Dodecahedral Graph

Author

Eric W. Weisstein
June 21, 2023

This notebook downloaded from http://mathworld.wolfram.com/notebooks/GraphTheory/DodecahedralGraph.nb.

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

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

Embeddings

<<MathWorld`Graphs`

Names

GraphData["DodecahedralGraph","StandardName"]

GraphData["DodecahedralGraph","StandardNames"]

Primary

GraphData["DodecahedralGraph"]//StyleGraphs[#,ImageSize->Small]&

Construction from Read and Wilson

v=Join[UnitCirclePoints[5],c=CirclePoints[1/2,5]];
v=Join[v,Mean/@Partition[c,2,1,1],CirclePoints[{1/4,-Pi/2},5]];
g=Graph[Range@Length@v,UndirectedEdge@@@{e[{1,5},1,6,11,7,12,8,13,9,14,10,15,6],e[15,16,17,18,19,20,16],{11,17},{12,18},{13,19},{14,20},{2,7},{3,8},{4,9},{5,10}},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph@g

All

GraphData["DodecahedralGraph","Graph","All"]//StyleGraphs

Inexact

Select[GraphData["DodecahedralGraph","Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData["DodecahedralGraph","EdgeList"]]

Default embedding

GraphPlot@GraphData["DodecahedralGraph","Edges"]

v={};
g=Graph[Range[Length[v]],UndirectedEdge@@@{},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

v=Join[UnitCirclePoints[5],CirclePoints[{1/2,-Pi/2},10],CirclePoints[{1/4,-Pi/2},5]];
g=Graph[Range[Length[v]],UndirectedEdge@@@{e[{1,5},1,{7,15},6,7],e[6,{16,20},16],{8,17},{2,9},{10,18},{3,11},{12,19},{4,13},{14,20},{5,15}},VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

Degenerate

GraphData["DodecahedralGraph","Graph","Degenerate"]//StyleGraphs

Bilateral

GraphData["DodecahedralGraph","Graph","Bilateral"]//StyleGraphs

GeneralizedPetersen

GraphData["DodecahedralGraph","NotationRules","GeneralizedPetersen"]

GraphData["DodecahedralGraph","Graph","GeneralizedPetersen"]//StyleGraphs

IntegerCoordinates

GraphData["DodecahedralGraph","Graph","IntegerCoordinates"]//StyleGraphs

Integral

GraphData["DodecahedralGraph","Graph","Integral"]//StyleGraphs

IntegralDrawing/@GraphData["DodecahedralGraph","Graph","Integral"]//StyleGraphs

IGraph

GraphData["DodecahedralGraph","NotationRules","IGraph"]

GraphData["DodecahedralGraph","Graph","IGraph"]//StyleGraphs

Circular

GraphData["DodecahedralGraph","Graph","Circular"]//StyleGraphs

Planar

GraphData["DodecahedralGraph","Planar"]

GraphData["DodecahedralGraph","Graph","Planar"]//StyleGraphs

Tutte embeddings

faces=ReverseSort@SymmetricallyDistinctFaces[GraphData["DodecahedralGraph"]]

Tally[ReverseSort[Length/@faces]]

IGLayoutTutte[GraphData["DodecahedralGraph"],"OuterFace"->#]&/@faces

gs=Annotate[#,{VertexLabels->None,ImageSize->Tiny}]&/@{}

RecognizeGraph/@gs

MinimalCrossing

GraphData["DodecahedralGraph",{"RCN","CN"}]

GraphData["DodecahedralGraph","Graph","MinimalCrossing"]//StyleGraphs

MinimalIntegral

GraphData["DodecahedralGraph","Graph","MinimalIntegral"]//StyleGraphs

IntegralDrawing/@GraphData["DodecahedralGraph","Graph","MinimalIntegral"]//StyleGraphs

MinimalPlanarIntegral

IntegralDrawing/@GraphData["DodecahedralGraph","Graph","MinimalPlanarIntegral"]//StyleGraphs

Circulant

GraphData["DodecahedralGraph","Circulant"]

GraphData["DodecahedralGraph","Graph","Circulant"]//StyleGraphs

Halin

GraphData["DodecahedralGraph","Halin"]

GraphData["DodecahedralGraph","Graph","Halin"]//StyleGraphs

LCF

GraphData["DodecahedralGraph","LCF"]

GraphData["DodecahedralGraph","HamiltonianCycleCount"]

GraphData["DodecahedralGraph","LCFSignature"]

GraphData["DodecahedralGraph","Graph","LCF"]//StyleGraphs

Tally[LCFFromEmbedding[#][[2]]&/@%]

LCF compute

Length[FindHamiltonianCycle[GraphData["DodecahedralGraph"],All][[All,All,1]]]//Timing

Tally[(lcf=LCF[GraphData["DodecahedralGraph"],All])[[All,2]]]//Timing

LCFGraph@@@lcf//StyleGraphs

gs=Flatten[{
(* all LCF's with order >1, sorted by order and with bilateral graphs first *)
Join[DeleteCases[#,{_Missing,_}][[All,1]],Cases[#,{_Missing,_}][[All,2]]]&/@Map[{BilateralLCFEmbedding[#],#}&,Map[LCFGraph@@#&,SplitBy[DeleteCases[lcf,{_,1}],Last],{2}],{2}],
(* order-1 bilateral only *)
DeleteMissing[BilateralLCFEmbedding/@LCFGraph@@@Cases[lcf,{_,1}]]}]

Perspective

GraphData["DodecahedralGraph","Graph","Perspective"]//StyleGraphs

UnitDistance

GraphData["DodecahedralGraph","UnitDistance"]

UnitDistanceGraphQ[GraphData["DodecahedralGraph"],Debug->True]//Timing

GraphData["DodecahedralGraph","Graph","UnitDistance"]//StyleGraphs

Select[%,!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&]

UnitDistance by refinement

UnitDistanceEmbeddingRefine[Graph[GraphData["DodecahedralGraph","VertexList"],GraphData["DodecahedralGraph","EdgeList"]]]

UnitDistanceGraphEmbeddingQ[%]

UnitDistanceEmbeddingRefine[Annotate[#,VertexCoordinates->N[GraphEmbedding[#]]]]&/@Complement@@(GraphData["DodecahedralGraph","Graph",#]&/@{"All","UnitDistance"})

Select[%,UnitDistanceGraphEmbeddingQ]

UnitDistance by FindUnitDistanceEmbedding

FindUnitDistanceEmbedding[GraphData["DodecahedralGraph"],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

FindUnitDistanceEmbedding[GraphData["DodecahedralGraph"],MaxIterations->5000,"ReturnWithMessage"->False]//Quiet//Timing

UnitDistance by FindSubgraphUnitDistanceEmbedding

{"DodecahedralGraph",l=FindSubgraphUnitDistanceEmbedding[GraphData["DodecahedralGraph"],TimeConstraint->300,"Results"->All]//Timing}

gs=SortBy[UniqueEmbeddings[l],Replace[RectilinearCrossingCount[#],Infinity->0]&]

Matchstick

GraphData["DodecahedralGraph","Matchstick"]

GraphData["DodecahedralGraph","Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData["DodecahedralGraph","Edges"]]

3D All

GraphData["DodecahedralGraph","Graph",{"3D","All"}]

3D UnitDistance

GraphData["DodecahedralGraph","Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData["DodecahedralGraph","Polyhedral"]

GraphData["DodecahedralGraph","Graph",{"3D","Polyhedron"}]

Graphics3D[{FaceForm[Green,Red],GraphData["DodecahedralGraph","Polyhedron"]}]

3D Canonical

GraphData["DodecahedralGraph","Graph",{"3D","Canonical"}]

Graphics3D[{Opacity[.5],Red,GraphData["DodecahedralGraph","Polyhedron","Canonical"],Yellow,Sphere[]}]

3D Canonical construction

Graphics3D[{FaceForm[Green,Red],poly=GraphPolyhedron[GraphData["DodecahedralGraph"],-1]}]

(can=FindCanonicalPolyhedron[poly,WorkingPrecision->50,MaxIterations->50000,PrecisionGoal->20,"ProgressInterval"->1000,"TangentFactor"->1/10,"PlanarFactor"->1/10,Debug->True])//Timing

Graphics3D[{Opacity[.5],Red,can,Yellow,Sphere[]}]

(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]

g=PolyhedronGraph[can]

Projections

Manipulate[PolyhedronProjectionGraph[PolyhedronData["Dodecahedron","Polyhedron"],{a,b,c}],{a,-1,1},{b,-1,1},{c,-1,1}]

{1, 0, 0}

g=PolyhedronProjectionGraph[PolyhedronData["Dodecahedron","Polyhedron"],{1,0,.0001,π/2},VertexLabels->"Name"]

PolyhedronProjectionGraph[PolyhedronData["Dodecahedron","Polyhedron"],{1,0,z,π/2},VertexLabels->"Name"]

v=Lookup[Options[%],VertexCoordinates];

Limit[v,z->0,Direction->"FromBelow"]//FullSimplify

SetProperty[g,VertexCoordinates->%49]

{1,0,GoldenRatio/2-1}

PolyhedronData["Dodecahedron","VertexCoordinates"][[1]]

v=FullSimplify[%/(Sqrt[1+(2/(Sqrt[5]))])]

N[%]

(1-GoldenRatio/2)

PolyhedronProjectionGraph[PolyhedronData["Dodecahedron","Polyhedron"],{1,0,GoldenRatio/2-1,π/2},VertexLabels->"Name"]

