Trapezo-Rhombic Dodecahedron

Author

Eric W. Weisstein
October 13, 2023

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Polyhedra/Trapezo-RhombicDodecahedron.nb.

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

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

Source

https://en.wikipedia.org/wiki/Trapezo-rhombic_dodecahedron

In geometry, the trapezo-rhombic dodecahedron or rhombo-trapezoidal dodecahedron is a convex dodecahedron with 6 rhombic and 6 trapezoidal faces. It has D3h symmetry.

Solid

PolyhedronData["TrapezoRhombicDodecahedron","Orientations"]

PolyhedronData["TrapezoRhombicDodecahedron","DefaultOrientation"]

PolyhedronData[{"TrapezoRhombicDodecahedron",#}]&/@PolyhedronData["TrapezoRhombicDodecahedron","Orientations"]

Graphics3D[{Opacity[.7],PolyhedronData[{"TrapezoRhombicDodecahedron","HCP"},"GraphicsComplex"]},Boxed->False,ViewPoint->{0.87, 1.96, 2.60},ViewVertical->{0.52, -0.75, 0.39},Method->{"ShrinkWrap"->True}]

Graphics3D[{Opacity[.7],PolyhedronData["TrapezoRhombicDodecahedron","GraphicsComplex"]},Axes->True,AxesOrigin->{0,0,0}]

Graphics3D[{Opacity[.7],PolyhedronData["TrapezoRhombicDodecahedron","GraphicsComplex"]},Axes->False,AxesOrigin->{0,0,0},ViewPoint->{0,0,2},Boxed->False]

Graphics3D[{PointSize[.04],Point[{0,0,0}],FaceForm[Blue,Red],Opacity[.8],PolyhedronData["TrapezoRhombicDodecahedron","GraphicsComplex"]},Boxed->False]

RegionPlot3D

RegionPlot3D[PolyhedronData["TrapezoRhombicDodecahedron","RegionFunction"][x,y,z],{x,-1.7,1.7},{y,-1.7,1.7},{z,-1.7,1.7},Boxed->False,Axes->False,PlotPoints->25,MaxRecursion->5]

Region

PolyhedronData["TrapezoRhombicDodecahedron","Region"]

ConvexHullMesh

v=Sort[{-1,1,-1} #&/@(6PolyhedronData[{"TrapezoRhombicDodecahedron","HCP"},"VertexCoordinates"]/Sqrt[3])]

edges=Select[Subsets[v/6,{2}],MemberQ[{1,3/2,2}/Sqrt[3],EuclideanDistance@@#]&];

Graphics3D[Tube[edges],Boxed->False]

ch=ConvexHullMesh[v]

ch["VertexCount"]

ch["EdgeCount"]

ch["FaceCount"]

Net

PolyhedronData["TrapezoRhombicDodecahedron","Net"]

PolyhedronData["TrapezoRhombicDodecahedron","Net","Colored"]

Orientations

C3

Graphics3D[{
MapIndexed[Text[#2[[1]],#,Background->White]&,v=PolyhedronData["TrapezoRhombicDodecahedron","VertexCoordinates"]],
PolyhedronData["TrapezoRhombicDodecahedron","GraphicsComplex"]
},Boxed->False]

m=RotationMatrix[{v[[10]],{0,0,1}}]

v2=FullSimplify[m.#&/@v]

Graphics3D[{MapIndexed[Text[#2[[1]],#,Background->White]&,v2],GraphicsComplex[v2,Polygon[PolyhedronData["TrapezoRhombicDodecahedron","FaceIndices"]]]}]

?ArcTan

rot=ArcTan@@Most[v2[[5]]]

<<MathWorld`PolyhedronOperations`

mrot=RotationMatrixZ[rot]

v3=FullSimplify[mrot.#&/@v2]

Graphics3D[{Opacity[.5],
MapIndexed[Text[#2[[1]],#,Background->White]&,v3],gcc3=GraphicsComplex[v3,Polygon[PolyhedronData["TrapezoRhombicDodecahedron","FaceIndices"]]]
}]

HCP

Constructed from hexagonal close packing

If spheres packed in a cubic lattice, face-centered cubic lattice, and hexagonal lattice are allowed to expand uniformly until running into each other, they form cubes, hexagonal prisms, and rhombic dodecahedra, respectively.  In particular, if the spheres of cubic close packing are expanded until they fill up the gaps, they form a solid rhombic dodecahedron, and if the spheres of hexagonal close packing are expanded, they form a second irregular dodecahedron consisting of six rhombi and six trapezoids (right figure above; Steinhaus 1999, p. 206) known as the trapezo-rhombic dodecahedron.  The latter can be obtained from the former by slicing in half and rotating the two halves 60° with respect to each other.

(Not quite) dual of triangular orthobicupola

https://en.wikipedia.org/wiki/Trapezo-rhombic_dodecahedron

It is the polyhedral dual of the triangular orthobicupola.

Graphics3D[{
Opacity[.5],
PolyhedronData["TriangularOrthobicupola","GraphicsComplex"],
DualPolyhedron[PolyhedronData["TriangularOrthobicupola","Polyhedron"]]
}]

poly0=DualPolyhedron[PolyhedronData["TriangularOrthobicupola","Polyhedron"]]

Orientation

Graphics3D[{FaceForm[Green,Red],poly0}]

EdgeLengths

e0=Sqrt[FullSimplify[SquaredEuclideanDistance@@@(PolyhedronEdges[poly]/.i_Integer:>PolyhedronCoordinates[poly0][[i]])]]//Union

N[%]

Graphics3D[poly=Polyhedron[#1/e0[[1]],#2]&@@poly0]

Constructed from dual:

e=Sqrt[FullSimplify[SquaredEuclideanDistance@@@(PolyhedronEdges[poly]/.i_Integer:>PolyhedronCoordinates[poly][[i]])]]//Union

N[e]

Squashed dodecahedron:

N[{1,3/2,2}]

Net from wikipedia:

{2.4,3.5,4.8}/2.4

Properties: {"TrapezoRhombicDodecahedron", "C3"}

Initialization

<<MathWorld`Polyhedra`

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

vol=PolyhedronData[pname,"Volume"]

Centroid

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

PolyhedronData[pname,"Centroid"]

RegionCentroid[p]//FullSimplify

PolyhedronCentroid[p]//FullSimplify

Circumsphere

PolyhedronData[pname,"Circumsphere"]

Circumsphere[p]

(sphere=MyCircumsphere[p])//Timing

Convex

PolyhedronData[pname,"Convex"]

ConvexPolyhedronQ[p]

DehnInvariant

PolyhedronData[pname,"DehnInvariant"]

(dehn=DehnInvariant[p,Debug->True])//Timing

DihedralAngles

PolyhedronData[pname,"DihedralAngles"]//Tally

DihedralAngles[p]//FullSimplify//Tally

EdgeLengths

PolyhedronData[pname,"EdgeLengths"]

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

Faces

GeneralizedDiameter

PolyhedronData[pname,"GeneralizedDiameter"]

GeneralizedDiameter[p]//FullSimplify//Quiet

InertiaTensor

PolyhedronData[pname,"InertiaTensor"]

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

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[%]

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

Integrate over polyhedron

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

CAD over  1/XX

reg=FullSimplify[PolyhedronData[pname,"RegionFunction"][x,y,z]&&0<y<Tan[Pi/3]x&&z>0]

Show[Graphics3D[{Opacity[.2],p}],RegionPlot3D[ImplicitRegion[reg,{x,y,z}],PlotPoints->50]]

((Volume[ImplicitRegion[reg,{x,y,z}]])/vol)//FullSimplify

cad=CylindricalDecomposition[reg,{x,y,z},"Function"]

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

(rxy=12((Integrate[(Sqrt[(x^2)+(y^2)])Boole[cad],{x,y,z}∈FullRegion[3]])/vol))//Timing

N[rxy]

FullSimplify[rxy]

Divergence theorem [bad imaginary part]

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

N[rxy]

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

Divergence theorem "IntegrationIntervals" -> {0, 1/3, 1/2, 2/3, 1}

(rxy=MeanCylindricalRadius[p,"IntegrationIntervals"->{0,1/3,1/2,2/3,1},"Simplification"->FullSimplify,Debug->True])//Timing

N[rxy]

ResourceFunction["LogSimplify"][rxy]

FullSimplify[%]

N[((5 (4+3Log[3]))/(32 (Sqrt[2])))]

MeanSquareCylindricalRadius

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

Rule[#,((Integrate[(x^2)+(y^2),{x,y,z}∈PolyhedronData[{pname,#},"Polyhedron"]])/vol)]&/@PolyhedronData[pname,"Orientations"]

MeanSphericalRadius

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

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

N[i]

MeanSquareSphericalRadius

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

Rule[#,((Integrate[(x^2)+(y^2)+(z^2),{x,y,z}∈PolyhedronData[{pname,#},"Polyhedron"]])/vol)]&/@PolyhedronData[pname,"Orientations"]

N[i]

Midsphere

sphere=PolyhedronData[pname,"Midsphere"]

(sphere=Midsphere[p])//Timing

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

SurfaceArea

PolyhedronData[pname,"SurfaceArea"]

RootReduce[%]

SurfaceArea[p]//RootReduce

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

Vertex subset hulls

Platonic

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

g=Function[poly,Graphics3D[{
{Gray,Opacity[.2],EdgeForm[Thick],PolyhedronData[p,"Faces"]},ConvexHull3D[N[PolyhedronData[p,"VertexCoordinates"]][[#]]]&/@poly},Boxed->False]]/@h[[All,2]]

GraphicsGrid[{g},Spacings->-100]

Volume

PolyhedronData[pname,"Volume"]

Volume[p]//FullSimplify

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

Properties: {"TrapezoRhombicDodecahedron","HCP"}

Initialization

<<MathWorld`Polyhedra`

p=PolyhedronData[pname={"TrapezoRhombicDodecahedron","HCP"},"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]//FullSimplify

PolyhedronCentroid[p]//FullSimplify//Quiet

Circumsphere

sphere=PolyhedronData[pname,"Circumsphere"]

sphere=Circumsphere[p]

(sphere=MyCircumsphere[p])//Timing

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

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"]

(dehn=DehnInvariant[p,Debug->True])//Timing

ToConwayBasis[dehn]

DihedralAngles

PolyhedronData[pname,"DihedralAngles"]//Tally

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

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[%]

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

Integrate over polyhedron

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

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

N[rxy]

ByteCount[rxylog=LogSimplify[rxy,"ZeroTestNumericalPrecision"->10000,WorkingPrecision->10000,Debug->False]]//Timing

rxylog

ByteCount[rxycollect=Collect[TrigToExp[rxylog],Log[_]]//RootReduce]//Timing

rxylog//FullSimplify//Expand

N[%]

rxycollect//FullSimplify

pname

Quit

Divergence theorem "Simplification"->FullSimplify [wrong imaginary part]

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

N[rxy]

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

Divergence theorem "Simplification"->FullSimplify, "IntegrationIntervals"->{0,1/3,1/2,2/3,1} [small wrong imaginary part]

(rxy=MeanCylindricalRadius[p,"IntegrationIntervals"->{0,1/3,1/2,2/3,1},"Simplification"->FullSimplify,Debug->True])//Timing

N[rxy]

