distinctlens->colors],EuclideanDistance@@v[[#]]],Line[v[[#]]]}&/@PolyhedronEdges[p],Boxed->False]//Quiet

FaceCount

PolyhedronData[pname,"FaceCount"]

GeneralizedDiameter

PolyhedronData[pname,"GeneralizedDiameter"]

GeneralizedDiameter[p]//FullSimplify//Quiet

Hull

PolyhedronData[pname,"Hull","Name"]

PolyhedronData[pname,"Hull","Scale"]

PolyhedronData[pname,"Hull"]

Max[Norm/@PolyhedronData[pname,"Hull","Polyhedron"][[1]]]

Max[Norm/@p[[1]]]

{Graphics3D[{Opacity[.5],Red,p}],Graphics3D[{Opacity[.5],Blue,PolyhedronData[pname,"Hull"][[1]]}]}

Show[%]

InertiaTensor

PolyhedronData[pname,"InertiaTensor"]

(i=((MomentOfInertia[p,{0,0,0}])/(Volume[p])))//Timing

Map[ToAlgebraicRoot,i,{2}]

PolyhedronInertiaTensor[p]//Timing

Insphere

PolyhedronData[pname,"Insphere"]

Insphere[p]

sphere=MyInsphere[p]

Graphics3D[{{Opacity[.5],p},{Red,sphere}},Boxed->False]

MeanCylindricalRadius

PolyhedronData[pname,"MeanCylindricalRadius"]

N[%]

Integrate over polyhedron

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

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

Divergence theorem

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

N[rxy]

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

N[%]

MeanSquareCylindricalRadius

PolyhedronData[pname,"MeanSquareCylindricalRadius"]

N[%]

Integrate over Polyhedron

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

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

Divergence theorem

(rxy=MeanSquareCylindricalRadius[p,Debug->False])//Timing

MeanSphericalRadius

PolyhedronData[pname,"MeanSphericalRadius"]

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

Integrate

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

Divergence theorem

(rxyz=MeanSphericalRadius[p,Debug->True])//Timing

N[rxyz]

ResourceFunction["LogSimplify"][rxyz]//FullSimplify

CAD

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)

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

Boole Integrate doing z integral first

{{xmin,xmax},{ymin,ymax},{zmin,zmax}}=CoordinateBounds[p]

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

(i1=Integrate[Boole[ineq](Sqrt[(x^2)+(y^2)+(z^2)]),{z,zmin,zmax}])//Timing

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

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

MeanSquareSphericalRadius

PolyhedronData[pname,"MeanSquareSphericalRadius"]

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

Integrate

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

Divergence theorem

(rxy2=MeanSquareSphericalRadius[p,Debug->False])

Midsphere

sphere=PolyhedronData[pname,"Midsphere"]

(sphere=Midsphere[p])//Timing

(sphere=Midsphere[p,Method->"PointLineDistance"])//Timing

Net

PolyhedronData[pname,"Net"]

Total[Area/@Flatten[PolyhedronData[pname,"Net","Polygon"]]]//FullSimplify

PolyhedronData[pname,"SurfaceArea"]

poly3d=PolyhedronData[pname,"Net","Polygons"]/.Polygon[v_]:>Polygon[Append[#,0]&/@v]

Total[Area/@Flatten[poly3d]]//RootReduce//FullSimplify

Graphics3D[{FaceForm[Green,Red],poly3d}]

Polygons

PolyhedronData[pname,"FaceCount"]

Graphics3D/@PolyhedronData[pname,"Polygons"]

Show[%]

Polyhedra

PolyhedronData[pname,"Polyhedra"]

Graphics3D[%]

PolyhedronData[pname,"PolyhedronIndices"]

Region

PolyhedronData[pname,"Region"]

RegionFunction

PolyhedronData[pname,"RegionFunction"]

Collect[#,{x,y,z},RootReduce]&/@PolyhedronInequalities[p,{x,y,z}]