{XXX}

Mean[PolyhedronData["Dodecahedron","Faces"][[1]][[{1,3}]]]//FullSimplify

PolyhedronProjectionGraph[PolyhedronData["Dodecahedron","Polyhedron"],{-(1/40) (Sqrt[650+290 (Sqrt[5])]),(1/8) (-3-(Sqrt[5])),(NumericalApproximation[Root, (RowDefault["0.263", FEExpressionsNumericalApproximationElider]), Root[1-20 ((#1)^2)+80 ((#1)^4)&,3], ]),π/10},VertexLabels->"Name"]

{0, 1, 0}

<<MathWorld`Graphs`

PolyhedronProjectionGraph[PolyhedronData["Dodecahedron","Polyhedron"],{0,1,0,-π/6},VertexLabels->"Name"]

{0, 0, 1}

PolyhedronProjectionGraph[PolyhedronData["Dodecahedron","Polyhedron"],{0,0,1},VertexLabels->"Name"]

{1, 0, 1}

PolyhedronProjectionGraph[PolyhedronData["Dodecahedron","Polyhedron"],{1,0,1},VertexLabels->"Name"]

{1, 0, -1}

PolyhedronProjectionGraph[PolyhedronData["Dodecahedron","Polyhedron"],{1,0,-1},VertexLabels->"Name"]

{1.,5/2,5/2,-π/8}

PolyhedronProjectionGraph[PolyhedronData["Dodecahedron","Polyhedron"],{1.,5/2,5/2,-π/8},VertexLabels->"Name"]

Stereoscopic projections

PolyhedronData["Dodecahedron","Orientations"]

Quiet@GraphPlot[SkeletonGraph[PolyhedronData[{"Dodecahedron","C2"},"Faces"]//Normal,#],Method->None]&/@Range[.2,1.5,.1]

GraphPlot[SkeletonGraph[PolyhedronData[{"Dodecahedron","C3"},"Faces"]//Normal,#],Method->None]&/@Range[.2,2,.1]

Quiet@GraphPlot[SkeletonGraph[PolyhedronData[{"Dodecahedron","C5"},"Faces"]//Normal,#],Method->None]&/@Range[.2,1.5,.1]

Properties

Classes

GraphData["DodecahedralGraph","Classes"]

Matrices

GraphicsRow[GraphMatrixPlot[GraphData["DodecahedralGraph"]]]

Properties

TableForm[GraphDataTable["DodecahedralGraph",x],TableDepth->2]//TraditionalForm

Graceful labelings

GraphData["DodecahedralGraph","Graceful"]

GraphData["DodecahedralGraph","GracefulLabelings"]

lab=%[[2]]

SetProperty[GraphData["DodecahedralGraph"],VertexLabels->Thread[Range[20]->lab]]

GracefullyLabeledGraphQ[%]

Partial population

Bert Dobbelaere 2020-10-06:

Over 12 million "fundamentally different" solutions after running overnight.  I estimate the program could run for a month or so.

adj=1+{{1,4,5},{0,2,7},{1,3,9},{2,4,11},{0,3,13},{0,6,14},{5,7,15},{1,6,8},{7,9,16},{2,8,10},{9,11,17},{3,10,12},{11,13,19},{4,12,14},{5,13,18},{6,16,18},{8,15,17},{10,16,19},{14,15,19},{12,17,18}};

r=RecognizeGraph@(dodec=FromAdjacencyLists1[adj])

iso=FindGraphIsomorphism[GraphData[r],dodec][[1]]

vorder=Normal[iso][[All,2]]

vlab={0,30,1,3,28,18,2,23,10,27,8,12,4,14,17,29,5,20,6,26}[[vorder]]

glab=SetProperty[GraphData[r],VertexLabels->Thread[Range[20]->vlab]]

GracefullyLabeledGraphQ[glab]

Full computation

On 10/21/20 12:50 PM, Bert Dobbelaere wrote:

> I couldn't resist starting the dodecahedron run anyway, split in multiple ranges so I could compute them in parallel.
> 
> The total number of graceful labelings I obtained for the dodecahedral graph is 784298856 * 240 = 188231725440.
> 
> Quite a number compared to the icosahedron...

2GraphData["DodecahedralGraph","AutomorphismCount"]

Construction

distanceregular.org

g=DistanceRegularGraph["http://www.distanceregular.org/graphdata/dodecahedron.am"]

ToEntity[g]

VertexCount[g]

GraphDiameter[g]//Timing

RegularParameters[g]//Timing

DistanceRegularGraphQ[g]//Timing

IntersectionArray[g]//Timing

IntegerSpectrum[g]//SpectrumForm

GroupOrder[GraphAutomorphismGroup[g]]

HamiltonianGraphQ[g]

MinimalPlanarIntegral [old]

v={{405.799`,20.1835`},{1261.27`,76.588`},{1480.62`,919.522`},{750.493`,1389.56`},{79.9064`,838.049`},{446.536`,51.5194`},{1258.13`,132.993`},{1417.95`,925.789`},{719.157`,1333.15`},{114.376`,800.446`},{418.333`,490.221`},{828.833`,230.134`},{1201.73`,549.759`},{1010.58`,1010.4`},{509.207`,975.926`},{528.009`,903.854`},{484.139`,465.152`},{869.569`,289.672`},{1167.26`,612.431`},{944.775`,988.461`}};

Length[v]

ShowLabeledGraph[g=Graph[List/@{
Sequence@@Partition[Range[5],2,1,1],
Sequence@@Partition[Range[16,20],2,1,1],
{1,7},{7,13},{13,8},{4,10},{9,15},{15,10},{10,11},{11,6},{6,12},{12,7},{12,17},{18,13},{14,8},{9,14},{15,20},{16,11},{14,19},{2,8},{3,9},{5,6}
},List/@v],VertexStyle->Disk[.02]]

Degrees[g]

RecognizeGraph[g]

θ=ArcTan@@Subtract@@v[[{2,1}]]

v=Vertices[g];

s=Norm[Subtract@@v[[{2,1}]]]

v2=RotationMatrix[-θ].#&/@((2/s)(#-v[[1]])&/@v)

c=Mean[Take[v2,5]]

v3=(#-c)&/@v2

GraphPlot[g2=ChangeVertices[g,v3],Method->None,Axes->True]

With[{
θ=-1.8,ϕ=.03,ψ=-.68,
r=1.1,R=.94
},ShowLabeledGraph[g3=ChangeVertices[g2,Join[
Table[Csc[(π/5)]Through[{Cos,Sin}[2π (k+1/4)/5+4π/5]],{k,5}],
Table[R Csc[(π/5)]Through[{Cos,Sin}[2π (k+1/4)/5+4π/5+ϕ]],{k,5}],
Table[(1/2)r Csc[(π/5)]Through[{Cos,Sin}[2π (k+1/4)/5+4π/5+ψ]],{k,5}],
Table[(1/2)Csc[(π/5)]Through[{Cos,Sin}[2π (k+1/4)/5+4π/5+θ]],{k,5}]
]
],VertexStyle->Disk[.01]]]

g3//InputForm

v=Vertices[g3];
Norm[Subtract@@v[[#]]]&/@Edges[g3]//FullSimplify

vv=Join[
Table[Csc[(π/5)]Through[{Cos,Sin}[2π (k+1/4)/5+4π/5]],{k,5}],
Table[R Csc[(π/5)]Through[{Cos,Sin}[2π (k+1/4)/5+4π/5+ϕ]],{k,5}],
Table[(1/2)r Csc[(π/5)]Through[{Cos,Sin}[2π (k+1/4)/5+4π/5+ψ]],{k,5}],
Table[(1/2)Csc[(π/5)]Through[{Cos,Sin}[2π (k+1/4)/5+4π/5+θ]],{k,5}]
]/.ϕ->π/100

n[x_,y_]:=#.#&[vv[[x]]-vv[[y]]]

FindRoot

soln=FindRoot[{
(*n[9,14]==1,*)
n[4,10]==2^2,
n[20,15]==1,
n[10,15]==1,
n[14,8]==1 (* added *)
},{θ,-1.8,-1.9},{ψ,-.68,-.7},
{r,1.1,1.2},{R,.94,.96},MaxIterations->10^4,WorkingPrecision->50]

Select[Edges[g3],Norm[Subtract@@Vertices[g3][[#]]]<.95&]

vnew=vv/.soln;

ShowLabeledGraph[g4=ChangeVertices[g3,vnew],VertexStyle->Disk[.01]]

s=Norm[Subtract@@vnew[[#]]]&/@Edges[g4]//FullSimplify

ToCommonEdges[g4,"DodecahedralGraph"]//N

Equations

{ϕ->π/100,
θ->-1.70560978792577961038601596146083756671243650575033830256890571236629180161806`50.,ψ->-0.59690260418206071530790224282310554799754850595296608734085729074044126728344`50.,r->1.20786944362003084727646891808969176051871599141625694902742932477722499564001`50.,R->0.95707815686611063222867711536412461546114343529004809765978406494528295603317`50.}

Through[{Cos,Sin}[-1.70560978792577961038601596146083756671243650575033830256890571236629180161806`50.]]