Attempt to fix up bad term

rxy=(1/(6 (Sqrt[3])))((265/1152)+(4517359/(2628288 (Sqrt[2])))+((517 (Sqrt[5]))/5616)+((Sqrt[(13/2)])/288)+((7 (Sqrt[17]))/864)-((3 (Sqrt[(37/2)]))/40)+(199/(48 (Sqrt[26])))+((Sqrt[41])/20)+(7/864) (Sqrt[97-8 (Sqrt[85])])+((I (-4394+621 (Sqrt[26])) π)/4100129280)-((173195 ArcCoth[4])/186624)+((6125 ArcCoth[4])/(292032 (Sqrt[26])))-((7 ArcCoth[5])/729)-((3 ArcCoth[5])/(2 (Sqrt[2])))+((112 ArcCoth[23])/27)-((ArcCoth[(Sqrt[2])])/3)+((15 ArcCoth[(Sqrt[2])])/(8 (Sqrt[2])))-((11421 ArcCoth[5 (Sqrt[2])])/(700000 (Sqrt[2])))-((5 ArcCoth[(Sqrt[5])])/63)-((11421 ArcCoth[((5 (Sqrt[10]))/13)])/(700000 (Sqrt[2])))-((21 ArcCoth[(Sqrt[10])])/(16 (Sqrt[2])))+((3 ArcCoth[(Sqrt[82])])/(2 (Sqrt[2])))+((3 ArcCoth[((10-2 (Sqrt[10]))/(5 (Sqrt[2])-4 (Sqrt[5])))])/3200)+((3788537 ArcCsch[2])/2555280)+((72168517 ArcCsch[2])/(6300000 (Sqrt[2])))-((112333 ArcCsch[3])/(26250 (Sqrt[2])))+((53 ArcCsch[4])/54)+((23 ArcCsch[5])/(5200 (Sqrt[26])))-((155 ArcCsch[6])/(12 (Sqrt[2])))+((81 ArcCsch[8])/(52 (Sqrt[26])))-((7619 ArcCsch[9])/(560 (Sqrt[2])))-(17/15) (Sqrt[(2/13)]) ArcCsch[18]+((21141 ArcCsch[18])/(17500000 (Sqrt[2])))+((8833637 ArcCsch[18])/(1687296 (Sqrt[26])))+(17/15) (Sqrt[(2/13)]) ArcSinh[(1/18)]-((21141 ArcSinh[(1/18)])/(17500000 (Sqrt[2])))-((8833637 ArcSinh[(1/18)])/(1687296 (Sqrt[26])))+((909193 ArcSinh[(1/9)])/(70000 (Sqrt[2])))-((81 ArcSinh[(1/8)])/(52 (Sqrt[26])))+((319 ArcSinh[(1/6)])/(24 (Sqrt[2])))-((672031 ArcSinh[(1/5)])/(8436480 (Sqrt[26])))-(53/54) ArcSinh[(1/4)]+((49499279 ArcSinh[(1/3)])/(10920000 (Sqrt[2])))+((1896237 ArcSinh[(5/12)])/(9139520 (Sqrt[26])))-((72727 ArcSinh[(1/2)])/36504)-((14562299 ArcSinh[(1/2)])/(1260000 (Sqrt[2])))-((1701 ArcSinh[(5/9)])/(10000 (Sqrt[2])))+((640047 ArcSinh[(2/3)])/(3640000 (Sqrt[2])))+((265492509 ArcSinh[(2/3)])/(182790400 (Sqrt[26])))-((22681017 ArcSinh[(7/9)])/(14060800 (Sqrt[26])))+((9071 ArcSinh[1])/17280)+((8263 ArcSinh[1])/(2016 (Sqrt[2])))+((488673 ArcSinh[(4/3)])/(45500000 (Sqrt[2])))+((488673 ArcSinh[(13/9)])/(45500000 (Sqrt[2])))-((648293 ArcSinh[2])/1572480)+((9 ArcSinh[2])/(128 (Sqrt[2])))+((7 ArcSinh[3])/48)+((17 ArcSinh[3])/(54 (Sqrt[2])))-((ArcSinh[(10/3)])/(16 (Sqrt[2])))-((75159 ArcSinh[(11/3)])/(36558080 (Sqrt[26])))-((8 ArcSinh[4])/45)+((13 ArcSinh[5])/36)-(288/65) (Sqrt[(2/13)]) ArcTanh[(5/13)]-(400/243) ArcTanh[(7/13)]+(133/108) ArcTanh[(5/9)]+((737 ArcTanh[(4/7)])/1458)-(5760/169) (Sqrt[(2/13)]) ArcTanh[(4/7)]+((27 ArcTanh[(4/5)])/(32 (Sqrt[2])))-((11421 ArcTanh[(1/(5 (Sqrt[2])))])/(1400000 (Sqrt[2])))+((12401 ArcTanh[(1/(Sqrt[2]))])/3600)-((15 ArcTanh[(1/(Sqrt[2]))])/(8 (Sqrt[2])))+(389/768) ArcTanh[(1/(Sqrt[5]))]+((40979 ArcTanh[(2/(Sqrt[5]))])/57600)+((3 ArcTanh[(1/(Sqrt[10]))])/(4 (Sqrt[2])))-((11421 ArcTanh[(13/(5 (Sqrt[10])))])/(1400000 (Sqrt[2])))-(288/65) (Sqrt[(2/13)]) ArcTanh[(2/(Sqrt[13]))]-((9 ArcTanh[(2/(Sqrt[13]))])/(32 (Sqrt[2])))+((729 ArcTanh[(2/(Sqrt[13]))])/(4160 (Sqrt[26])))+(3/320) ArcTanh[(5/(Sqrt[26]))]-((27 ArcTanh[(10/(Sqrt[109]))])/(64 (Sqrt[2])))-((729 ArcTanh[(11/(Sqrt[130]))])/(4160 (Sqrt[26])))+((1333 ArcTanh[(13/(Sqrt[194]))])/1440)+(7/27) ArcTanh[2-(Sqrt[5])]+((9 ArcTanh[(3/2)-((Sqrt[13])/2)])/(2 (Sqrt[26])))+(103/351) ArcTanh[4-(Sqrt[17])]+((3 ArcTanh[5-(Sqrt[34])])/(16 (Sqrt[2])))-((ArcTanh[(NumericalApproximation[Root, (RowDefault["0.561", FEExpressionsNumericalApproximationElider]), Root[169-689 ((#1)^2)+484 ((#1)^4)&,3], ])])/(3042 (Sqrt[26])))-(19/27) Log[(22/21)]-Log[(614656/531441)]-((3 Log[(32/27)])/(16 (Sqrt[2])))+Log[(4/3)]-(9/32) Log[(7/5)]+((287 Log[(3/2)])/1152)+((3 Log[(3/2)])/(160 (Sqrt[2])))+((51 Log[(3/2)])/(52 (Sqrt[26])))-((13638679 Log[2])/5391360)+((1798 (Sqrt[(2/13)]) Log[2])/2197)-((1382697 Log[2])/(1400000 (Sqrt[2])))-((6010807 Log[2])/(36558080 (Sqrt[26])))+((Log[(8/3)])/(32 (Sqrt[2])))-((24689137 Log[3])/3919104)-((11957301 Log[3])/(22750000 (Sqrt[2])))-((152893951361 Log[3])/(4935340800 (Sqrt[26])))+((25 Log[(10/3)])/4374)+((Log[(27/8)])/(Sqrt[26]))+(73/432) Log[(7/2)]-(58/81) Log[(11/3)]+2 Log[4]+(23/27) Log[(13/3)]-(4/27) Log[(14/3)]-((1724741 Log[5])/373248)-((56557621 Log[5])/(379641600 (Sqrt[26])))-Log[(177147/35152)]+((8 Log[6])/21)+((Log[6])/(8 (Sqrt[2])))-((11603 Log[7])/13824)-((442989 Log[7])/(1400000 (Sqrt[2])))+((135 Log[7])/(1352 (Sqrt[26])))-Log[8]+((29187 Log[9])/(1400000 (Sqrt[2])))+((2983101 Log[9])/(5624320 (Sqrt[26])))+((38671 Log[10])/21870)+((703 Log[11])/2916)+(7/169) (Sqrt[(2/13)]) Log[11]+((1890 (Sqrt[(2/13)]) Log[12])/2197)-((23 Log[13])/27)-(259/845) (Sqrt[(2/13)]) Log[13]-((5589 Log[13])/(108160 (Sqrt[26])))-((3 Log[(27/2)])/(8 (Sqrt[2])))+((4 Log[14])/27)-Log[16]-((1701 Log[18])/(50000 (Sqrt[2])))+((52041 Log[18])/(11881376 (Sqrt[26])))-(298/845) (Sqrt[(2/13)]) Log[26]-((95827 Log[27])/1119744)+((3 Log[(243/8)])/(32 (Sqrt[2])))+Log[32]+(Sqrt[(2/13)]) Log[32]-((26037 Log[(81/2)])/(87500 (Sqrt[2])))-((389 Log[97])/1728)-((Log[288])/(Sqrt[26]))-((Log[384])/(32 (Sqrt[2])))-Log[625]+(Sqrt[(2/13)]) Log[676]-Log[(6250/9)]+Log[1331]+((235 Log[1728])/6912)-Log[2197]+(Sqrt[(2/13)]) Log[4096]-((45 Log[(27783/4)])/(1352 (Sqrt[26])))-(Sqrt[(2/13)]) Log[12636]+((Log[13824])/(Sqrt[26]))-(Sqrt[(2/13)]) Log[28561]+Log[38416]+(Sqrt[(2/13)]) Log[(3688372195443/33554432)]-((1701 Log[314928])/(100000 (Sqrt[2])))+(Sqrt[(2/13)]) Log[505447028499293771]+(4/405) Log[1970-1393 (Sqrt[2])]+((Log[17-12 (Sqrt[2])])/11664)+((217 Log[10-7 (Sqrt[2])])/1458)-((251 Log[3-2 (Sqrt[2])])/2160)-((905 Log[2-(Sqrt[2])])/2916)+((35721 Log[10-(Sqrt[2])])/(200000 (Sqrt[2])))-(23/288) Log[-1+(Sqrt[2])]-((79 Log[-1+(Sqrt[2])])/(480 (Sqrt[2])))-(3/128) Log[1+(Sqrt[2])]-((8381 Log[1+(Sqrt[2])])/(2520 (Sqrt[2])))-((419273 Log[2+(Sqrt[2])])/116640)-(28/135) Log[3+2 (Sqrt[2])]-((565 Log[-1+5 (Sqrt[2])])/3072)-((11421 Log[-1+5 (Sqrt[2])])/(350000 (Sqrt[2])))-((565 Log[1+5 (Sqrt[2])])/3072)+((119313 Log[1+5 (Sqrt[2])])/(700000 (Sqrt[2])))+(16/135) Log[10+7 (Sqrt[2])]-(1/80) Log[9-4 (Sqrt[5])]+((35 Log[-2+(Sqrt[5])])/2304)-((648683 Log[-1+(Sqrt[5])])/449280)-((2423 Log[1+(Sqrt[5])])/3360)-((5103 Log[1+(Sqrt[5])])/(100000 (Sqrt[2])))-((911 Log[2+(Sqrt[5])])/7560)-((9 Log[2+(Sqrt[5])])/(128 (Sqrt[2])))-2 Log[(Sqrt[2])+2 (Sqrt[5])]+((59691 Log[(Sqrt[2])+2 (Sqrt[5])])/(100000 (Sqrt[2])))-((907 Log[3 (Sqrt[2])+2 (Sqrt[5])])/2016)-((3 Log[3 (Sqrt[2])+2 (Sqrt[5])])/(4 (Sqrt[2])))-(32/135) Log[7+3 (Sqrt[5])]-((5103 Log[13 (Sqrt[2])+10 (Sqrt[5])])/(200000 (Sqrt[2])))-((3 Log[19-6 (Sqrt[10])])/(32 (Sqrt[2])))+(9/32) Log[-3+(Sqrt[10])]-((5 Log[-3+(Sqrt[10])])/(36 (Sqrt[2])))+((15 Log[-1+(Sqrt[10])])/(16 (Sqrt[2])))-((30057093 Log[1+(Sqrt[10])])/(18200000 (Sqrt[2])))-(13/288) Log[3+(Sqrt[10])]+((5 Log[3+(Sqrt[10])])/(108 (Sqrt[2])))+((15309 Log[10+(Sqrt[2])+2 (Sqrt[5])+(Sqrt[10])])/(100000 (Sqrt[2])))+Log[11+2 (Sqrt[10])]-((5103 Log[11+2 (Sqrt[10])])/(100000 (Sqrt[2])))+((1785807 Log[13+5 (Sqrt[10])])/(45500000 (Sqrt[2])))+((545 Log[19+6 (Sqrt[10])])/1344)+((Log[19+6 (Sqrt[10])])/(32 (Sqrt[2])))-((3 Log[-31+13 (Sqrt[10])])/(16 (Sqrt[2])))+(13/216) Log[-117+37 (Sqrt[10])]-((27 Log[17-4 (Sqrt[13])])/(64 (Sqrt[2])))+((405 Log[17-4 (Sqrt[13])])/(10816 (Sqrt[26])))-(384/65) (Sqrt[(2/13)]) Log[13-2 (Sqrt[13])]+((355 Log[-3+(Sqrt[13])])/(562432 (Sqrt[26])))-(90/169) (Sqrt[(2/13)]) Log[-2+(Sqrt[13])]+((9 Log[-2+(Sqrt[13])])/(16 (Sqrt[2])))-((40958651 Log[-2+(Sqrt[13])])/(24676704 (Sqrt[26])))-(138/845) (Sqrt[(2/13)]) Log[2+(Sqrt[13])]+((9597453 Log[2+(Sqrt[13])])/(3640000 (Sqrt[2])))+((47996579 Log[2+(Sqrt[13])])/(75928320 (Sqrt[26])))+((160261 Log[3+(Sqrt[13])])/(8225568 (Sqrt[26])))+((5589 Log[13+2 (Sqrt[13])])/(54080 (Sqrt[26])))-(1/720) Log[-5 (Sqrt[2])+2 (Sqrt[13])]+(625/144) Log[-(Sqrt[2])+2 (Sqrt[13])]+(125/144) Log[(Sqrt[2])+2 (Sqrt[13])]-((237551 Log[5 (Sqrt[2])+2 (Sqrt[13])])/13063680)+((630 (Sqrt[(2/13)]) Log[1+5 (Sqrt[13])])/2197)-(384/65) (Sqrt[(2/13)]) Log[13 (Sqrt[10])+7 (Sqrt[13])]-(4/45) Log[33-8 (Sqrt[17])]-(7/54) Log[-1+(Sqrt[17])]+(2/117) Log[1+(Sqrt[17])]+((3 Log[5 (Sqrt[2])+2 (Sqrt[17])])/(4 (Sqrt[2])))+((101 Log[51-10 (Sqrt[26])])/1008)-(384/65) (Sqrt[(2/13)]) Log[26 (Sqrt[5])-3 (Sqrt[26])]+((33353 Log[-1+(Sqrt[26])])/(1054560 (Sqrt[26])))+((4507699 Log[1+(Sqrt[26])])/(42182400 (Sqrt[26])))+(125/72) Log[27+2 (Sqrt[26])]+(1440/169) (Sqrt[(2/13)]) Log[26 (Sqrt[5])+7 (Sqrt[26])]-((405053 Log[51+10 (Sqrt[26])])/5225472)+(1/221184)(70848+20736 (Sqrt[2])+7776 I π-10368 ArcCoth[19]-2048 (Sqrt[2]) ArcCoth[(3/(Sqrt[5]))]-10368 ArcCoth[(3/(Sqrt[10]))]-62208 ArcCsch[3]-5184 (Sqrt[2]) ArcCsch[3]+62208 ArcSinh[(1/3)]+5184 (Sqrt[2]) ArcSinh[(1/3)]+8968 ArcSinh[1]+41472 (Sqrt[2]) ArcSinh[1]-2048 ArcSinh[2]+1024 (Sqrt[2]) ArcSinh[2]-31104 ArcSinh[3]-10240 ArcSinh[5]-15552 ArcTanh[(Sqrt[2])]+41472 ArcTanh[((Sqrt[10])/3)]-2048 (Sqrt[2]) ArcTanh[13-6 (Sqrt[5])]-2048 Log[(3/2)]+2048 (Sqrt[2]) Log[(3/2)]-18337 Log[2]+3072 (Sqrt[2]) Log[2]+5184 Log[(9/4)]+8192 