Császár Polyhedron

Author

Eric W. Weisstein
May 15, 2023

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

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

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

Graphic

Show[PolyhedronData["CsaszarPolyhedron"],Boxed->False]

PolyhedronData["CsaszarPolyhedron","Region"]

Properties

<<MathWorld`Polyhedra`

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

vol=(125/(6 (Sqrt[6])))

Centroid

PolyhedronData["CsaszarPolyhedron","Centroid"]

RegionCentroid [wrong]

RegionCentroid[p]//FullSimplify

Direct

PolyhedronCentroid[p,Method->"Direct"]//Quiet//Simplify

GaussLaw

PolyhedronCentroid[p,Method->"GaussLaw"]//Quiet//Simplify

Difference

Subtract@@{((5789733343518731992834565-63241437852392674397320 (Sqrt[6]))/(44092272 (5593760458166203+30653517590942358 (Sqrt[6])))),(97/(25 (Sqrt[6])))}//N

Circumsphere

sphere=PolyhedronData[pname,"Circumsphere"]

sphere=Circumsphere[p]

(sphere=MyCircumsphere[p])//Timing

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

Compound

PolyhedronData[pname,"Compound"]

Convex

PolyhedronData[pname,"Convex"]

ConvexPolyhedronQ[p]

DihedralAngles

PolyhedronData[pname,"DihedralAngles"]//Tally

SimplifyNumericallyDistinct[DihedralAngles[p]]//Tally

EdgeCount

PolyhedronData[pname,"EdgeCount"]

EdgeLengths

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"]

Faces

GeneralizedDiameter

PolyhedronData[pname,"GeneralizedDiameter"]

GeneralizedDiameter[p]//FullSimplify//Quiet

InertiaTensor

PolyhedronData[pname,"InertiaTensor"]

MomentOfInertia [wrong]

((MomentOfInertia[p,{0,0,0}])/vol)//FullSimplify

GaussLaw

PolyhedronInertiaTensor[p]

Insphere

PolyhedronData[pname,"Insphere"]

Insphere[p]

sphere=MyInsphere[p]

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

MeanCylindricalRadius

PolyhedronData[pname,"MeanCylindricalRadius"]

Integrate over polyhedron

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

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

Divergence theorem

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

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)

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

Inequalities

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

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

(i1=Integrate[Boole[ineq](Sqrt[(x^2)+(y^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

MeanSquareCylindricalRadius

PolyhedronData[pname,"MeanSquareCylindricalRadius"]

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

Integrate [wrong]

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

Divergence theorem

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

N[(791/1500)]

Difference

Subtract@@{(791/1500),((7785404927214859109169256019299866+322337269939549615239394514286691 (Sqrt[6]))/16260998919687564796102672816542000)}//N

MeanSphericalRadius

PolyhedronData[pname,"MeanSphericalRadius"]

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

Integrate

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

Divergence theorem

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

Integrate over z first

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

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

Divergence theorem

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

Difference

Midsphere

sphere=PolyhedronData[pname,"Midsphere"]

(sphere=Midsphere[p])//Timing

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

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

Region

PolyhedronData[pname,"Region"]

RegionFunction

PolyhedronData[pname,"RegionFunction"]

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

FullSimplify[%]

SimplePolyhedron

PolyhedronData[pname,"Simple"]

bug(388294): SimplePolyhedronQ[PolyhedronData[poly, "Polyhedron"]] incorrectly returns True for "CsazarPolyhedron" and "SzilassiPolyhedron"

SimplePolyhedronQ[p]

Skeleton

PolyhedronData["CsaszarPolyhedron","Skeleton"]

PolyhedronData["CsaszarPolyhedron","Skeleton","Name"]

Graph[UndirectedEdge@@@PolyhedronData["CsaszarPolyhedron","Edges"]]

RecognizeGraph[%]

SurfaceArea

PolyhedronData["CsaszarPolyhedron","SurfaceArea"]

SurfaceArea[PolyhedronData["CsaszarPolyhedron","Region"]]//Simplify

N[%]

SurfaceArea[PolyhedronData["CsaszarPolyhedron","Polyhedron"]]//Simplify

PolyhedronSurfaceArea[PolyhedronData["CsaszarPolyhedron","Polyhedron"]]//Simplify

Volume

PolyhedronData["CsaszarPolyhedron","Volume"]

Volume V12.2 [wrong]

bug(407136): Volume returns "an outlandish number" when applied to PolyhedronData["SzilassiPolyhedron", "Polyhedron"]

Volume[p]//Simplify

N[%]

Volume

$Version

Volume[p]

N[%]

Volume by PolyhedronDecomposition V12.2 [wrong]

Length[pieces=PolyhedronDecomposition[p]]

Length[PolyhedronCoordinates[#]]&/@pieces//Union

Total[Volume/@pieces]//FullSimplify

Volume by PolyhedronDecomposition V13.3 [wrong]

$Version

Length[pieces=PolyhedronDecomposition[p]]

Length[PolyhedronCoordinates[#]]&/@pieces//Union

Total[Volume/@pieces]//FullSimplify

NIntegrate

NIntegrate[1,{x,y,z}∈PolyhedronData["CsaszarPolyhedron","Polyhedron"]]//Timing

Direct

PolyhedronVolume[PolyhedronData["CsaszarPolyhedron","Polyhedron"],Method->"Direct"]//FullSimplify

GaussLaw

PolyhedronVolume[PolyhedronData["CsaszarPolyhedron","Polyhedron"],Method->"GaussLaw"]//FullSimplify

N[%]

Subtract@@{(125/(6 (Sqrt[6]))),(5593760458166203/9485902291752936)+(349/(18 (Sqrt[6])))}//N

PolyhedronData

PolyhedronDataString

PolyhedronDataString[{Polyhedron[Csaszar],"CsaszarPolyhedron","Csaszar polyhedron"},{"Concave"},Simplify]

Show[Normal[PolyhedronData["CsaszarPolyhedron","SkeletonImage"]]/.Annotation[x_,y__]:>x/.Tooltip[x_,y__]:>Sequence@@{}];