Through[{Cos,Sin}[-0.59690260418206071530790224282310554799754850595296608734085729074044126728344`50]]

eqn=FullSimplify[TrigExpand@{
n[4,10]==2^2,
n[20,15]==1,
n[10,15]==1,
n[14,8]==1
}/.{Cos[θ]->u,Sin[θ]->-(Sqrt[1-(u^2)]),Cos[ψ]->v,Sin[ψ]->-(Sqrt[1-(v^2)])}]

NSolve π/100

soln=NSolve[eqn,{u,v,r,R},Reals]

soln={{R->-0.3990959447876521`,v->0.8270805742745618`,r->0.4321405727249934`,u->-0.7343747174881446`},{R->-0.3990959447876521`,v->0.8270805742745606`,r->-1.7236421796023367`,u->-0.9922859882503167`},{R->0.9570781568661106`,v->0.827080574274562`,r->1.8893005317789116`,u->0.39583991014193765`},{R->0.9570781568661106`,v->0.827080574274562`,r->1.8893005317789116`,u->0.9995502986010655`},{R->0.9570781568661106`,v->0.827080574274562`,r->1.2078694436200312`,u->-0.13440546705303744`}};

soln[[-1]]

{θ->ArcCos[u],ψ->ArcCos[v]}/.soln[[-1]]

r=.

vv/.{θ->ArcCos[u],ψ->ArcCos[v]}/.soln[[1]]

e=Select[Subsets[Range[20],{2}],Abs[Round[d=EuclideanDistance@@(vv[[#]]/.{θ->-ArcCos[u],ψ->-ArcCos[v]}/.soln[[-1]])]-d]<=1*^-2&]

e2={{1,2},{1,5},{1,7},{2,3},{2,8},{3,4},{3,9},{4,5},{4,10},{5,6},{6,11},{6,12},{7,12},{7,13},{8,13},{8,14},{9,14},{9,15},{10,11},{10,15},{11,16},{12,17},{13,18},{14,19},{15,20},{16,17},{16,20},{17,18},{18,19},{19,20}};

vc=(vv/.{θ->-ArcCos[u],ψ->-ArcCos[v]}/.soln[[-1]])

g=Graph[Range[20],UndirectedEdge@@@e2,VertexCoordinates->vc,VertexLabels->"Name"]

RecognizeGraph[g]

EdgeLengths[g]//FullSimplify

NSolve ArcSin[1/32]

π/100.

Convergents[Sin[(π/100)],5]

vv=Join[
Table[Csc[(π/5)]Through[{Cos,Sin}[2π (k+1/4)/5+4π/5]],{k,5}],
Table[R Csc[(π/5)]Through[{Cos,Sin}[2π (k+1/4)/5+4π/5+ϕ]],{k,5}],
Table[(1/2)r Csc[(π/5)]Through[{Cos,Sin}[2π (k+1/4)/5+4π/5+ψ]],{k,5}],
Table[(1/2)Csc[(π/5)]Through[{Cos,Sin}[2π (k+1/4)/5+4π/5+θ]],{k,5}]
]/.ϕ->ArcSin[1/32]

eqn=FullSimplify[TrigExpand@{
n[4,10]==2^2,
n[20,15]==1,
n[10,15]==1,
n[14,8]==1
}/.{Cos[θ]->u,Sin[θ]->-(Sqrt[1-(u^2)]),Cos[ψ]->v,Sin[ψ]->-(Sqrt[1-(v^2)])}]

soln=NSolve[eqn,{u,v,r,R},Reals]

vc=(vv/.{θ->-ArcCos[u],ψ->-ArcCos[v]}/.soln[[-1]])

g=Graph[Range[20],UndirectedEdge@@@e2,VertexCoordinates->vc,VertexLabels->"Name"]

RecognizeGraph[g]

EdgeLengths[g]//FullSimplify

Solve

eqn//Column

(soln=Solve[eqn,{u,v,r,R},Reals])//Timing

Solve[(Sqrt[1023]) (-1+(Sqrt[5])) R==32 (-3+(Sqrt[5]))+R ((Sqrt[2 (5+(Sqrt[5]))])+64 R),R,Reals]//FullSimplify

N[%]

NSolve[eqn/.R->(1/128) ((Sqrt[1023]) (-1+(Sqrt[5]))-(Sqrt[2 (5+(Sqrt[5]))])+2 (Sqrt[7681-2559 (Sqrt[5])-(Sqrt[2046 (5-(Sqrt[5]))])])),{u,v,r},Reals]

NSolve[Append[eqn,u<0]/.R->(1/128) ((Sqrt[1023]) (-1+(Sqrt[5]))-(Sqrt[2 (5+(Sqrt[5]))])+2 (Sqrt[7681-2559 (Sqrt[5])-(Sqrt[2046 (5-(Sqrt[5]))])])),{u,v,r},Reals]

Solve[Append[eqn,u<0]/.R->(1/128) ((Sqrt[1023]) (-1+(Sqrt[5]))-(Sqrt[2 (5+(Sqrt[5]))])+2 (Sqrt[7681-2559 (Sqrt[5])-(Sqrt[2046 (5-(Sqrt[5]))])])),{u,v,r},Reals]

NSolve[{
 (1-(u^2) )(1-(v^2))==((((1/2)(((3-(Sqrt[5]))/(2r))-r)+u v))^2),
(r^2) (R^2) (1-(v^2))==(((20+(Sqrt[1023]) r R v-(8 (r^2)+4 ((Sqrt[5])+8 (R^2)))))^2),
16 (-5+(Sqrt[5])+8 (R^2))+r (32 r-((Sqrt[1023]) (-1+(Sqrt[5]))+(Sqrt[2 (5+(Sqrt[5]))])) R v+(-1+(Sqrt[5])-(Sqrt[2046 (5+(Sqrt[5]))])) R (Sqrt[1-(v^2)]))==0
}/.R->(1/128) ((Sqrt[1023]) (-1+(Sqrt[5]))-(Sqrt[2 (5+(Sqrt[5]))])+2 (Sqrt[7681-2559 (Sqrt[5])-(Sqrt[2046 (5-(Sqrt[5]))])])),{u,v,r},Reals]

NSolve[{
 (1-(u^2) )(1-(v^2))==((((1/2)(((3-(Sqrt[5]))/(2r))-r)+u v))^2),
(r^2) (R^2) (1-(v^2))==(((20+(Sqrt[1023]) r R v-(8 (r^2)+4 ((Sqrt[5])+8 (R^2)))))^2),
 (R^2) (1-(v^2))==(((((((-16 (-5+(Sqrt[5])+8 (R^2)))/r)- 32 r+((Sqrt[1023]) (-1+(Sqrt[5]))+(Sqrt[2 (5+(Sqrt[5]))])) R v)/((-1+(Sqrt[5])-(Sqrt[2046 (5+(Sqrt[5]))]))))))^2)
}/.R->(1/128) ((Sqrt[1023]) (-1+(Sqrt[5]))-(Sqrt[2 (5+(Sqrt[5]))])+2 (Sqrt[7681-2559 (Sqrt[5])-(Sqrt[2046 (5-(Sqrt[5]))])])),{u,v,r},Reals]

FullSimplify[(((((((-16 (-5+(Sqrt[5])+8 (R^2)))/r)- 32 r+((Sqrt[1023]) (-1+(Sqrt[5]))+(Sqrt[2 (5+(Sqrt[5]))])) R v)/((-1+(Sqrt[5])-(Sqrt[2046 (5+(Sqrt[5]))]))))))^2)]

NSolve[{
 (1-(u^2) )(1-(v^2))==((((1/2)(((3-(Sqrt[5]))/(2r))-r)+u v))^2),
(r^2) (R^2) (1-(v^2))==(((20+(Sqrt[1023]) r R v-(8 (r^2)+4 ((Sqrt[5])+8 (R^2)))))^2),
 (R^2) (1-(v^2))==((((32 r+((16 (-5+(Sqrt[5])+8 (R^2)))/r)-((Sqrt[1023]) (-1+(Sqrt[5]))+(Sqrt[2 (5+(Sqrt[5]))])) R v))^2)/(((1-(Sqrt[5])+(Sqrt[2046 (5+(Sqrt[5]))])))^2)),
v>0,u<0,r>1.2
}/.R->(1/128) ((Sqrt[1023]) (-1+(Sqrt[5]))-(Sqrt[2 (5+(Sqrt[5]))])+2 (Sqrt[7681-2559 (Sqrt[5])-(Sqrt[2046 (5-(Sqrt[5]))])])),{u,v,r},Reals]

{v->0.8269901591903418`,r->1.2090696436948747`,u->-0.1337265555509568`}}

Solve[{
 (1-(u^2) )(1-(v^2))==((((1/2)(((3-(Sqrt[5]))/(2r))-r)+u v))^2),
(r^2) (R^2) (1-(v^2))==(((20+(Sqrt[1023]) r R v-(8 (r^2)+4 ((Sqrt[5])+8 (R^2)))))^2),
(((1-(Sqrt[5])+(Sqrt[2046 (5+(Sqrt[5]))])))^2) (R^2) (1-(v^2))==(((32 r+((16 (-5+(Sqrt[5])+8 (R^2)))/r)-((Sqrt[1023]) (-1+(Sqrt[5]))+(Sqrt[2 (5+(Sqrt[5]))])) R v))^2),
v>0,u<0,r>1.2
}/.R->(1/128) ((Sqrt[1023]) (-1+(Sqrt[5]))-(Sqrt[2 (5+(Sqrt[5]))])+2 (Sqrt[7681-2559 (Sqrt[5])-(Sqrt[2046 (5-(Sqrt[5]))])])),{u,v,r},Reals]