Log[(5/2)]+13440 Log[3]-3200 (Sqrt[2]) Log[3]+1296 Log[4]+5184 Log[(40/9)]-3456 Log[8]-5184 Log[9]-32 Log[81]+32 (Sqrt[2]) Log[81]-3456 Log[243]+1728 Log[512]-2592 Log[2-(Sqrt[2])]+6584 Log[1+(Sqrt[2])]-23328 Log[2+(Sqrt[2])]+6912 Log[840+594 (Sqrt[2])]+8192 Log[2-(2/(Sqrt[5]))]-8192 Log[5-(Sqrt[5])]-16384 Log[(Sqrt[2])+((2 (Sqrt[5]))/3)]+8192 Log[1+(Sqrt[5])]+2048 Log[2+(Sqrt[5])]-14848 Log[3 (Sqrt[2])+2 (Sqrt[5])]+3456 Log[(19/2)-3 (Sqrt[10])]-8192 Log[(Sqrt[2])+(Sqrt[10])]-6561 Log[11+2 (Sqrt[10])]+6561 Log[22+4 (Sqrt[10])]+3456 Log[(2/9) (19+6 (Sqrt[10]))]+6912 Log[19+6 (Sqrt[10])]+8704 Log[38+12 (Sqrt[10])]-10368 Log[5 (Sqrt[2])+2 (Sqrt[13])]+10240 Log[5+(Sqrt[26])]+5184 Log[51+10 (Sqrt[26])])+(1/110592)(-42336+31104 (Sqrt[2])+124416 ArcCoth[(Sqrt[2])]-290304 ArcCoth[(Sqrt[5])]+165888 ArcCoth[(Sqrt[10])]-40448 ArcCsch[2]+321408 (Sqrt[2]) ArcCsch[2]-285552 (Sqrt[2]) ArcCsch[3]+250992 (Sqrt[2]) ArcSinh[(1/3)]-321408 (Sqrt[2]) ArcSinh[(1/2)]+38016 (Sqrt[2]) ArcSinh[(2/3)]+36288 ArcSinh[1]+82944 (Sqrt[2]) ArcSinh[1]+290304 ArcTanh[(1/(Sqrt[5]))]-165888 ArcTanh[(1/(Sqrt[10]))]-44928 (Sqrt[2]) ArcTanh[3-(Sqrt[10])]+126239 Log[2]-16640 Log[3]+25920 (Sqrt[2]) Log[3]-145152 Log[4]+41472 Log[5]+27648 Log[(2048/81)]+2656 Log[27]+864 Log[64]+8 Log[81]-24192 Log[243]+3456 Log[512]-10368 Log[625]-3456 Log[(33554432/6561)]+27648 Log[124416]-171072 Log[2+(Sqrt[2])]+4160 Log[1+(Sqrt[5])]-207360 Log[(Sqrt[2])+2 (Sqrt[5])]+20736 Log[1+(Sqrt[10])]+12096 (Sqrt[2]) Log[1+(Sqrt[10])]+41472 Log[8 (1+(Sqrt[10]))]+36288 Log[(Sqrt[2])+(Sqrt[10])]+10368 Log[22+4 (Sqrt[10])]-81 Log[19+6 (Sqrt[10])]+81 Log[38+12 (Sqrt[10])]+13824 Log[((6561 (186961+59449 (Sqrt[10])))/33554432)]-38016 (Sqrt[2]) Log[2+(Sqrt[13])]+212544 Log[(Sqrt[2])+2 (Sqrt[13])]+27648 Log[1+(Sqrt[26])]-18144 Log[27+2 (Sqrt[26])]-93312 Log[54+4 (Sqrt[26])]+16 Log[51+10 (Sqrt[26])]-16 Log[102+20 (Sqrt[26])]-1728 Log[36279387+6852362 (Sqrt[26])])+(1/64) Log[-2+(Sqrt[29])]+(1/64) Log[2+(Sqrt[29])]+(8/405) Log[727+135 (Sqrt[29])]-(4/405) Log[528527+98145 (Sqrt[29])]+((9 Log[-5+(Sqrt[34])])/(32 (Sqrt[2])))-((3 Log[5+(Sqrt[34])])/(8 (Sqrt[2])))+((3 Log[-1+(Sqrt[37])])/(10 (Sqrt[2])))-(7/320) Log[5 (Sqrt[2])+2 (Sqrt[37])]-((8875 Log[7 (Sqrt[2])+2 (Sqrt[37])])/20412)-((Log[14+5 (Sqrt[37])])/(40 (Sqrt[2])))+((5103 Log[23 (Sqrt[2])+10 (Sqrt[41])])/(50000 (Sqrt[2])))-(13/108) Log[-7+(Sqrt[53])]+(13/36) Log[7+(Sqrt[53])]-(13/216) Log[6754799+927843 (Sqrt[53])]-2 Log[-(Sqrt[2])+2 (Sqrt[61])]+(2/13) (Sqrt[(2/13)]) Log[-1+(Sqrt[65])]-((630 (Sqrt[(2/13)]) Log[1+65 (Sqrt[5])+5 (Sqrt[13])+(Sqrt[65])])/2197)-(7/144) Log[123-14 (Sqrt[74])]+(7/432) Log[-7+(Sqrt[74])]+(7/640) Log[99+10 (Sqrt[74])]+((8875 Log[123+14 (Sqrt[74])])/40824)+(7/432) Log[-462287+53741 (Sqrt[74])]+((3 Log[-1+(Sqrt[82])])/(16 (Sqrt[2])))+((3807 Log[1+(Sqrt[82])])/(70000 (Sqrt[2])))-(2/5) (Sqrt[2]) Log[1+(Sqrt[82])]+((Log[83+2 (Sqrt[82])])/(40 (Sqrt[2])))-((5103 Log[23+5 (Sqrt[82])])/(50000 (Sqrt[2])))+(1/54) Log[-11 (Sqrt[2])+2 (Sqrt[85])]+(275/432) Log[7 (Sqrt[2])+2 (Sqrt[85])]+(1/144) Log[99 (Sqrt[2])-44 (Sqrt[10])-40 (Sqrt[17])+18 (Sqrt[85])]-((389 Log[-13 (Sqrt[2])+2 (Sqrt[97])])/1080)-(625/486) Log[13 (Sqrt[2])+2 (Sqrt[97])]+((1701 Log[5+(Sqrt[106])])/(10000 (Sqrt[2])))+(1440/169) (Sqrt[(2/13)]) Log[17 (Sqrt[13])+13 (Sqrt[106])]-((Log[209-20 (Sqrt[109])])/(8 (Sqrt[2])))-((27 Log[109-10 (Sqrt[109])])/(128 (Sqrt[2])))+((3 Log[-10+(Sqrt[109])])/(16 (Sqrt[2])))+((27 Log[109+10 (Sqrt[109])])/(128 (Sqrt[2])))+Log[123-2 (Sqrt[122])]+(192/65) (Sqrt[(2/13)]) Log[139-6 (Sqrt[130])]-((302619 Log[-11+(Sqrt[130])])/(1406080 (Sqrt[26])))-((21313 Log[-7+(Sqrt[130])])/(562432 (Sqrt[26])))-((11737529 Log[7+(Sqrt[130])])/(36558080 (Sqrt[26])))+((759141 Log[11+(Sqrt[130])])/(36558080 (Sqrt[26])))+((207 Log[14+13 (Sqrt[10])+7 (Sqrt[13])+2 (Sqrt[130])])/(20800 (Sqrt[26])))-((32775 Log[-21+13 (Sqrt[10])-7 (Sqrt[13])+3 (Sqrt[130])])/(11881376 (Sqrt[26])))-(1104/845) (Sqrt[(2/13)]) Log[179+14 (Sqrt[130])]-((5589 Log[251+22 (Sqrt[130])])/(108160 (Sqrt[26])))-(1/108) Log[291-22 (Sqrt[170])]-(275/864) Log[219+14 (Sqrt[170])]-(1/288) Log[291+22 (Sqrt[170])]+(625/432) Log[363-26 (Sqrt[194])]+(389/864) Log[194+13 (Sqrt[194])]+((2375 Log[363+26 (Sqrt[194])])/1944)+(1/373248)(217404+52488 (Sqrt[2])+127008 (Sqrt[5])-58968 (Sqrt[26])-720896 ArcCoth[7]+567 ArcCoth[(Sqrt[2])]+62208 ArcCoth[(Sqrt[5])]-298080 ArcCsch[2]+393984 (Sqrt[2]) ArcCsch[2]-489321 (Sqrt[2]) ArcCsch[3]-23328 (Sqrt[2]) ArcCsch[9]+323109 (Sqrt[2]) ArcSinh[(1/3)]-209952 (Sqrt[2]) ArcSinh[(4/9)]-393984 (Sqrt[2]) ArcSinh[(1/2)]+236196 (Sqrt[2]) ArcSinh[(2/3)]+27216 ArcSinh[1]+268272 (Sqrt[2]) ArcSinh[1]-40986 ArcSinh[2]-60831 ArcTanh[(1/(Sqrt[2]))]-62208 ArcTanh[(1/(Sqrt[5]))]+31104 ArcTanh[(2/(Sqrt[5]))]-1944 ArcTanh[(5/(Sqrt[26]))]+31104 Log[(15/14)]+248832 Log[(7/6)]-248832 Log[(3/2)]-13824 Log[(6561/4096)]+1272062 Log[2]+373248 Log[(9/4)]+439712 Log[(7/3)]-165888 Log[(12/5)]-365634 Log[(5/2)]+11664 (Sqrt[2]) Log[(8/3)]+55296 Log[(25/9)]-776744 Log[3]+154548 (Sqrt[2]) Log[3]-55296 Log[(256/81)]+32832 Log[(6561/2048)]-62208 Log[4]-82944 Log[(1024/225)]-346302 Log[5]-781856 Log[7]-73360 Log[8]+139968 (Sqrt[2]) Log[9]+5184 Log[16]+331776 Log[18]+62208 Log[25]-11664 Log[27]+11664 (Sqrt[2]) Log[(243/8)]+27648 Log[(1024/27)]+62208 Log[49]-6048 Log[(117649/2187)]+41472 Log[64]+65664 Log[(15625/243)]+82944 Log[72]-11664 (Sqrt[2]) Log[81]-1296 Log[243]-11664 (Sqrt[2]) Log[243]-6048 Log[(43046721/117649)]-27216 Log[512]+5184 Log[625]-20736 Log[729]-46656 (Sqrt[2]) Log[729]+13824 Log[(10125/8)]+5400 Log[1728]+55296 Log[4096]+13824 Log[(129140163/4096)]+7560 Log[59049]-6912 Log[43046721]+373248 Log[((2 (2^(7/9)))/(3 (3^(7/27))))]-5184 Log[-1+(Sqrt[2])]+93312 (Sqrt[2]) Log[-1+(Sqrt[2])]-72576 Log[2+(Sqrt[2])]+82944 Log[3 (2+(Sqrt[2]))]+18144 Log[9-4 (Sqrt[5])]-9504 Log[6-2 (Sqrt[5])]+567 Log[-2+(Sqrt[5])]+19008 Log[-1+(Sqrt[5])]+202176 Log[1+(Sqrt[5])]+81 Log[2+(Sqrt[5])]-292896 Log[(Sqrt[2])+2 (Sqrt[5])]-174960 (Sqrt[2]) Log[-1+(Sqrt[10])]+373248 Log[1+(Sqrt[10])]+192456 (Sqrt[2]) Log[1+(Sqrt[10])]-131220 (Sqrt[2]) Log[3 (1+(Sqrt[10]))]+95904 Log[(Sqrt[2])+(Sqrt[10])]-40176 Log[11+2 (Sqrt[10])]+34992 (Sqrt[2]) Log[-31+13 (Sqrt[10])]-61236 (Sqrt[2]) Log[2+(Sqrt[13])]-208288 Log[(Sqrt[2])+2 (Sqrt[13])]-18144 Log[5 (Sqrt[2])+2 (Sqrt[13])]-9504 Log[((36279387-6852362 (Sqrt[26]))/59049)]-76032 Log[-1+(Sqrt[26])]+167088 Log[27+2 (Sqrt[26])]-148480 Log[54+4 (Sqrt[26])]+10368 Log[102+20 (Sqrt[26])]-34992 (Sqrt[2]) Log[14+5 (Sqrt[37])]+34992 (Sqrt[2]) Log[(1/9) (-1+(Sqrt[82]))]+69984 (Sqrt[2]) Log[(1/3) (-1+(Sqrt[82]))]-34992 (Sqrt[2]) Log[-1+(Sqrt[82])]+297432 (Sqrt[2]) Log[1+(Sqrt[82])]-262440 (Sqrt[2]) Log[2 (1+(Sqrt[82]))]-11664 (Sqrt[2]) Log[3 (1+(Sqrt[82]))]+34992 (Sqrt[2]) Log[83+2 (Sqrt[82])]-34992 (Sqrt[2]) Log[(1/9) (89-4 (Sqrt[85]))]-262440 (Sqrt[2]) Log[2+(Sqrt[85])]+262440 (Sqrt[2]) Log[2 (2+(Sqrt[85]))]-75952 Log[11 (Sqrt[2])+2 (Sqrt[85])]+69984 (Sqrt[2]) Log[(1/27) (89+4 (Sqrt[85]))]-34992 (Sqrt[2]) Log[89+4 (Sqrt[85])]+34992 (Sqrt[2]) Log[((729 (14+5 (Sqrt[37])))/(9281+712 (Sqrt[85])))]-69984 (Sqrt[2]) Log[-4+(Sqrt[97])]+52304 Log[13 (Sqrt[2])+2 (Sqrt[97])]-18144 Log[(9-4 (Sqrt[5])) (13 (Sqrt[2])+2 (Sqrt[97]))]+34992 (Sqrt[2]) Log[18977+1808 (Sqrt[97])]+37976 Log[582+44 (Sqrt[170])]-17080 Log[726+52 (Sqrt[194])])-((57 Log[-60+20 (Sqrt[13])+13 (Sqrt[37])-3 (Sqrt[481])])/(35152 (Sqrt[26])))-((57 Log[-20+(Sqrt[481])])/(35152 (Sqrt[26])))+((1665 Log[20+(Sqrt[481])])/(1827904 (Sqrt[26])))+((Log[35+(Sqrt[1261])])/(6084 (Sqrt[26])))-(720/169) (Sqrt[(2/13)]) Log[1667+34 (Sqrt[1378])]-((968 (Sqrt[(2/13)]) Log[-11+11 (Sqrt[65])+13 (Sqrt[545])-(Sqrt[1417])])/2197)-((968 (Sqrt[(2/13)]) Log[-11+(Sqrt[1417])])/2197)-((Log[(NumericalApproximation[Root, (RowDefault["14.5", FEExpressionsNumericalApproximationElider]), Root[6561+3402 #1-1087 ((#1)^2)+42 ((#1)^3)+((#1)^4)&,4], ])])/(832 (Sqrt[26])))-((119 Log[(NumericalApproximation[Root, (RowDefault["0.619", FEExpressionsNumericalApproximationElider]), Root[81+378 #1-1087 ((#1)^2)+378 ((#1)^3)+81 ((#1)^4)&,3], ])])/(562432 (Sqrt[26])))-((422977 Log[(NumericalApproximation[Root, (RowDefault["1.62", FEExpressionsNumericalApproximationElider]), Root[81+378 #1-1087 ((#1)^2)+378 ((#1)^3)+81 ((#1)^4)&,4], ])])/(23762752 (Sqrt[26])))+((1665 Log[(NumericalApproximation[Root, (RowDefault["0.709", FEExpressionsNumericalApproximationElider]), Root[81+1080 #1-2491 ((#1)^2)+1080 ((#1)^3)+81 ((#1)^4)&,3], ])])/(1827904 (Sqrt[26])))+((405 Log[(NumericalApproximation[Root, (RowDefault["101.", FEExpressionsNumericalApproximationElider]), Root[6561-481585588 #1+28158305814 ((#1)^2)-351075893652 ((#1)^3)+3486784401 ((#1)^4)&,4], ])])/(10816 (Sqrt[26])))+((NumericalApproximation[Root, (RowDefault["0.138", FEExpressionsNumericalApproximationElider]), Root[25-1322 ((#1)^2)+((#1)^4)&,3], ])/(40 (Sqrt[2]))));