FullSimplify[%]

SimplePolyhedron

PolyhedronData[pname,"Simple"]

SimplePolyhedronQ[p]

Skeleton

PolyhedronData[pname,"Skeleton"]

gn=PolyhedronData[pname,"Skeleton","Name"]

emb=GraphData[gn,"PolyhedralEmbeddings"]

SurfaceArea

PolyhedronData[pname,"SurfaceArea"]

SurfaceArea[p]//FullSimplify

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

SurfaceArea[PolyhedronData[pname,"Region"]]

VertexCount

PolyhedronData[pname,"VertexCount"]

VertexSubsetHulls

subsethulls=PolyhedronData[pname,"VertexSubsetHulls"]

subsethulls2=Flatten[Function[{pn,l},Rule[pn,#]&/@l]@@@subsethulls]

v=PolyhedronData[pname,"Vertices"];

Rule[#,Graphics3D[{{Green,p},Blue,Opacity[.5],ConvexHullRegion[v[[#2]]]}]]&@@@subsethulls2

Platonic

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

Volume

PolyhedronData[pname,"Volume"]

Volume[p]//FullSimplify

PolyhedronVolume[p]//FullSimplify

Volume[PolyhedronData[pname,"Region"]]

Perspective projections

Manipulate[PolyhedronProjectionGraph[p,{a,b,c}],{a,-1,1},{b,-1,1},{c,-1,1}]

XXX

PolyhedronProjectionGraph[p,#]&/@{}

Net construction

Table[{poly=UnfoldPolyhedron[p,WorkingPrecision->1000],Graphics[{FaceForm[LightYellow],EdgeForm[Black],poly}]},100]

{Polygon[],}

net=Polygon[]

Graphics[{FaceForm[LightYellow],EdgeForm[Black],net,MapIndexed[Text[#2[[1]],#,Background->White]&,v=net[[1]]]}]

rot=RotationMatrix[{v[[13]]-v[[12]],{1,0}}];

v2=(rot.(#-Mean[v[[{3,12}]]]))&/@v;

Graphics[{FaceForm[LightYellow],EdgeForm[Black],Polygon[v2,net[[2]]],MapIndexed[Text[#2[[1]],#,Background->White]&,v2]}]

vexact=Map[ToAlgebraicRoot,v2,{2}]

vexact[[{3,12}]]//Mean

net[[2]]//Sort

Construction

p=ConvexHullRegion[{{0,-2,−1},{0,2,-1},{-2,0,1},{2,0,1},{-1,0,−1},{1,0,-1},{0,-1,1},{0,1,1}}]

Min[PolyhedronEdgeLengths[p]]

Volume[p]

PolyhedronData[pname,"Volume"]((((Sqrt[5])))^3)

Tally[FullSimplify[DihedralAngles[p]]]

DehnInvariant[p,Debug->True]

Canonical construction

(can=FindCanonicalPolyhedron[p,WorkingPrecision->1000,MaxIterations->50000,PrecisionGoal->500,"ProgressInterval"->1000,Debug->True])//Timing

SimplePolyhedronQ[can]

(lens=Mean/@Split[Sort[PolyhedronEdgeLengths[can]],Abs[#1-#2]<1.*^-10&])//N

v=PolyhedronCoordinates[can];

colors=With[{n=Length[lens]},Table[Hue[i/n],{i,0,n-1}]]

Graphics3D[{Thick,Sequence@@Nearest[Thread[lens->colors],EuclideanDistance@@v[[#]]],Line[v[[#]]]}&/@PolyhedronEdges[can],Boxed->False]

v=PolyhedronCoordinates[can];

vexact=Map[ToAlgebraicRoot,v,{2}]

poly=Polyhedron[v,can[[2]]]

PolyhedronEdgeLengths[poly]//RootReduce//Union

sphere=Midsphere[poly]

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