FullSimplify@RootReduce[{
 (1-(u^2) )(1-(v^2))==((((1/2)(((3-(Sqrt[5]))/(2r))-r)+u v))^2),
(r^2) (R^2) (1-(v^2))==(((20+(Sqrt[1023]) r R v-(8 (r^2)+4 ((Sqrt[5])+8 (R^2)))))^2),
(((1-(Sqrt[5])+(Sqrt[2046 (5+(Sqrt[5]))])))^2) (R^2) (1-(v^2))==(((32 r+((16 (-5+(Sqrt[5])+8 (R^2)))/r)-((Sqrt[1023]) (-1+(Sqrt[5]))+(Sqrt[2 (5+(Sqrt[5]))])) R v))^2)
}/.R->(1/128) ((Sqrt[1023]) (-1+(Sqrt[5]))-(Sqrt[2 (5+(Sqrt[5]))])+2 (Sqrt[7681-2559 (Sqrt[5])-(Sqrt[2046 (5-(Sqrt[5]))])]))]

(eqn3={16 (r^2)(-1+(u^2)) (-1+(v^2))==(((-3+(Sqrt[5])+2 r (r-2 u v)))^2),(r^2) (-1+(v^2)) (NumericalApproximation[Root, (RowDefault["-0.916", FEExpressionsNumericalApproximationElider]), Root[4294967296+77493960704 #1+486106071040 ((#1)^2)+1316531731456 ((#1)^3)+1807078258689 ((#1)^4)+1316021862656 ((#1)^5)+485599805440 ((#1)^6)+77326188544 ((#1)^7)+4294967296 ((#1)^8)&,6], ])==(((-8 (r^2)+(NumericalApproximation[Root, (RowDefault["-18.3", FEExpressionsNumericalApproximationElider]), Root[23512593043456+11096713075712 #1-866613204800 ((#1)^2)-594154633808 ((#1)^3)+12972975489 ((#1)^4)+5445419528 ((#1)^5)+204128320 ((#1)^6)+1704448 ((#1)^7)+4096 ((#1)^8)&,4], ])+r v (NumericalApproximation[Root, (RowDefault["30.6", FEExpressionsNumericalApproximationElider]), Root[71776531109707776+4486033194356736 #1-492783949380096 ((#1)^2)-13087004217408 ((#1)^3)+1371707537409 ((#1)^4)-12876492816 ((#1)^5)-469565184 ((#1)^6)+4190208 ((#1)^7)+65536 ((#1)^8)&,5], ])))^2),(r^2)(-1+(v^2)) (NumericalApproximation[Root, (RowDefault["-1.33", FEExpressionsNumericalApproximationElider, " × ", (Superscript["10", "4"])]), Root[3194814997072315910768797521339744256+4695788797132717630576692857667584 #1+2643725544601800570663830814720 ((#1)^2)+731760025692743188537716736 ((#1)^3)+108432958666577573375489 ((#1)^4)+8775149549656432576 ((#1)^5)+379805682501120 ((#1)^6)+8071921664 ((#1)^7)+65536 ((#1)^8)&,4], ])==(((32 (r^2)+(NumericalApproximation[Root, (RowDefault["73.1", FEExpressionsNumericalApproximationElider]), Root[6019223819124736-710189636845568 #1-13865811276800 ((#1)^2)+2376618535232 ((#1)^3)+12972975489 ((#1)^4)-1361354882 ((#1)^5)+12758020 ((#1)^6)-26632 ((#1)^7)+16 ((#1)^8)&,5], ])+v r (NumericalApproximation[Root, (RowDefault["-41.5", FEExpressionsNumericalApproximationElider]), Root[70380813052542976-6706695785938944 #1-57086057619456 ((#1)^2)+11464396046272 ((#1)^3)+86904077569 ((#1)^4)-2812789776 ((#1)^5)-3215264 ((#1)^6)+98048 ((#1)^7)+256 ((#1)^8)&,3], ])))^2)})//Column

NSolve[Join[eqn3,{v>0,u<0,r>1.2}],{u,v,r},Reals]

Take[eqn3,-2]

Solve[Join[
{
(r^2) (-1+(v^2)) (NumericalApproximation[Root, (RowDefault["-0.916", FEExpressionsNumericalApproximationElider]), Root[4294967296+77493960704 #1+486106071040 ((#1)^2)+1316531731456 ((#1)^3)+1807078258689 ((#1)^4)+1316021862656 ((#1)^5)+485599805440 ((#1)^6)+77326188544 ((#1)^7)+4294967296 ((#1)^8)&,6], ])==(((-8 (r^2)+(NumericalApproximation[Root, (RowDefault["-18.3", FEExpressionsNumericalApproximationElider]), Root[23512593043456+11096713075712 #1-866613204800 ((#1)^2)-594154633808 ((#1)^3)+12972975489 ((#1)^4)+5445419528 ((#1)^5)+204128320 ((#1)^6)+1704448 ((#1)^7)+4096 ((#1)^8)&,4], ])+r v (NumericalApproximation[Root, (RowDefault["30.6", FEExpressionsNumericalApproximationElider]), Root[71776531109707776+4486033194356736 #1-492783949380096 ((#1)^2)-13087004217408 ((#1)^3)+1371707537409 ((#1)^4)-12876492816 ((#1)^5)-469565184 ((#1)^6)+4190208 ((#1)^7)+65536 ((#1)^8)&,5], ])))^2),
(r^2) (-1+(v^2)) (NumericalApproximation[Root, (RowDefault["-1.33", FEExpressionsNumericalApproximationElider, " × ", (Superscript["10", "4"])]), Root[3194814997072315910768797521339744256+4695788797132717630576692857667584 #1+2643725544601800570663830814720 ((#1)^2)+731760025692743188537716736 ((#1)^3)+108432958666577573375489 ((#1)^4)+8775149549656432576 ((#1)^5)+379805682501120 ((#1)^6)+8071921664 ((#1)^7)+65536 ((#1)^8)&,4], ])==(((32 (r^2)+(NumericalApproximation[Root, (RowDefault["73.1", FEExpressionsNumericalApproximationElider]), Root[6019223819124736-710189636845568 #1-13865811276800 ((#1)^2)+2376618535232 ((#1)^3)+12972975489 ((#1)^4)-1361354882 ((#1)^5)+12758020 ((#1)^6)-26632 ((#1)^7)+16 ((#1)^8)&,5], ])+r v (NumericalApproximation[Root, (RowDefault["-41.5", FEExpressionsNumericalApproximationElider]), Root[70380813052542976-6706695785938944 #1-57086057619456 ((#1)^2)+11464396046272 ((#1)^3)+86904077569 ((#1)^4)-2812789776 ((#1)^5)-3215264 ((#1)^6)+98048 ((#1)^7)+256 ((#1)^8)&,3], ])))^2)
},{v>0,r>1.2}],{r,v},Reals]

gbr=GroebnerBasis[{
(r^2) (-1+(v^2)) (NumericalApproximation[Root, (RowDefault["-0.916", FEExpressionsNumericalApproximationElider]), Root[4294967296+77493960704 #1+486106071040 ((#1)^2)+1316531731456 ((#1)^3)+1807078258689 ((#1)^4)+1316021862656 ((#1)^5)+485599805440 ((#1)^6)+77326188544 ((#1)^7)+4294967296 ((#1)^8)&,6], ])-(((-8 (r^2)+(NumericalApproximation[Root, (RowDefault["-18.3", FEExpressionsNumericalApproximationElider]), Root[23512593043456+11096713075712 #1-866613204800 ((#1)^2)-594154633808 ((#1)^3)+12972975489 ((#1)^4)+5445419528 ((#1)^5)+204128320 ((#1)^6)+1704448 ((#1)^7)+4096 ((#1)^8)&,4], ])+r v (NumericalApproximation[Root, (RowDefault["30.6", FEExpressionsNumericalApproximationElider]), Root[71776531109707776+4486033194356736 #1-492783949380096 ((#1)^2)-13087004217408 ((#1)^3)+1371707537409 ((#1)^4)-12876492816 ((#1)^5)-469565184 ((#1)^6)+4190208 ((#1)^7)+65536 ((#1)^8)&,5], ])))^2),
(r^2) (-1+(v^2)) (NumericalApproximation[Root, (RowDefault["-1.33", FEExpressionsNumericalApproximationElider, " × ", (Superscript["10", "4"])]), Root[3194814997072315910768797521339744256+4695788797132717630576692857667584 #1+2643725544601800570663830814720 ((#1)^2)+731760025692743188537716736 ((#1)^3)+108432958666577573375489 ((#1)^4)+8775149549656432576 ((#1)^5)+379805682501120 ((#1)^6)+8071921664 ((#1)^7)+65536 ((#1)^8)&,4], ])-(((32 (r^2)+(NumericalApproximation[Root, (RowDefault["73.1", FEExpressionsNumericalApproximationElider]), Root[6019223819124736-710189636845568 #1-13865811276800 ((#1)^2)+2376618535232 ((#1)^3)+12972975489 ((#1)^4)-1361354882 ((#1)^5)+12758020 ((#1)^6)-26632 ((#1)^7)+16 ((#1)^8)&,5], ])+r v (NumericalApproximation[Root, (RowDefault["-41.5", FEExpressionsNumericalApproximationElider]), Root[70380813052542976-6706695785938944 #1-57086057619456 ((#1)^2)+11464396046272 ((#1)^3)+86904077569 ((#1)^4)-2812789776 ((#1)^5)-3215264 ((#1)^6)+98048 ((#1)^7)+256 ((#1)^8)&,3], ])))^2)
},{r},{v},MonomialOrder->EliminationOrder];

Length[gbr]