ints={(InactiveIntegrate[(1/24) ((((3+\[FormalS]-4 \[FormalT]) (Sqrt[13 (\[FormalS]^2)+(((3-4 \[FormalT]))^2)+4 \[FormalS] (-3+4 \[FormalT])]))/(Sqrt[2]))+(1/4) (((-3+5 \[FormalS]+4 \[FormalT]))^2) (Log[12]-2 Log[-3-\[FormalS]+4 \[FormalT]+(Sqrt[26 (\[FormalS]^2)+2 (((3-4 \[FormalT]))^2)+8 \[FormalS] (-3+4 \[FormalT])])])), \[FormalT], 0, (1/3), \[FormalS], 0, 1-\[FormalT]]),(InactiveIntegrate[(1/24) ((((3+\[FormalS]-4 \[FormalT]) (Sqrt[13 (\[FormalS]^2)+(((3-4 \[FormalT]))^2)+4 \[FormalS] (-3+4 \[FormalT])]))/(Sqrt[2]))+(1/4) (((-3+5 \[FormalS]+4 \[FormalT]))^2) (Log[12]-2 Log[-3-\[FormalS]+4 \[FormalT]+(Sqrt[26 (\[FormalS]^2)+2 (((3-4 \[FormalT]))^2)+8 \[FormalS] (-3+4 \[FormalT])])])), \[FormalT], (1/3), (1/2), \[FormalS], 0, 1-\[FormalT]]),(InactiveIntegrate[(1/24) ((((3+\[FormalS]-4 \[FormalT]) (Sqrt[13 (\[FormalS]^2)+(((3-4 \[FormalT]))^2)+4 \[FormalS] (-3+4 \[FormalT])]))/(Sqrt[2]))+(1/4) (((-3+5 \[FormalS]+4 \[FormalT]))^2) (Log[12]-2 Log[-3-\[FormalS]+4 \[FormalT]+(Sqrt[26 (\[FormalS]^2)+2 (((3-4 \[FormalT]))^2)+8 \[FormalS] (-3+4 \[FormalT])])])), \[FormalT], (1/2), (2/3), \[FormalS], 0, 1-\[FormalT]]),(InactiveIntegrate[(1/24) ((((3+\[FormalS]-4 \[FormalT]) (Sqrt[13 (\[FormalS]^2)+(((3-4 \[FormalT]))^2)+4 \[FormalS] (-3+4 \[FormalT])]))/(Sqrt[2]))+(1/4) (((-3+5 \[FormalS]+4 \[FormalT]))^2) (Log[12]-2 Log[-3-\[FormalS]+4 \[FormalT]+(Sqrt[26 (\[FormalS]^2)+2 (((3-4 \[FormalT]))^2)+8 \[FormalS] (-3+4 \[FormalT])])])), \[FormalT], (2/3), 1, \[FormalS], 0, 1-\[FormalT]])};