NSolve[gbr[[1]]==0]

Solve[gbr[[1]]==0&&12/10<r<13/10]

N[%238]

gbv=GroebnerBasis[{
(r^2) (-1+(v^2)) (NumericalApproximation[Root, (RowDefault["-0.916", FEExpressionsNumericalApproximationElider]), Root[4294967296+77493960704 #1+486106071040 ((#1)^2)+1316531731456 ((#1)^3)+1807078258689 ((#1)^4)+1316021862656 ((#1)^5)+485599805440 ((#1)^6)+77326188544 ((#1)^7)+4294967296 ((#1)^8)&,6], ])-(((-8 (r^2)+(NumericalApproximation[Root, (RowDefault["-18.3", FEExpressionsNumericalApproximationElider]), Root[23512593043456+11096713075712 #1-866613204800 ((#1)^2)-594154633808 ((#1)^3)+12972975489 ((#1)^4)+5445419528 ((#1)^5)+204128320 ((#1)^6)+1704448 ((#1)^7)+4096 ((#1)^8)&,4], ])+r v (NumericalApproximation[Root, (RowDefault["30.6", FEExpressionsNumericalApproximationElider]), Root[71776531109707776+4486033194356736 #1-492783949380096 ((#1)^2)-13087004217408 ((#1)^3)+1371707537409 ((#1)^4)-12876492816 ((#1)^5)-469565184 ((#1)^6)+4190208 ((#1)^7)+65536 ((#1)^8)&,5], ])))^2),
(r^2) (-1+(v^2)) (NumericalApproximation[Root, (RowDefault["-1.33", FEExpressionsNumericalApproximationElider, " × ", (Superscript["10", "4"])]), Root[3194814997072315910768797521339744256+4695788797132717630576692857667584 #1+2643725544601800570663830814720 ((#1)^2)+731760025692743188537716736 ((#1)^3)+108432958666577573375489 ((#1)^4)+8775149549656432576 ((#1)^5)+379805682501120 ((#1)^6)+8071921664 ((#1)^7)+65536 ((#1)^8)&,4], ])-(((32 (r^2)+(NumericalApproximation[Root, (RowDefault["73.1", FEExpressionsNumericalApproximationElider]), Root[6019223819124736-710189636845568 #1-13865811276800 ((#1)^2)+2376618535232 ((#1)^3)+12972975489 ((#1)^4)-1361354882 ((#1)^5)+12758020 ((#1)^6)-26632 ((#1)^7)+16 ((#1)^8)&,5], ])+r v (NumericalApproximation[Root, (RowDefault["-41.5", FEExpressionsNumericalApproximationElider]), Root[70380813052542976-6706695785938944 #1-57086057619456 ((#1)^2)+11464396046272 ((#1)^3)+86904077569 ((#1)^4)-2812789776 ((#1)^5)-3215264 ((#1)^6)+98048 ((#1)^7)+256 ((#1)^8)&,3], ])))^2)
},{v},{r},MonomialOrder->EliminationOrder];

NSolve[gbv[[1]]==0]

Solve[gbv[[1]]==0&&v<-81/100]

Solve[16 (r^2) (-1+(u^2)) (-1+(v^2))==(((-3+(Sqrt[5])+2 r (r-2 u v)))^2)/.{r->(NumericalApproximation[Root, (RowDefault["1.21", FEExpressionsNumericalApproximationElider]), Root[{70380813052542976-6706695785938944 #1-57086057619456 ((#1)^2)+11464396046272 ((#1)^3)+86904077569 ((#1)^4)-2812789776 ((#1)^5)-3215264 ((#1)^6)+98048 ((#1)^7)+256 ((#1)^8)&,«5»,1024 ((#1)^6) ((#2)^3) ((#5)^2) ((#6)^2)+4096 ((#1)^5) ((#2)^4) ((#5)^2) ((#6)^2)-3072 ((#1)^4) ((#2)^3) #3 ((#5)^2) ((#6)^2)-8192 ((#1)^3) ((#2)^4) #3 ((#5)^2) ((#6)^2)+3072 ((#1)^2) ((#2)^3) ((#3)^2) ((#5)^2) ((#6)^2)+4096 #1 ((#2)^4) ((#3)^2) ((#5)^2) ((#6)^2)-1024 ((#2)^3) ((#3)^3) ((#5)^2) ((#6)^2)-4096 ((#1)^5) ((#2)^2) #4 ((#5)^2) ((#6)^2)+«522»+1610612736 ((#1)^3) ((#2)^2) #3 #4 ((#7)^8)+6442450944 ((#1)^2) ((#2)^3) #3 #4 ((#7)^8)-805306368 #1 ((#2)^2) ((#3)^2) #4 ((#7)^8)+3221225472 ((#1)^4) #2 ((#4)^2) ((#7)^8)+12884901888 ((#1)^3) ((#2)^2) ((#4)^2) ((#7)^8)-3221225472 ((#1)^2) #2 #3 ((#4)^2) ((#7)^8)-4294967296 ((#1)^3) ((#4)^3) ((#7)^8)&},{3,5,4,6,4,5,6}], ]),v->(NumericalApproximation[Root, (RowDefault["-0.827", FEExpressionsNumericalApproximationElider]), Root[{6019223819124736-710189636845568 #1-13865811276800 ((#1)^2)+2376618535232 ((#1)^3)+12972975489 ((#1)^4)-1361354882 ((#1)^5)+12758020 ((#1)^6)-26632 ((#1)^7)+16 ((#1)^8)&,«11»+98048 ((#2)^7)+256 ((#2)^8)&,«1»&,«1»&,«1»&,4294967296+«7»+4294967296 ((#6)^8)&,4096 ((#1)^4) #2+16384 ((#1)^4) #3+32768 ((#1)^3) #2 #4+131072 ((#1)^3) #3 #4+65536 ((#1)^2) #2 ((#4)^2)+262144 ((#1)^2) #3 ((#4)^2)+128 ((#1)^3) #2 #5+512 ((#1)^3) #3 #5+512 ((#1)^2) #2 #4 #5+2048 ((#1)^2) #3 #4 #5+((#1)^2) #2 ((#5)^2)+«39»+16384 #2 ((#3)^2) ((#4)^2) #6 ((#7)^2)+4 ((#1)^2) ((#2)^5) ((#7)^4)+16 ((#1)^2) ((#2)^4) #3 ((#7)^4)-128 #1 ((#2)^4) #3 #4 ((#7)^4)-512 #1 ((#2)^3) ((#3)^2) #4 ((#7)^4)+1024 ((#2)^3) ((#3)^2) ((#4)^2) ((#7)^4)+4096 ((#2)^2) ((#3)^3) ((#4)^2) ((#7)^4)-4 ((#1)^2) ((#2)^3) #5 ((#7)^4)-16 ((#1)^2) ((#2)^2) #3 #5 ((#7)^4)+128 #1 ((#2)^2) #3 #4 #5 ((#7)^4)+512 #1 #2 ((#3)^2) #4 #5 ((#7)^4)-1024 #2 ((#3)^2) ((#4)^2) #5 ((#7)^4)-4096 ((#3)^3) ((#4)^2) #5 ((#7)^4)&},{5,3,«3»,6,1}], ])},u]

N[%]

{{u->-(1/(32 ((NumericalApproximation[Root, (RowDefault["1.21", FEExpressionsNumericalApproximationElider]), Root[{70380813052542976-6706695785938944 #1-57086057619456 ((#1)^2)+11464396046272 ((#1)^3)+86904077569 ((#1)^4)-2812789776 ((#1)^5)-3215264 ((#1)^6)+98048 ((#1)^7)+256 ((#1)^8)&,«5»,1024 ((#1)^6) ((#2)^3) ((#5)^2) ((#6)^2)+4096 ((#1)^5) ((#2)^4) ((#5)^2) ((#6)^2)-3072 ((#1)^4) ((#2)^3) #3 ((#5)^2) ((#6)^2)-8192 ((#1)^3) ((#2)^4) #3 ((#5)^2) ((#6)^2)+3072 ((#1)^2) ((#2)^3) ((#3)^2) ((#5)^2) ((#6)^2)+4096 #1 ((#2)^4) ((#3)^2) ((#5)^2) ((#6)^2)-1024 ((#2)^3) ((#3)^3) ((#5)^2) ((#6)^2)-4096 ((#1)^5) ((#2)^2) #4 ((#5)^2) ((#6)^2)+«522»+1610612736 ((#1)^3) ((#2)^2) #3 #4 ((#7)^8)+6442450944 ((#1)^2) ((#2)^3) #3 #4 ((#7)^8)-805306368 #1 ((#2)^2) ((#3)^2) #4 ((#7)^8)+3221225472 ((#1)^4) #2 ((#4)^2) ((#7)^8)+12884901888 ((#1)^3) ((#2)^2) ((#4)^2) ((#7)^8)-3221225472 ((#1)^2) #2 #3 ((#4)^2) ((#7)^8)-4294967296 ((#1)^3) ((#4)^3) ((#7)^8)&},{3,5,4,6,4,5,6}], ])^2)))(-24 (NumericalApproximation[Root, (RowDefault["-0.827", FEExpressionsNumericalApproximationElider]), Root[{6019223819124736-710189636845568 #1-13865811276800 ((#1)^2)+2376618535232 ((#1)^3)+12972975489 ((#1)^4)-1361354882 ((#1)^5)+12758020 ((#1)^6)-26632 ((#1)^7)+16 ((#1)^8)&,«11»+98048 ((#2)^7)+256 ((#2)^8)&,«1»&,«1»&,«1»&,4294967296+«7»+4294967296 ((#6)^8)&,4096 ((#1)^4) #2+16384 ((#1)^4) #3+32768 ((#1)^3) #2 #4+131072 ((#1)^3) #3 #4+65536 ((#1)^2) #2 ((#4)^2)+262144 ((#1)^2) #3 ((#4)^2)+128 ((#1)^3) #2 #5+512 ((#1)^3) #3 #5+512 ((#1)^2) #2 #4 #5+2048 ((#1)^2) #3 #4 #5+((#1)^2) #2 ((#5)^2)+«39»+16384 #2 ((#3)^2) ((#4)^2) #6 ((#7)^2)+4 ((#1)^2) ((#2)^5) ((#7)^4)+16 ((#1)^2) ((#2)^4) #3 ((#7)^4)-128 #1 ((#2)^4) #3 #4 ((#7)^4)-512 #1 ((#2)^3) ((#3)^2) #4 ((#7)^4)+1024 ((#2)^3) ((#3)^2) ((#4)^2) ((#7)^4)+4096 ((#2)^2) ((#3)^3) ((#4)^2) ((#7)^4)-4 ((#1)^2) ((#2)^3) #5 ((#7)^4)-16 ((#1)^2) ((#2)^2) #3 #5 ((#7)^4)+128 #1 ((#2)^2) #3 #4 #5 ((#7)^4)+512 #1 #2 ((#3)^2) #4 #5 ((#7)^4)-1024 #2 ((#3)^2) ((#4)^2) #5 ((#7)^4)-4096 ((#3)^3) ((#4)^2) #5 ((#7)^4)&},{5,3,«3»,6,1}], ]) (NumericalApproximation[Root, (RowDefault["1.21", FEExpressionsNumericalApproximationElider]), Root[{70380813052542976-6706695785938944 #1-57086057619456 ((#1)^2)+11464396046272 ((#1)^3)+86904077569 ((#1)^4)-2812789776 ((#1)^5)-3215264 ((#1)^6)+98048 ((#1)^7)+256 ((#1)^8)&,«5»,1024 ((#1)^6) ((#2)^3) ((#5)^2) ((#6)^2)+4096 ((#1)^5) ((#2)^4) ((#5)^2) ((#6)^2)-3072 ((#1)^4) ((#2)^3) #3 ((#5)^2) ((#6)^2)-8192 ((#1)^3) ((#2)^4) #3 ((#5)^2) ((#6)^2)+3072 ((#1)^2) ((#2)^3) ((#3)^2) ((#5)^2) ((#6)^2)+4096 #1 ((#2)^4) ((#3)^2) ((#5)^2) ((#6)^2)-1024 ((#2)^3) ((#3)^3) ((#5)^2) ((#6)^2)-4096 ((#1)^5) ((#2)^2) #4 ((#5)^2) ((#6)^2)+«522»+1610612736 ((#1)^3) ((#2)^2) #3 #4 ((#7)^8)+6442450944 ((#1)^2) ((#2)^3) #3 #4 ((#7)^8)-805306368 #1 ((#2)^2) ((#3)^2) #4 ((#7)^8)+3221225472 ((#1)^4) #2 ((#4)^2) ((#7)^8)+12884901888 ((#1)^3) ((#2)^2) ((#4)^2) ((#7)^8)-3221225472 ((#1)^2) #2 #3 ((#4)^2) ((#7)^8)-4294967296 ((#1)^3) ((#4)^3) ((#7)^8)&},{3,5,4,6,4,5,6}], ])+8 (Sqrt[5]) (NumericalApproximation[Root, (RowDefault["-0.827", FEExpressionsNumericalApproximationElider]), Root[{6019223819124736-710189636845568 #1-13865811276800 ((#1)^2)+2376618535232 ((#1)^3)+12972975489 ((#1)^4)-1361354882 ((#1)^5)+12758020 ((#1)^6)-26632 ((#1)^7)+16 ((#1)^8)&,«11»+98048 ((#2)^7)+256 ((#2)^8)&,«1»&,«1»&,«1»&,4294967296+«7»+4294967296 ((#6)^8)&,4096 ((#1)^4) #2+16384 ((#1)^4) #3+32768 ((#1)^3) #2 #4+131072 ((#1)^3) #3 #4+65536 ((#1)^2) #2 ((#4)^2)+262144 ((#1)^2) #3 ((#4)^2)+128 ((#1)^3) #2 #5+512 ((#1)^3) #3 #5+512 ((#1)^2) #2 #4 #5+2048 ((#1)^2) #3 #4 #5+((#1)^2) #2 ((#5)^2)+«39»+16384 #2 ((#3)^2) ((#4)^2) #6 ((#7)^2)+4 ((#1)^2) ((#2)^5) ((#7)^4)+16 ((#1)^2) ((#2)^4) #3 ((#7)^4)-128 #1 ((#2)^4) #3 #4 ((#7)^4)-512 #1 ((#2)^3) ((#3)^2) #4 ((#7)^4)+1024 ((#2)^3) ((#3)^2) ((#4)^2) ((#7)^4)+4096 ((#2)^2) ((#3)^3) ((#4)^2) ((#7)^4)-4 ((#1)^2) ((#2)^3) #5 ((#7)^4)-16 ((#1)^2) ((#2)^2) #3 #5 ((#7)^4)+128 #1 ((#2)^2) #3 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FullSimplify[-(1/(32 ((NumericalApproximation[Root, (RowDefault["1.21", FEExpressionsNumericalApproximationElider]), Root[{70380813052542976-6706695785938944 #1-57086057619456 ((#1)^2)+11464396046272 ((#1)^3)+86904077569 ((#1)^4)-2812789776 ((#1)^5)-3215264 ((#1)^6)+98048 ((#1)^7)+256 ((#1)^8)&,«5»,1024 ((#1)^6) ((#2)^3) ((#5)^2) ((#6)^2)+4096 ((#1)^5) ((#2)^4) ((#5)^2) ((#6)^2)-3072 ((#1)^4) ((#2)^3) #3 ((#5)^2) ((#6)^2)-8192 ((#1)^3) ((#2)^4) #3 ((#5)^2) ((#6)^2)+3072 ((#1)^2) ((#2)^3) ((#3)^2) ((#5)^2) ((#6)^2)+4096 #1 ((#2)^4) ((#3)^2) ((#5)^2) ((#6)^2)-1024 ((#2)^3) ((#3)^3) ((#5)^2) ((#6)^2)-4096 ((#1)^5) ((#2)^2) #4 ((#5)^2) ((#6)^2)+«522»+1610612736 ((#1)^3) ((#2)^2) #3 #4 ((#7)^8)+6442450944 ((#1)^2) ((#2)^3) #3 #4 ((#7)^8)-805306368 #1 ((#2)^2) ((#3)^2) #4 ((#7)^8)+3221225472 ((#1)^4) #2 ((#4)^2) ((#7)^8)+12884901888 ((#1)^3) ((#2)^2) ((#4)^2) ((#7)^8)-3221225472 ((#1)^2) #2 #3 ((#4)^2) ((#7)^8)-4294967296 ((#1)^3) ((#4)^3) ((#7)^8)&},{3,5,4,6,4,5,6}], ])^2)))(-24 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((#1)^3) ((#4)^3) ((#7)^8)&},{3,5,4,6,4,5,6}], ])+8 (Sqrt[5]) (NumericalApproximation[Root, (RowDefault["-0.827", FEExpressionsNumericalApproximationElider]), Root[{6019223819124736-710189636845568 #1-13865811276800 ((#1)^2)+2376618535232 ((#1)^3)+12972975489 ((#1)^4)-1361354882 ((#1)^5)+12758020 ((#1)^6)-26632 ((#1)^7)+16 ((#1)^8)&,«11»+98048 ((#2)^7)+256 ((#2)^8)&,«1»&,«1»&,«1»&,4294967296+«7»+4294967296 ((#6)^8)&,4096 ((#1)^4) #2+16384 ((#1)^4) #3+32768 ((#1)^3) #2 #4+131072 ((#1)^3) #3 #4+65536 ((#1)^2) #2 ((#4)^2)+262144 ((#1)^2) #3 ((#4)^2)+128 ((#1)^3) #2 #5+512 ((#1)^3) #3 #5+512 ((#1)^2) #2 #4 #5+2048 ((#1)^2) #3 #4 #5+((#1)^2) #2 ((#5)^2)+«39»+16384 #2 ((#3)^2) ((#4)^2) #6 ((#7)^2)+4 ((#1)^2) ((#2)^5) ((#7)^4)+16 ((#1)^2) ((#2)^4) #3 ((#7)^4)-128 #1 ((#2)^4) #3 #4 ((#7)^4)-512 #1 ((#2)^3) ((#3)^2) #4 ((#7)^4)+1024 ((#2)^3) ((#3)^2) ((#4)^2) ((#7)^4)+4096 ((#2)^2) ((#3)^3) ((#4)^2) ((#7)^4)-4 ((#1)^2) ((#2)^3) #5 ((#7)^4)-16 ((#1)^2) ((#2)^2) #3 #5 ((#7)^4)+128 #1 ((#2)^2) #3 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N[%]

soln=NSolve[Join[eqn3,{v>0,u<0,r>1.2}],{u,v,r},Reals]