ints[[-1]]/.Inactive[Integrate][args__]:>NIntegrate[args]

vals={(3017/559872)+(1855/(75816 (Sqrt[2])))+((11 (Sqrt[5]))/936)-((305 (Sqrt[(37/2)]))/303264)+((7 ArcSinh[2])/72)+((2463 ArcSinh[(20/9)])/(913952 (Sqrt[26])))-((2463 ArcSinh[(11/3)])/(913952 (Sqrt[26])))-(65/768) ArcTanh[(2/(Sqrt[5]))]+(65/768) ArcTanh[(7/(Sqrt[74]))]+((731 Log[2])/7680)+((3655 Log[3])/46656)+((11907 Log[3])/(108160 (Sqrt[26])))-((65 Log[5])/768)-((6125 Log[5])/(584064 (Sqrt[26])))+((431 Log[1728])/279936)-((9 Log[6-3 (Sqrt[2])])/1280)+((125 Log[-(5/3) (-2+(Sqrt[2]))])/186624)+((9 Log[-2+(Sqrt[5])])/1280)+((8533 Log[-2+(Sqrt[13])])/(365040 (Sqrt[26])))+((1855 Log[2+(Sqrt[13])])/(24336 (Sqrt[26])))-((125 Log[-7 (Sqrt[2])+2 (Sqrt[37])])/186624)+((325 Log[-7+(Sqrt[74])])/46656)+(7/432) Log[((-462287+53741 (Sqrt[74]))/15552)]-((3645 Log[-7+(Sqrt[130])])/(43264 (Sqrt[26])))-((5589 Log[7+(Sqrt[130])])/(216320 (Sqrt[26])))-((1665 Log[3 (11+(Sqrt[130]))])/(1827904 (Sqrt[26])))+((3125 Log[-16+(Sqrt[481])])/(389376 (Sqrt[26])))+((2875 Log[16+(Sqrt[481])])/(1168128 (Sqrt[26])))+((1665 Log[20+(Sqrt[481])])/(1827904 (Sqrt[26]))),(7861/8957952)+(4319/(4852224 (Sqrt[2])))+((305 (Sqrt[(37/2)]))/303264)+((6125 ArcCoth[4])/(292032 (Sqrt[26])))+((7 ArcSinh[1])/288)+((2463 ArcSinh[(3/2)])/(913952 (Sqrt[26])))-((2463 ArcSinh[(20/9)])/(913952 (Sqrt[26])))+(65/768) ArcTanh[(1/(Sqrt[2]))]-(65/768) ArcTanh[(7/(Sqrt[74]))]-((2761 Log[(3/2)])/11520)-((125 Log[(5/3)])/186624)-((107695 Log[2])/373248)-((147 Log[2])/(108160 (Sqrt[26])))+((365573 Log[3])/1492992)+((17297 Log[3])/(1460160 (Sqrt[26])))+((65 Log[5])/768)+((18019 Log[2-(Sqrt[2])])/186624)+((125 Log[-1+(Sqrt[2])])/9216)+((45 Log[-3+(Sqrt[13])])/(43264 (Sqrt[26])))+((391 Log[-2+(Sqrt[13])])/(182520 (Sqrt[26])))+((85 Log[2+(Sqrt[13])])/(12168 (Sqrt[26])))+((69 Log[3+(Sqrt[13])])/(216320 (Sqrt[26])))+((125 Log[-7 (Sqrt[2])+2 (Sqrt[37])])/186624)-(7/144) Log[123-14 (Sqrt[74])]+((431 Log[-7+(Sqrt[74])])/46656)-((3125 Log[-16+(Sqrt[481])])/(389376 (Sqrt[26])))-((2875 Log[16+(Sqrt[481])])/(1168128 (Sqrt[26])))+((1665 Log[(NumericalApproximation[Root, (RowDefault["0.709", FEExpressionsNumericalApproximationElider]), Root[81+1080 #1-2491 ((#1)^2)+1080 ((#1)^3)+81 ((#1)^4)&,3], ])])/(1827904 (Sqrt[26]))),(3899/8957952)+(8161/(4852224 (Sqrt[2])))+((Sqrt[5])/37908)+((2463 ArcSinh[(7/9)])/(913952 (Sqrt[26])))-((7 ArcSinh[1])/288)-((2463 ArcSinh[(3/2)])/(913952 (Sqrt[26])))-(65/768) ArcTanh[(1/(Sqrt[2]))]+(65/768) ArcTanh[(1/(Sqrt[5]))]-(7/144) Log[(9/8)]+(235/864) Log[(3/2)]+((2697479 Log[2])/3732480)+((147 Log[2])/(108160 (Sqrt[26])))-((353845 Log[3])/746496)+((693437 Log[3])/(18279040 (Sqrt[26])))+((557 Log[1728])/4478976)+(1/6) Log[(3/8) ((((3/2)))^(7/9))]+((Log[3-(3/(Sqrt[2]))])/11520)-((1135 Log[2-(Sqrt[2])])/11664)-((125 Log[-1+(Sqrt[2])])/9216)+((79999 Log[-1+(Sqrt[5])])/933120)-((45 Log[-3+(Sqrt[13])])/(43264 (Sqrt[26])))+((23 Log[-2+(Sqrt[13])])/(73008 (Sqrt[26])))+((25 Log[2+(Sqrt[13])])/(24336 (Sqrt[26])))+((706619 Log[3+(Sqrt[13])])/(36558080 (Sqrt[26])))+((5 Log[-11+(Sqrt[130])])/(389376 (Sqrt[26])))-((17957 Log[2 (7+(Sqrt[130]))])/(913952 (Sqrt[26])))+((23 Log[11+(Sqrt[130])])/(5840640 (Sqrt[26])))+((32775 Log[(18/(-21+13 (Sqrt[10])-7 (Sqrt[13])+3 (Sqrt[130])))])/(11881376 (Sqrt[26])))-((422977 Log[(NumericalApproximation[Root, (RowDefault["1.62", FEExpressionsNumericalApproximationElider]), Root[81+378 #1-1087 ((#1)^2)+378 ((#1)^3)+81 ((#1)^4)&,4], ])])/(23762752 (Sqrt[26]))),(217/559872)+(263/(3942432 (Sqrt[2])))-((Sqrt[5])/37908)+((I (-4394+621 (Sqrt[26])) π)/4100129280)-((ArcCoth[(Sqrt[2])])/3)-((ArcCoth[(Sqrt[5])])/3)-((708963 ArcSinh[(5/12)])/(9139520 (Sqrt[26])))-((368229 ArcSinh[(2/3)])/(4569760 (Sqrt[26])))-((368229 ArcSinh[(7/9)])/(4569760 (Sqrt[26])))+((5 ArcSinh[1])/144)+(19/60) ArcTanh[(1/(Sqrt[2]))]+(19/60) ArcTanh[(1/(Sqrt[5]))]+((729 ArcTanh[(2/(Sqrt[13]))])/(4160 (Sqrt[26])))-((729 ArcTanh[(11/(Sqrt[130]))])/(4160 (Sqrt[26])))-(1/80) Log[(81/64)]-(7/54) Log[(3/2)]-(7/432) Log[(128/81)]-((4303 Log[2])/7290)-((708963 Log[2])/(9139520 (Sqrt[26])))+(1/6) Log[(8/3)]+((263681 Log[3])/933120)+((15677993 Log[3])/(36558080 (Sqrt[26])))+((16767 Log[9])/(432640 (Sqrt[26])))-((5 Log[144])/432)+((2191 Log[1728])/279936)-(3/64) Log[1+(Sqrt[2])]+((853 Log[2+(Sqrt[2])])/29160)-((259 Log[-1+(Sqrt[5])])/14580)+((405 Log[51-12 (Sqrt[13])])/(10816 (Sqrt[26])))+((3895 Log[-2+(Sqrt[13])])/(43264 (Sqrt[26])))+((1665 Log[3 (-2+(Sqrt[13]))])/(1827904 (Sqrt[26])))-((1521109 Log[2+(Sqrt[13])])/(5840640 (Sqrt[26])))+((5589 Log[13+2 (Sqrt[13])])/(54080 (Sqrt[26])))-((655 Log[-11+(Sqrt[130])])/(3042 (Sqrt[26])))-((80793 Log[7+(Sqrt[130])])/(2284880 (Sqrt[26])))+((943 Log[11+(Sqrt[130])])/(36504 (Sqrt[26])))-((5589 Log[3263+286 (Sqrt[130])])/(108160 (Sqrt[26])))+((405 Log[(NumericalApproximation[Root, (RowDefault["101.", FEExpressionsNumericalApproximationElider]), Root[6561-481585588 #1+28158305814 ((#1)^2)-351075893652 ((#1)^3)+3486784401 ((#1)^4)&,4], ])])/(10816 (Sqrt[26])))};

vals[[-1]]

N[%]

N[rxy]

N[rxy,20]

N[rxy-vals[[-1]]+ints[[-1]]/.Inactive[Integrate][args__]:>NIntegrate[args],20]

Divergence theorem "Simplification"->FullSimplify, "IntegrationIntervals"->{0,1/4,1/3,1/2,2/3,3/4,1} [computed but too complicated to simplify]

(rxy=MeanCylindricalRadius[p,"IntegrationIntervals"->{0,1/4,1/3,1/2,2/3,3/4,1},"Simplification"->FullSimplify,Debug->True])//Timing

N[rxy]

N[rxy,20]

ResourceFunction["LogSimplify"][rxy]

FullSimplify[%]

needs 50+ GB

rxysimp=((607 (Sqrt[(5/3)]))/8424)+(1/(8 (Sqrt[3])))+((7 (Sqrt[(17/3)]))/5184)+(11/(24 (Sqrt[6])))-((Sqrt[(37/6)])/80)+((Sqrt[(41/3)])/120)+(1/(72 (Sqrt[78])))+((7 (Sqrt[(1/3) (97-8 (Sqrt[85]))]))/5184)-((I Log[1-((2 I)/3)])/(512 (Sqrt[3])))+((I Log[1+((2 I)/3)])/(512 (Sqrt[3])))+((I Log[1-((3 I)/4)])/(512 (Sqrt[3])))-((I Log[1+((3 I)/4)])/(512 (Sqrt[3])))+((I Log[1-((4 I)/3)])/(512 (Sqrt[3])))-((I Log[1+((4 I)/3)])/(512 (Sqrt[3])))-((I Log[1-((3 I)/2)])/(512 (Sqrt[3])))+((I Log[1+((3 I)/2)])/(512 (Sqrt[3])))+((181835516321 Log[2])/(130427781120 (Sqrt[3])))-((1319892293 Log[2])/(4914000000 (Sqrt[6])))-((53006252101 Log[2])/(171091814400 (Sqrt[78])))-((1819 Log[3])/(20736 (Sqrt[3])))-((15987415989719 Log[3])/(24530688000000 (Sqrt[6])))+((15375181447 Log[3])/(5922408960 (Sqrt[78])))-((32689 Log[5])/(51840 (Sqrt[3])))-((401893 Log[5])/(16872960 (Sqrt[78])))-((25407 (Sqrt[(3/2)]) Log[7])/2800000)-((222959 Log[7])/(414720 (Sqrt[3])))-((17 Log[17])/(27 (Sqrt[3])))-((5 Log[97])/(10368 (Sqrt[3])))+((Log[(243/32) ((((3/2)))^(1/3))])/(3 (Sqrt[78])))+((Log[4 (2^(2/27))])/(6 (Sqrt[3])))+((Log[((2^(3/4))/(Sqrt[3]))])/(6 (Sqrt[6])))+((Log[(1/(288 (2^(330/2197)) (3^(132/2197))))])/(6 (Sqrt[78])))+((Log[(1/((2^(67/576)) (3^(67/1152))))])/(6 (Sqrt[3])))+((Log[9 (3^(9/40))])/(6 (Sqrt[6])))+((Log[13824 (2^(94/169)) (3^(139/169))])/(6 (Sqrt[78])))+((Log[(1/(625 (5^(17/72))))])/(6 (Sqrt[3])))+((Log[38416 (14^(4/27))])/(6 (Sqrt[3])))+((Log[((4913 (17^(7/9)))/(128 (2^(5/9))))])/(6 (Sqrt[3])))-((9553 Log[1-(1/(Sqrt[2]))])/(27648 (Sqrt[3])))+((3807 (Sqrt[(3/2)]) Log[1-(1/(5 (Sqrt[2])))])/5600000)+(1/32) (Sqrt[3]) Log[1-(1/(5 (Sqrt[2])))]-((3807 (Sqrt[(3/2)]) Log[1+(1/(5 (Sqrt[2])))])/5600000)-(1/32) (Sqrt[3]) Log[1+(1/(5 (Sqrt[2])))]+((9553 Log[1+(1/(Sqrt[2]))])/(27648 (Sqrt[3])))+((Log[19601-13860 (Sqrt[2])])/(405 (Sqrt[3])))-((2 Log[1970-1393 (Sqrt[2])])/(1215 (Sqrt[3])))+((2445623 Log[10-7 (Sqrt[2])])/(37158912 (Sqrt[3])))-((4051 Log[3-2 (Sqrt[2])])/(207360 (Sqrt[3])))-((10573 Log[2-(Sqrt[2])])/(207360 (Sqrt[3])))-(1/32) (Sqrt[3]) Log[10-(Sqrt[2])]+((Log[-1+(Sqrt[2])])/(1728 (Sqrt[3])))-((119 Log[-1+(Sqrt[2])])/(3456 (Sqrt[6])))+((1861 Log[1+(Sqrt[2])])/(22680 (Sqrt[3])))+((2041 Log[1+(Sqrt[2])])/(3456 (Sqrt[6])))-((49645 Log[2+(Sqrt[2])])/(72576 (Sqrt[3])))+((157 Log[10+(Sqrt[2])])/(864 (Sqrt[3])))-((14 Log[3+2 (Sqrt[2])])/(405 (Sqrt[3])))-((17 Log[-7+5 (Sqrt[2])])/(576 (Sqrt[3])))+((27 (Sqrt[(3/2)]) Log[-1+5 (Sqrt[2])])/7000)-((1781 Log[-1+5 (Sqrt[2])])/(18432 (Sqrt[3])))+((14607 (Sqrt[(3/2)]) Log[1+5 (Sqrt[2])])/2800000)-((565 Log[1+5 (Sqrt[2])])/(18432 (Sqrt[3])))+((53 Log[7+5 (Sqrt[2])])/(1728 (Sqrt[3])))+((128531 Log[10+7 (Sqrt[2])])/(185794560 (Sqrt[3])))-((19 Log[963551+896285 (Sqrt[2])])/(864 (Sqrt[3])))+((9191 Log[1-(2/(Sqrt[5]))])/(138240 (Sqrt[3])))+((Log[2-(2/(Sqrt[5]))])/(162 (Sqrt[3])))-((7241 Log[1-(1/(Sqrt[5]))])/(241920 (Sqrt[3])))+((7241 Log[1+(1/(Sqrt[5]))])/(241920 (Sqrt[3])))-((9191 Log[1+(2/(Sqrt[5]))])/(138240 (Sqrt[3])))-((Log[14-6 (Sqrt[5])])/(648 (Sqrt[6])))-((73 Log[9-4 (Sqrt[5])])/(4320 (Sqrt[3])))-((Log[5-(Sqrt[5])])/(162 (Sqrt[3])))-(1/512) (Sqrt[3]) Log[-1+((Sqrt[5])/2)]-((1652663 Log[(1/2)+((Sqrt[5])/2)])/(4717440 (Sqrt[3])))+((186271 Log[(1/2)+((Sqrt[5])/2)])/(5400000 (Sqrt[6])))+(1/512) (Sqrt[3]) Log[1+((Sqrt[5])/2)]-((Log[(Sqrt[2])+((2 (Sqrt[5]))/3)])/(81 (Sqrt[3])))-((3647 Log[-2+(Sqrt[5])])/(27648 (Sqrt[3])))-((24905 Log[-1+(Sqrt[5])])/(134784 (Sqrt[3])))+((251 Log[-1+(Sqrt[5])])/(5184 (Sqrt[6])))+((5479 Log[1+(Sqrt[5])])/(51840 (Sqrt[3])))-((70927 Log[1+(Sqrt[5])])/(5400000 (Sqrt[6])))-((1165 Log[2+(Sqrt[5])])/(27648 (Sqrt[3])))-((34373 Log[(Sqrt[2])+2 (Sqrt[5])])/(92160 (Sqrt[3])))+((4549 Log[(Sqrt[2])+2 (Sqrt[5])])/(200000 (Sqrt[6])))-((3743 Log[3 (Sqrt[2])+2 (Sqrt[5])])/(36288 (Sqrt[3])))-((Log[3 (Sqrt[2])+2 (Sqrt[5])])/(32 (Sqrt[6])))+((Log[-12+6 (Sqrt[5])])/(648 (Sqrt[6])))+((1701 (Sqrt[(3/2)]) Log[13 (Sqrt[2])+10 (Sqrt[5])])/200000)+((3807 (Sqrt[(3/2)]) Log[1-(13/(5 (Sqrt[10])))])/5600000)+(1/64) (Sqrt[(3/2)]) Log[1-(1/(Sqrt[10]))]-(1/64) (Sqrt[(3/2)]) Log[1+(1/(Sqrt[10]))]-((3807 (Sqrt[(3/2)]) Log[1+(13/(5 (Sqrt[10])))])/5600000)+((143803 Log[4-(Sqrt[10])])/(614400 (Sqrt[6])))-((389401 Log[(1/3)+((Sqrt[10])/3)])/(2730000 (Sqrt[6])))-((853443 (Sqrt[(3/2)]) Log[(13/9)+((5 (Sqrt[10]))/9)])/2366000000)+((22759 Log[-3+(Sqrt[10])])/(442368 (Sqrt[3])))-((143803 Log[-2+(Sqrt[10])])/(614400 (Sqrt[6])))-((38999 Log[-1+(Sqrt[10])])/(1277952 (Sqrt[6])))+((53 Log[1+(Sqrt[10])])/(192 (Sqrt[3])))+((1043071 Log[1+(Sqrt[10])])/(430080000 (Sqrt[6])))+((3149 Log[3+(Sqrt[10])])/(147456 (Sqrt[3])))+((Log[3+(Sqrt[10])])/(48 (Sqrt[6])))+((1701 (Sqrt[(3/2)]) Log[10+(Sqrt[2])+2 (Sqrt[5])+(Sqrt[10])])/200000)-((567 (Sqrt[(3/2)]) Log[11+2 (Sqrt[10])])/200000)-((19753 Log[11+2 (Sqrt[10])])/(737280 (Sqrt[3])))+((27 (Sqrt[3]) Log[22+4 (Sqrt[10])])/16384)-((5991057 (Sqrt[(3/2)]) Log[13+5 (Sqrt[10])])/2366000000)+((Log[(2/9) (19+6 (Sqrt[10]))])/(384 (Sqrt[3])))+((6011 Log[19+6 (Sqrt[10])])/(72576 (Sqrt[3])))-((17 Log[38+12 (Sqrt[10])])/(10368 (Sqrt[3])))-((Log[31+13 (Sqrt[10])])/(64 (Sqrt[3])))+((13 Log[-117+37 (Sqrt[10])])/(1296 (Sqrt[3])))+(1/128) (Sqrt[(3/2)]) Log[161+44 (Sqrt[10])]-((1701 (Sqrt[(3/2)]) Log[419+130 (Sqrt[10])])/800000)+((Log[721+228 (Sqrt[10])])/(384 (Sqrt[6])))+((Log[186961+59449 (Sqrt[10])])/(48 (Sqrt[3])))+(8/65) (Sqrt[(6/13)]) Log[1-(2/(Sqrt[13]))]-(1/64) (Sqrt[(3/2)]) Log[1-(2/(Sqrt[13]))]-(8/65) (Sqrt[(6/13)]) Log[1+(2/(Sqrt[13]))]+(1/64) (Sqrt[(3/2)]) Log[1+(2/(Sqrt[13]))]-(64/65) (Sqrt[(2/39)]) Log[13-2 (Sqrt[13])]+((23 Log[(16/3)-((4 (Sqrt[13]))/3)])/(31200 (Sqrt[78])))+((2599 Log[(5/2)-((Sqrt[13])/2)])/(20280 (Sqrt[6])))+(1/32) (Sqrt[(3/26)]) Log[1-((4 (Sqrt[13]))/17)]-(1/32) (Sqrt[(3/26)]) Log[1+((4 (Sqrt[13]))/17)]+((144662729 Log[(2/3)+((Sqrt[13])/3)])/(851760000 (Sqrt[6])))+((4315081 Log[(2/3)+((Sqrt[13])/3)])/(16872960 (Sqrt[78])))-((2599 Log[-(1/2)+((Sqrt[13])/2)])/(20280 (Sqrt[6])))+((109 Log[-3+(Sqrt[13])])/(1124864 (Sqrt[78])))+(43/640) (Sqrt[(3/2)]) Log[-2+(Sqrt[13])]-((1179393187 Log[-2+(Sqrt[13])])/(2851530240 (Sqrt[78])))+((22641371 Log[2+(Sqrt[13])])/(32760000 (Sqrt[6])))+((119179609 Log[2+(Sqrt[13])])/(1188137600 (Sqrt[78])))+((573103 Log[3+(Sqrt[13])])/(175478784 (Sqrt[78])))-((23 Log[-(10/3)+((4 (Sqrt[13]))/3)])/(31200 (Sqrt[78])))-((Log[-5 (Sqrt[2])+2 (Sqrt[13])])/(4320 (Sqrt[3])))+((311 Log[-(Sqrt[2])+2 (Sqrt[13])])/(864 (Sqrt[3])))-((140 Log[(Sqrt[2])+2 (Sqrt[13])])/(729 (Sqrt[3])))-((10165 Log[5 (Sqrt[2])+2 (Sqrt[13])])/(193536 (Sqrt[3])))-((Log[-18+5 (Sqrt[13])])/(346112 (Sqrt[78])))-((1701 (Sqrt[(3/2)]) Log[-1+5 (Sqrt[13])])/200000)-((1701 (Sqrt[(3/2)]) Log[1+5 (Sqrt[13])])/200000)-(80/169) (Sqrt[(6/13)]) Log[13 (Sqrt[10])+7 (Sqrt[13])]+((517 Log[3-(5/(Sqrt[17]))])/(175000 (Sqrt[6])))-((517 Log[-1+(5/(Sqrt[17]))])/(175000 (Sqrt[6])))-((2 Log[33-8 (Sqrt[17])])/(135 (Sqrt[3])))+((103 Log[5-(Sqrt[17])])/(4212 (Sqrt[3])))-((7 (Sqrt[(3/2)]) Log[(9/5)-((Sqrt[17])/5)])/400000)+((7 (Sqrt[(3/2)]) Log[(1/5)+((Sqrt[17])/5)])/400000)-((17 Log[(1/4)+((Sqrt[17])/4)])/(80 (Sqrt[6])))-((103 Log[-3+(Sqrt[17])])/(4212 (Sqrt[3])))-((7 Log[-1+(Sqrt[17])])/(324 (Sqrt[3])))+((Log[(1/4) (1+(Sqrt[17]))])/(16 (Sqrt[6])))+(1/20) (Sqrt[(3/2)]) Log[1+(Sqrt[17])]+((Log[1+(Sqrt[17])])/(351 (Sqrt[3])))-((4 Log[4+(Sqrt[17])])/(135 (Sqrt[3])))-((319 Log[3 (Sqrt[2])+2 (Sqrt[17])])/(864 (Sqrt[3])))-((9411 (Sqrt[3]) Log[5 (Sqrt[2])+2 (Sqrt[17])])/458752)+((Log[5 (Sqrt[2])+2 (Sqrt[17])])/(32 (Sqrt[6])))+((91 (Sqrt[(3/2)]) Log[-13+5 (Sqrt[17])])/400000)-((1627 Log[13+5 (Sqrt[17])])/(700000 (Sqrt[6])))-((Log[1-(5/(Sqrt[26]))])/(2560 (Sqrt[3])))-((5 Log[1-(1/(Sqrt[26]))])/(432 (Sqrt[3])))+((5 Log[1+(1/(Sqrt[26]))])/(432 (Sqrt[3])))+((Log[1+(5/(Sqrt[26]))])/(2560 (Sqrt[3])))+((Log[26+13 (Sqrt[2])-5 (Sqrt[13])-5 (Sqrt[26])])/(11520 (Sqrt[3])))+((167 Log[27-2 (Sqrt[26])])/(864 (Sqrt[3])))-((672031 Log[(1/5)+((Sqrt[26])/5)])/(50618880 (Sqrt[78])))+((33353 Log[-1+(Sqrt[26])])/(6327360 (Sqrt[78])))+((187771 Log[1+(Sqrt[26])])/(10123776 (Sqrt[78])))-((7 Log[5+(Sqrt[26])])/(3456 (Sqrt[3])))+((21355 Log[27+2 (Sqrt[26])])/(46656 (Sqrt[3])))+(64/65) (Sqrt[(2/39)]) Log[26 (Sqrt[5])+7 (Sqrt[26])]+((6593 