vc=(vv/.{θ->-ArcCos[u],ψ->-ArcCos[v]}/.soln[[-1]])/.R->(1/128) ((Sqrt[1023]) (-1+(Sqrt[5]))-(Sqrt[2 (5+(Sqrt[5]))])+2 (Sqrt[7681-2559 (Sqrt[5])-(Sqrt[2046 (5-(Sqrt[5]))])]))//RootReduce//FullSimplify

vc={{-1,-(Sqrt[1+(2/(Sqrt[5]))])},{1,-(Sqrt[1+(2/(Sqrt[5]))])},{(1/2) (1+(Sqrt[5])),(NumericalApproximation[Root, (RowDefault["0.526", FEExpressionsNumericalApproximationElider]), Root[1-5 ((#1)^2)+5 ((#1)^4)&,3], ])},{0,(Sqrt[2+(2/(Sqrt[5]))])},{(1/2) (-1-(Sqrt[5])),(NumericalApproximation[Root, (RowDefault["0.526", FEExpressionsNumericalApproximationElider]), Root[1-5 ((#1)^2)+5 ((#1)^4)&,3], ])},{(NumericalApproximation[Root, (RowDefault["-0.916", FEExpressionsNumericalApproximationElider]), Root[106537612801-212240618245 #1-750776492795 ((#1)^2)+640064865285 ((#1)^3)+2147481021445 ((#1)^4)+643399686400 ((#1)^5)-753293721600 ((#1)^6)-213909504000 ((#1)^7)+107374182400 ((#1)^8)&,4], ]),(NumericalApproximation[Root, (RowDefault["-1.35", FEExpressionsNumericalApproximationElider]), Root[17598220999335736321-539111626071872353265 ((#1)^2)+6269521563436040371285 ((#1)^4)-35734932813943308364825 ((#1)^6)+108820923885449897484825 ((#1)^8)-180651487477897355264000 ((#1)^10)+159824446714883866624000 ((#1)^12)-69355434261505638400000 ((#1)^14)+11529215046068469760000 ((#1)^16)&,2], ])},{(NumericalApproximation[Root, (RowDefault["0.998", FEExpressionsNumericalApproximationElider]), Root[106537612801+215161907195 #1-754115575025 ((#1)^2)-633789772815 ((#1)^3)+2139107152645 ((#1)^4)-645501747200 ((#1)^5)-749941555200 ((#1)^6)+215167795200 ((#1)^7)+107374182400 ((#1)^8)&,6], ]),(NumericalApproximation[Root, (RowDefault["-1.29", FEExpressionsNumericalApproximationElider]), Root[17598220999335736321-540528768140269969925 ((#1)^2)+6269411700808243814485 ((#1)^4)-35627336894415949311925 ((#1)^6)+108386758107061028364825 ((#1)^8)-180110760013281099776000 ((#1)^10)+159827258715871903744000 ((#1)^12)-69535402324740014080000 ((#1)^14)+11529215046068469760000 ((#1)^16)&,2], ])},{(NumericalApproximation[Root, (RowDefault["1.53", FEExpressionsNumericalApproximationElider]), Root[106537612801+426150777605 #1+210571399695 ((#1)^2)-858980052465 ((#1)^3)-532674325755 ((#1)^4)+860669542400 ((#1)^5)+213072281600 ((#1)^6)-429496729600 ((#1)^7)+107374182400 ((#1)^8)&,6], ]),(Sqrt[(NumericalApproximation[Root, (RowDefault["0.304", FEExpressionsNumericalApproximationElider]), Root[17598220999335736321-1100703701881579435525 #1+21366179173843941862485 ((#1)^2)-153773296557369462079925 ((#1)^3)+500538150850720122380825 ((#1)^4)-772147051737011732480000 ((#1)^5)+538130681574451052544000 ((#1)^6)-138170788411447705600000 ((#1)^7)+11529215046068469760000 ((#1)^8)&,4], ])])},{(NumericalApproximation[Root, (RowDefault["-0.0509", FEExpressionsNumericalApproximationElider]), Root[1+5 #1-3835 ((#1)^2)-8965 ((#1)^3)+3927045 ((#1)^4)+4915200 ((#1)^5)-1258291200 ((#1)^6)-419430400 ((#1)^7)+107374182400 ((#1)^8)&,2], ]),(Sqrt[(NumericalApproximation[Root, (RowDefault["2.65", FEExpressionsNumericalApproximationElider]), Root[1199513305477529806561281-9022732048533325377799665 #1+25962141421217984341452885 ((#1)^2)-36712060727845224685260825 ((#1)^3)+27796592096551279797708825 ((#1)^4)-11496063758699253989376000 ((#1)^5)+2543846604318770724864000 ((#1)^6)-276610913191235420160000 ((#1)^7)+11529215046068469760000 ((#1)^8)&,5], ])])},{(NumericalApproximation[Root, (RowDefault["-1.56", FEExpressionsNumericalApproximationElider]), Root[106537612801-426984732160 #1+212240940810 ((#1)^2)+858988220160 ((#1)^3)-532674325755 ((#1)^4)-860661356800 ((#1)^5)+211396198400 ((#1)^6)+428657868800 ((#1)^7)+107374182400 ((#1)^8)&,3], ]),(Sqrt[(NumericalApproximation[Root, (RowDefault["0.207", FEExpressionsNumericalApproximationElider]), Root[17598220999335736321-1075468191904076864020 #1+21281804568408463782510 ((#1)^2)-155921619528698397990500 ((#1)^3)+507419710433470154252825 ((#1)^4)-778608770214113828864000 ((#1)^5)+540290301079358275584000 ((#1)^6)-138530724537916456960000 ((#1)^7)+11529215046068469760000 ((#1)^8)&,3], ])])},{-0.9677481949315356`,-0.348234973437605`},{0.032140502180179956`,-1.0279937517504376`},{0.987612117694377`,-0.2871001053666874`},{0.5782373542562063`,0.8505561284601488`},{-0.6302417791992277`,0.8127727020945811`},{-0.6151465359790604`,0.5875385408830143`},{-0.7488730915297469`,-0.40347972753616523`},{0.15231751215346578`,-0.8369027262719113`},{0.843010491122414`,-0.11375460257732557`},{0.36869162423292756`,0.7665985155023878`}};

g=Graph[Range[20],UndirectedEdge@@@e2,VertexCoordinates->vc,VertexLabels->"Name"]

EdgeLengths[g]//RootReduce

Max[Abs[#-Round[#]]&/@%256]