Log[51+10 (Sqrt[26])])/(215040 (Sqrt[3])))-((7 Log[-515+101 (Sqrt[26])])/(51840 (Sqrt[3])))-((8 Log[-5+(Sqrt[29])])/(243 (Sqrt[3])))-((8 Log[5+(Sqrt[29])])/(405 (Sqrt[3])))-((2 Log[528527+98145 (Sqrt[29])])/(1215 (Sqrt[3])))+((Log[6-(Sqrt[34])])/(64 (Sqrt[6])))-((Log[-4+(Sqrt[34])])/(64 (Sqrt[6])))+((19 Log[-3+(Sqrt[34])])/(864 (Sqrt[3])))+((11 Log[3+(Sqrt[34])])/(128 (Sqrt[3])))-((13 Log[5+(Sqrt[34])])/(864 (Sqrt[3])))-((Log[5+(Sqrt[34])])/(64 (Sqrt[6])))+((1604531 Log[59+10 (Sqrt[34])])/(24772608 (Sqrt[3])))-((Log[118+20 (Sqrt[34])])/(256 (Sqrt[3])))+((13 Log[-635+109 (Sqrt[34])])/(864 (Sqrt[3])))+((5 Log[26763+4621 (Sqrt[34])])/(128 (Sqrt[3])))+((19 Log[454971-321156 (Sqrt[2])-110904 (Sqrt[17])+78557 (Sqrt[34])])/(864 (Sqrt[3])))+((47 (Sqrt[(3/2)]) Log[7-(Sqrt[37])])/1664)-((31 Log[(1/6)+((Sqrt[37])/6)])/(320 (Sqrt[6])))-((47 (Sqrt[(3/2)]) Log[-5+(Sqrt[37])])/1664)-((11 Log[-1+(Sqrt[37])])/(80 (Sqrt[6])))+((157 (Sqrt[(3/2)]) Log[1+(Sqrt[37])])/8320)-(((Sqrt[3]) Log[-7 (Sqrt[2])+2 (Sqrt[37])])/2048)-((1841 Log[5 (Sqrt[2])+2 (Sqrt[37])])/(17280 (Sqrt[3])))-((151 Log[7 (Sqrt[2])+2 (Sqrt[37])])/(1728 (Sqrt[3])))-((7021 (Sqrt[(3/2)]) Log[-14+5 (Sqrt[37])])/13000000)-((225689 Log[14+5 (Sqrt[37])])/(39000000 (Sqrt[6])))+((567 (Sqrt[(3/2)]) Log[23 (Sqrt[2])+10 (Sqrt[41])])/100000)-((13 Log[-7+(Sqrt[53])])/(648 (Sqrt[3])))+((13 Log[7+(Sqrt[53])])/(216 (Sqrt[3])))-((567 (Sqrt[(3/2)]) Log[7 (Sqrt[2])+10 (Sqrt[53])])/100000)-((13 Log[6754799+927843 (Sqrt[53])])/(1296 (Sqrt[3])))-((7 Log[969-306 (Sqrt[10])+190 (Sqrt[26])-120 (Sqrt[65])])/(1728 (Sqrt[3])))+(1/39) (Sqrt[(2/39)]) Log[-1+(Sqrt[65])]+((Log[-1+(Sqrt[65])])/(162 (Sqrt[3])))+((Log[1+(Sqrt[65])])/(162 (Sqrt[3])))-((35 (Sqrt[(6/13)]) Log[1+65 (Sqrt[5])+5 (Sqrt[13])+(Sqrt[65])])/2197)+(1/32) (Sqrt[(3/2)]) Log[7 (Sqrt[2])+2 (Sqrt[65])]+((4779 (Sqrt[(3/2)]) Log[(131/71)-((5 (Sqrt[73]))/71)])/13000000)-((4779 (Sqrt[(3/2)]) Log[(11/71)+((5 (Sqrt[73]))/71)])/13000000)-((4779 (Sqrt[(3/2)]) Log[-23+5 (Sqrt[73])])/13000000)+(((Sqrt[3]) Log[123-14 (Sqrt[74])])/4096)-((7 Log[-7+(Sqrt[74])])/(576 (Sqrt[3])))+((1841 Log[99+10 (Sqrt[74])])/(34560 (Sqrt[3])))+((55 Log[123+14 (Sqrt[74])])/(1152 (Sqrt[3])))+((7 Log[-1386861-924574 (Sqrt[2])+214964 (Sqrt[37])+161223 (Sqrt[74])])/(1728 (Sqrt[3])))+((6577 (Sqrt[(3/2)]) Log[(1/9)+((Sqrt[82])/9)])/140000)-((47 Log[-1+(Sqrt[82])])/(480 (Sqrt[6])))-((17303 Log[1+(Sqrt[82])])/(70000 (Sqrt[6])))+((17 Log[83+2 (Sqrt[82])])/(480 (Sqrt[6])))-((567 (Sqrt[(3/2)]) Log[23+5 (Sqrt[82])])/100000)-((Log[7217+332 (Sqrt[82])])/(64 (Sqrt[6])))+(1/16) (Sqrt[(3/2)]) Log[(2/9)+((Sqrt[85])/9)]+((Log[-2+(Sqrt[85])])/(16 (Sqrt[6])))+((Log[-11 (Sqrt[2])+2 (Sqrt[85])])/(216 (Sqrt[3])))+((349 Log[7 (Sqrt[2])+2 (Sqrt[85])])/(3240 (Sqrt[3])))+((509 Log[11 (Sqrt[2])+2 (Sqrt[85])])/(23328 (Sqrt[3])))-((Log[89+4 (Sqrt[85])])/(32 (Sqrt[6])))+((Log[99 (Sqrt[2])-44 (Sqrt[10])-40 (Sqrt[17])+18 (Sqrt[85])])/(864 (Sqrt[3])))-((Log[9281+712 (Sqrt[85])])/(64 (Sqrt[6])))+((16 Log[(5/8)+((Sqrt[89])/8)])/(405 (Sqrt[3])))-((2 Log[-8+(Sqrt[89])])/(135 (Sqrt[3])))-((16 Log[129+13 (Sqrt[89])])/(405 (Sqrt[3])))+(368/845) (Sqrt[(2/39)]) Log[-17 (Sqrt[26])+26 (Sqrt[89])]-((2 Log[-805288+85361 (Sqrt[89])])/(405 (Sqrt[3])))-(1/16) (Sqrt[(3/2)]) Log[(4/9)+((Sqrt[97])/9)]-((Log[-4+(Sqrt[97])])/(16 (Sqrt[6])))-((Log[-13 (Sqrt[2])+2 (Sqrt[97])])/(1296 (Sqrt[3])))+((29135 Log[13 (Sqrt[2])+2 (Sqrt[97])])/(139968 (Sqrt[3])))+((Log[18977+1808 (Sqrt[97])])/(32 (Sqrt[6])))+((567 (Sqrt[(3/2)]) Log[7+5 (Sqrt[106])])/100000)+((Log[-(1/(6359666963 (Sqrt[2])-861462002 (Sqrt[109])))])/(405 (Sqrt[3])))-((Log[209-20 (Sqrt[109])])/(48 (Sqrt[6])))-((Log[10+(Sqrt[109])])/(96 (Sqrt[6])))+((Log[-13 (Sqrt[2])+2 (Sqrt[109])])/(27 (Sqrt[3])))-((Log[7 (Sqrt[2])+2 (Sqrt[109])])/(27 (Sqrt[3])))+((8 Log[13 (Sqrt[2])+2 (Sqrt[109])])/(405 (Sqrt[3])))-((Log[87281+8360 (Sqrt[109])])/(128 (Sqrt[6])))-((476577 (Sqrt[(3/26)]) Log[(7/9)+((Sqrt[130])/9)])/5624320)-((101659 Log[(11/3)+((Sqrt[130])/3)])/(296120448 (Sqrt[78])))+((79 Log[-11+(Sqrt[130])])/(5624320 (Sqrt[78])))+(1/128) (Sqrt[(3/2)]) Log[-7+(Sqrt[130])]-((7873 Log[-7+(Sqrt[130])])/(259584 (Sqrt[78])))-((27861581 Log[7+(Sqrt[130])])/(219348480 (Sqrt[78])))-((5001913 Log[11+(Sqrt[130])])/(5922408960 (Sqrt[78])))-((36233 Log[-21+13 (Sqrt[10])-7 (Sqrt[13])+3 (Sqrt[130])])/(71288256 (Sqrt[78])))+(184/845) (Sqrt[(2/39)]) Log[179+14 (Sqrt[130])]-(1/128) (Sqrt[(3/2)]) Log[3073+277 (Sqrt[130])]-((13 Log[13-(Sqrt[145])])/(192 (Sqrt[6])))-((Log[(1/12)+((Sqrt[145])/12)])/(64 (Sqrt[6])))+((13 Log[-11+(Sqrt[145])])/(192 (Sqrt[6])))+(1/64) (Sqrt[(3/2)]) Log[-1+(Sqrt[145])]-((5 Log[1+(Sqrt[145])])/(96 (Sqrt[6])))+((Log[109+37 (Sqrt[145])])/(64 (Sqrt[6])))-((Log[291-22 (Sqrt[170])])/(96 (Sqrt[3])))+((7 Log[-11+(Sqrt[170])])/(3456 (Sqrt[3])))-((349 Log[219+14 (Sqrt[170])])/(6480 (Sqrt[3])))-((67 Log[291+22 (Sqrt[170])])/(5832 (Sqrt[3])))+((7 Log[-2319046741+177862777 (Sqrt[170])])/(3456 (Sqrt[3])))-(7/256) (Sqrt[(3/2)]) Log[1-(13/(Sqrt[178]))]+(7/256) (Sqrt[(3/2)]) Log[1+(13/(Sqrt[178]))]+(1/256) (Sqrt[(3/2)]) Log[178-13 (Sqrt[178])]+((27 (Sqrt[(3/2)]) Log[-13+(Sqrt[178])])/1024)-(1/256) (Sqrt[(3/2)]) Log[178+13 (Sqrt[178])]-((Log[-9139+685 (Sqrt[178])])/(128 (Sqrt[6])))-((Log[9139+685 (Sqrt[178])])/(128 (Sqrt[6])))-((3 (Sqrt[(3/2)]) Log[4400200363+329808961 (Sqrt[178])])/1024)+(3/128) (Sqrt[(3/2)]) Log[-10+(Sqrt[181])]+(3/128) (Sqrt[(3/2)]) Log[10+(Sqrt[181])]+((19 Log[(Sqrt[2])+2 (Sqrt[181])])/(54 (Sqrt[3])))+((13 Log[(19/7)-((Sqrt[193])/7)])/(120 (Sqrt[6])))-((13 Log[-(5/7)+((Sqrt[193])/7)])/(120 (Sqrt[6])))-((13 Log[-7+(Sqrt[193])])/(120 (Sqrt[6])))-((1597 Log[19 (Sqrt[2])+2 (Sqrt[193])])/(2304 (Sqrt[3])))+(64/65) (Sqrt[(2/39)]) Log[22 (Sqrt[13])+13 (Sqrt[193])]-((Log[22 (Sqrt[13])+13 (Sqrt[193])])/(11520 (Sqrt[3])))+((Log[1-(13/(Sqrt[194]))])/(1152 (Sqrt[3])))-((Log[1+(13/(Sqrt[194]))])/(1152 (Sqrt[3])))+((5 Log[194+13 (Sqrt[194])])/(5184 (Sqrt[3])))-((29135 Log[363+26 (Sqrt[194])])/(279936 (Sqrt[3])))+(3/256) (Sqrt[(3/2)]) Log[1-(14/(Sqrt[205]))]-(3/256) (Sqrt[(3/2)]) Log[1+(14/(Sqrt[205]))]-(3/256) (Sqrt[(3/2)]) Log[205-14 (Sqrt[205])]+(1/128) (Sqrt[(3/2)]) Log[-14+(Sqrt[205])]-((17 Log[11 (Sqrt[2])+2 (Sqrt[205])])/(54 (Sqrt[3])))+(3/256) (Sqrt[(3/2)]) Log[205+14 (Sqrt[205])]+((Log[11354+793 (Sqrt[205])])/(128 (Sqrt[6])))+((Log[267+14 (Sqrt[218])])/(54 (Sqrt[3])))-((19 Log[363+2 (Sqrt[362])])/(108 (Sqrt[3])))-((125 Log[1-(19/(Sqrt[386]))])/(288 (Sqrt[3])))+((125 Log[1+(19/(Sqrt[386]))])/(288 (Sqrt[3])))+((875 Log[747-38 (Sqrt[386])])/(4608 (Sqrt[3])))+((59 Log[747+38 (Sqrt[386])])/(576 (Sqrt[3])))+((17 Log[531+22 (Sqrt[410])])/(108 (Sqrt[3])))+((703 (Sqrt[(3/26)]) Log[-60+20 (Sqrt[13])+13 (Sqrt[37])-3 (Sqrt[481])])/456976)-((17 Log[-20+(Sqrt[481])])/(64896 (Sqrt[78])))-((53489 Log[20+(Sqrt[481])])/(10967424 (Sqrt[78])))+((45 (Sqrt[(3/26)]) Log[-8+(Sqrt[793])])/5408)+((45 (Sqrt[(3/26)]) Log[8+(Sqrt[793])])/5408)-((484 (Sqrt[(2/39)]) Log[-11+11 (Sqrt[65])+13 (Sqrt[545])-(Sqrt[1417])])/6591)-((484 (Sqrt[(2/39)]) Log[-11+(Sqrt[1417])])/6591)-(184/845) (Sqrt[(2/39)]) Log[2603-34 (Sqrt[2314])]-(32/65) (Sqrt[(2/39)]) Log[2993+44 (Sqrt[2509])]+((Log[2993+44 (Sqrt[2509])])/(23040 (Sqrt[3])))-(1/39) (Sqrt[(2/39)]) Log[-19+(Sqrt[2665])]-((3911 Log[1+(1/43) (39-4 (Sqrt[130]))])/(64896 (Sqrt[78])))+((3911 Log[1+(1/43) (-39+4 (Sqrt[130]))])/(64896 (Sqrt[78])))-((35 (Sqrt[(6/13)]) Log[1+(1/27) (52-(Sqrt[2353]))])/2197)+((35 (Sqrt[(6/13)]) Log[1+(1/27) (-52+(Sqrt[2353]))])/2197)-((369877 Log[(NumericalApproximation[Root, (RowDefault["0.619", FEExpressionsNumericalApproximationElider]), Root[81+378 #1-1087 ((#1)^2)+378 ((#1)^3)+81 ((#1)^4)&,3], ])])/(190102016 (Sqrt[78])))-(1/39) (Sqrt[(2/39)]) Log[(NumericalApproximation[Root, (RowDefault["0.787", FEExpressionsNumericalApproximationElider]), Root[576+608 #1-2537 ((#1)^2)+608 ((#1)^3)+576 ((#1)^4)&,3], ])]+((35 (Sqrt[(6/13)]) Log[(NumericalApproximation[Root, (RowDefault["0.914", FEExpressionsNumericalApproximationElider]), Root[5184+168 #1-10873 ((#1)^2)+168 ((#1)^3)+5184 ((#1)^4)&,3], ])])/2197)+((NumericalApproximation[Root, (RowDefault["0.138", FEExpressionsNumericalApproximationElider]), Root[25-1322 ((#1)^2)+((#1)^4)&,3], ])/(240 (Sqrt[6])));

ResourceFunction["LogSimplify"][rxysimp]

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->True])//Timing

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

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

N[rxyz]

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

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])//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]((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

Midsphere

sphere=PolyhedronData[pname,"Midsphere"]

(sphere=Midsphere[p])//Timing

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

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

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},Axes->True,AxesLabel->{"x","y","z"}]

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[{Opacity[.5],{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[PolyhedronData["Cuboctahedron","Polyhedron"],{a,b,c}],{a,-1,1},{b,-1,1},{c,-1,1}]

XXX

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

Net construction

PolyhedronData["TrapezoRhombicDodecahedron","Net"]

h=With[{a=1,b=2,c=3/2},Sqrt[c^2-(b-a)^2/4]]

p=PolyhedronData["TrapezoRhombicDodecahedron","Polygons"][[1,1]]

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

alpha=ArcCos[Dot@@Normalize/@{p[[2]]-p[[1]],p[[4]]-p[[1]]}]//FullSimplify

N[ArcSec[3],20]

N[ArcSec[3]/Degree,20]

N[alpha]/Degree

beta=With[{a=1,b=2,c=3/2},ArcCos[(((b-a)/2)/c)]]//FullSimplify

v[[20]]+2RotationMatrix[-alpha].Normalize[Subtract@@v[[{21,20}]]]//FunctionExpand//FullSimplify

v[[21]]+(v[[25]]-v[[20]])/2

v=RotationMatrix[-π/2].#&/@{{0,0},{2,0},{1/2,h},{3/2,h},{-(7/6),((2 (Sqrt[2]))/3)},{-(2/3),((5 (Sqrt[2]))/3)},{-(23/18),-((5 (Sqrt[2]))/9)},{-(22/9),((Sqrt[2])/9)},{(34/81),-((112 (Sqrt[2]))/81)},{-(173/162),-((101 (Sqrt[2]))/81)},{-(149/54),((31 (Sqrt[2]))/27)},{-(40/27),((46 (Sqrt[2]))/27)},{-(82/27),((70 (Sqrt[2]))/27)},{-(191/54),((43 (Sqrt[2]))/27)},{-(29/18),-((11 (Sqrt[2]))/9)},{-(28/9),-((11 (Sqrt[2]))/9)},{-(125/486),-((566 (Sqrt[2]))/243)},{-(424/243),-((533 (Sqrt[2]))/243)},{-(695/486),-((785 (Sqrt[2]))/243)},{(14/243),-((818 (Sqrt[2]))/243)},{(649/486),-((683 (Sqrt[2]))/243)},{(248/243),-((431 (Sqrt[2]))/243)},{-(1109/486),-((875 (Sqrt[2]))/243)},{-(838/243),-((713 (Sqrt[2]))/243)},{(392/243),-((1034 (Sqrt[2]))/243)},{(1027/486),-((791 (Sqrt[2]))/243)}};
Graphics[{FaceForm[LightYellow],EdgeForm[Black],gc=GraphicsComplex[
v,
{
Polygon[{{1,2,4,3},{3,6,5,1},{5,8,7,1},{9,1,7,10},{5,12,11,8},{12,13,14,11},{7,8,16,15},{9,10,18,17},{17,18,19,20},{20,21,22,17},{18,24,23,19},{20,25,26,21}}]
}],MapIndexed[Text[#2[[1]],#+.1{1,1}]&,v]}]

v

EuclideanDistance@@@Flatten[Flatten[Normal[gc]]/.Polygon[l_]:>Partition[l,2,1,1],1]//Union

PolyhedronData["TrapezoRhombicDodecahedron","EdgeLengths"]

PolyhedronData

PolyhedronDataString[p,
"StandardName"->"TrapezoRhombicDodecahedron",
"Name"->"squashed dodecahedron",
"Information"->"Space-FillingPolyhedron",
"Classes"->{"Convex","SpaceFilling"},
Debug->True
]