vc=(vv/.r->(NumericalApproximation[Root, (RowDefault["1.21", FEExpressionsNumericalApproximationElider]), Root[{70380813052542976-6706695785938944 #1-57086057619456 ((#1)^2)+11464396046272 ((#1)^3)+86904077569 ((#1)^4)-2812789776 ((#1)^5)-3215264 ((#1)^6)+98048 ((#1)^7)+256 ((#1)^8)&,«5»,1024 ((#1)^6) ((#2)^3) ((#5)^2) ((#6)^2)+4096 ((#1)^5) ((#2)^4) ((#5)^2) ((#6)^2)-3072 ((#1)^4) ((#2)^3) #3 ((#5)^2) ((#6)^2)-8192 ((#1)^3) ((#2)^4) #3 ((#5)^2) ((#6)^2)+3072 ((#1)^2) ((#2)^3) ((#3)^2) ((#5)^2) ((#6)^2)+4096 #1 ((#2)^4) ((#3)^2) ((#5)^2) ((#6)^2)-1024 ((#2)^3) ((#3)^3) ((#5)^2) ((#6)^2)-4096 ((#1)^5) ((#2)^2) #4 ((#5)^2) ((#6)^2)+«522»+1610612736 ((#1)^3) ((#2)^2) #3 #4 ((#7)^8)+6442450944 ((#1)^2) ((#2)^3) #3 #4 ((#7)^8)-805306368 #1 ((#2)^2) ((#3)^2) #4 ((#7)^8)+3221225472 ((#1)^4) #2 ((#4)^2) ((#7)^8)+12884901888 ((#1)^3) ((#2)^2) ((#4)^2) ((#7)^8)-3221225472 ((#1)^2) #2 #3 ((#4)^2) ((#7)^8)-4294967296 ((#1)^3) ((#4)^3) ((#7)^8)&},{3,5,4,6,4,5,6}], ])/.{θ->-ArcCos[-(1/(32 ((NumericalApproximation[Root, (RowDefault["1.21", FEExpressionsNumericalApproximationElider]), Root[{70380813052542976-6706695785938944 #1-57086057619456 ((#1)^2)+11464396046272 ((#1)^3)+86904077569 ((#1)^4)-2812789776 ((#1)^5)-3215264 ((#1)^6)+98048 ((#1)^7)+256 ((#1)^8)&,«5»,1024 ((#1)^6) ((#2)^3) ((#5)^2) ((#6)^2)+4096 ((#1)^5) ((#2)^4) ((#5)^2) ((#6)^2)-3072 ((#1)^4) ((#2)^3) #3 ((#5)^2) ((#6)^2)-8192 ((#1)^3) ((#2)^4) #3 ((#5)^2) ((#6)^2)+3072 ((#1)^2) ((#2)^3) ((#3)^2) ((#5)^2) ((#6)^2)+4096 #1 ((#2)^4) ((#3)^2) ((#5)^2) ((#6)^2)-1024 ((#2)^3) ((#3)^3) ((#5)^2) ((#6)^2)-4096 ((#1)^5) ((#2)^2) #4 ((#5)^2) ((#6)^2)+«522»+1610612736 ((#1)^3) ((#2)^2) #3 #4 ((#7)^8)+6442450944 ((#1)^2) ((#2)^3) #3 #4 ((#7)^8)-805306368 #1 ((#2)^2) ((#3)^2) #4 ((#7)^8)+3221225472 ((#1)^4) #2 ((#4)^2) ((#7)^8)+12884901888 ((#1)^3) ((#2)^2) ((#4)^2) ((#7)^8)-3221225472 ((#1)^2) #2 #3 ((#4)^2) ((#7)^8)-4294967296 ((#1)^3) ((#4)^3) ((#7)^8)&},{3,5,4,6,4,5,6}], ])^2)))(-24 (NumericalApproximation[Root, (RowDefault["-0.827", FEExpressionsNumericalApproximationElider]), Root[{6019223819124736-710189636845568 #1-13865811276800 ((#1)^2)+2376618535232 ((#1)^3)+12972975489 ((#1)^4)-1361354882 ((#1)^5)+12758020 ((#1)^6)-26632 ((#1)^7)+16 ((#1)^8)&,«11»+98048 ((#2)^7)+256 ((#2)^8)&,«1»&,«1»&,«1»&,4294967296+«7»+4294967296 ((#6)^8)&,4096 ((#1)^4) #2+16384 ((#1)^4) #3+32768 ((#1)^3) #2 #4+131072 ((#1)^3) #3 #4+65536 ((#1)^2) #2 ((#4)^2)+262144 ((#1)^2) #3 ((#4)^2)+128 ((#1)^3) #2 #5+512 ((#1)^3) #3 #5+512 ((#1)^2) #2 #4 #5+2048 ((#1)^2) #3 #4 #5+((#1)^2) #2 ((#5)^2)+«39»+16384 #2 ((#3)^2) ((#4)^2) #6 ((#7)^2)+4 ((#1)^2) ((#2)^5) ((#7)^4)+16 ((#1)^2) ((#2)^4) #3 ((#7)^4)-128 #1 ((#2)^4) #3 #4 ((#7)^4)-512 #1 ((#2)^3) ((#3)^2) #4 ((#7)^4)+1024 ((#2)^3) ((#3)^2) ((#4)^2) ((#7)^4)+4096 ((#2)^2) ((#3)^3) ((#4)^2) ((#7)^4)-4 ((#1)^2) ((#2)^3) #5 ((#7)^4)-16 ((#1)^2) ((#2)^2) #3 #5 ((#7)^4)+128 #1 ((#2)^2) #3 #4 #5 ((#7)^4)+512 #1 #2 ((#3)^2) #4 #5 ((#7)^4)-1024 #2 ((#3)^2) ((#4)^2) #5 ((#7)^4)-4096 ((#3)^3) ((#4)^2) #5 ((#7)^4)&},{5,3,«3»,6,1}], ]) (NumericalApproximation[Root, (RowDefault["1.21", FEExpressionsNumericalApproximationElider]), Root[{70380813052542976-6706695785938944 #1-57086057619456 ((#1)^2)+11464396046272 ((#1)^3)+86904077569 ((#1)^4)-2812789776 ((#1)^5)-3215264 ((#1)^6)+98048 ((#1)^7)+256 ((#1)^8)&,«5»,1024 ((#1)^6) ((#2)^3) ((#5)^2) ((#6)^2)+4096 ((#1)^5) ((#2)^4) ((#5)^2) ((#6)^2)-3072 ((#1)^4) ((#2)^3) #3 ((#5)^2) ((#6)^2)-8192 ((#1)^3) ((#2)^4) #3 ((#5)^2) ((#6)^2)+3072 ((#1)^2) ((#2)^3) ((#3)^2) ((#5)^2) ((#6)^2)+4096 #1 ((#2)^4) ((#3)^2) ((#5)^2) ((#6)^2)-1024 ((#2)^3) ((#3)^3) ((#5)^2) ((#6)^2)-4096 ((#1)^5) ((#2)^2) #4 ((#5)^2) ((#6)^2)+«522»+1610612736 ((#1)^3) ((#2)^2) #3 #4 ((#7)^8)+6442450944 ((#1)^2) ((#2)^3) #3 #4 ((#7)^8)-805306368 #1 ((#2)^2) ((#3)^2) #4 ((#7)^8)+3221225472 ((#1)^4) #2 ((#4)^2) ((#7)^8)+12884901888 ((#1)^3) ((#2)^2) ((#4)^2) ((#7)^8)-3221225472 ((#1)^2) #2 #3 ((#4)^2) ((#7)^8)-4294967296 ((#1)^3) ((#4)^3) ((#7)^8)&},{3,5,4,6,4,5,6}], ])+8 (Sqrt[5]) (NumericalApproximation[Root, (RowDefault["-0.827", FEExpressionsNumericalApproximationElider]), Root[{6019223819124736-710189636845568 #1-13865811276800 ((#1)^2)+2376618535232 ((#1)^3)+12972975489 ((#1)^4)-1361354882 ((#1)^5)+12758020 ((#1)^6)-26632 ((#1)^7)+16 ((#1)^8)&,«11»+98048 ((#2)^7)+256 ((#2)^8)&,«1»&,«1»&,«1»&,4294967296+«7»+4294967296 ((#6)^8)&,4096 ((#1)^4) #2+16384 ((#1)^4) #3+32768 ((#1)^3) #2 #4+131072 ((#1)^3) #3 #4+65536 ((#1)^2) #2 ((#4)^2)+262144 ((#1)^2) #3 ((#4)^2)+128 ((#1)^3) #2 #5+512 ((#1)^3) #3 #5+512 ((#1)^2) #2 #4 #5+2048 ((#1)^2) #3 #4 #5+((#1)^2) #2 ((#5)^2)+«39»+16384 #2 ((#3)^2) ((#4)^2) #6 ((#7)^2)+4 ((#1)^2) ((#2)^5) ((#7)^4)+16 ((#1)^2) ((#2)^4) #3 ((#7)^4)-128 #1 ((#2)^4) #3 #4 ((#7)^4)-512 #1 ((#2)^3) ((#3)^2) #4 ((#7)^4)+1024 ((#2)^3) ((#3)^2) ((#4)^2) ((#7)^4)+4096 ((#2)^2) ((#3)^3) ((#4)^2) ((#7)^4)-4 ((#1)^2) ((#2)^3) #5 ((#7)^4)-16 ((#1)^2) ((#2)^2) #3 #5 ((#7)^4)+128 #1 ((#2)^2) #3 #4 #5 ((#7)^4)+512 #1 #2 ((#3)^2) #4 #5 ((#7)^4)-1024 #2 ((#3)^2) ((#4)^2) #5 ((#7)^4)-4096 ((#3)^3) ((#4)^2) #5 ((#7)^4)&},{5,3,«3»,6,1}], ]) (NumericalApproximation[Root, (RowDefault["1.21", FEExpressionsNumericalApproximationElider]), Root[{70380813052542976-6706695785938944 #1-57086057619456 ((#1)^2)+11464396046272 ((#1)^3)+86904077569 ((#1)^4)-2812789776 ((#1)^5)-3215264 ((#1)^6)+98048 ((#1)^7)+256 ((#1)^8)&,«5»,1024 ((#1)^6) ((#2)^3) ((#5)^2) ((#6)^2)+4096 ((#1)^5) ((#2)^4) ((#5)^2) ((#6)^2)-3072 ((#1)^4) ((#2)^3) #3 ((#5)^2) ((#6)^2)-8192 ((#1)^3) ((#2)^4) #3 ((#5)^2) ((#6)^2)+3072 ((#1)^2) ((#2)^3) ((#3)^2) ((#5)^2) ((#6)^2)+4096 #1 ((#2)^4) ((#3)^2) ((#5)^2) ((#6)^2)-1024 ((#2)^3) ((#3)^3) ((#5)^2) ((#6)^2)-4096 ((#1)^5) ((#2)^2) #4 ((#5)^2) ((#6)^2)+«522»+1610612736 ((#1)^3) ((#2)^2) #3 #4 ((#7)^8)+6442450944 ((#1)^2) ((#2)^3) #3 #4 ((#7)^8)-805306368 #1 ((#2)^2) ((#3)^2) #4 ((#7)^8)+3221225472 ((#1)^4) #2 ((#4)^2) ((#7)^8)+12884901888 ((#1)^3) ((#2)^2) ((#4)^2) ((#7)^8)-3221225472 ((#1)^2) #2 #3 ((#4)^2) ((#7)^8)-4294967296 ((#1)^3) ((#4)^3) ((#7)^8)&},{3,5,4,6,4,5,6}], ])+16 (NumericalApproximation[Root, (RowDefault["-0.827", FEExpressionsNumericalApproximationElider]), Root[{6019223819124736-710189636845568 #1-13865811276800 ((#1)^2)+2376618535232 ((#1)^3)+12972975489 ((#1)^4)-1361354882 ((#1)^5)+12758020 ((#1)^6)-26632 ((#1)^7)+16 ((#1)^8)&,«11»+98048 ((#2)^7)+256 ((#2)^8)&,«1»&,«1»&,«1»&,4294967296+«7»+4294967296 ((#6)^8)&,4096 ((#1)^4) #2+16384 ((#1)^4) #3+32768 ((#1)^3) #2 #4+131072 ((#1)^3) #3 #4+65536 ((#1)^2) #2 ((#4)^2)+262144 ((#1)^2) #3 ((#4)^2)+128 ((#1)^3) #2 #5+512 ((#1)^3) #3 #5+512 ((#1)^2) #2 #4 #5+2048 ((#1)^2) #3 #4 #5+((#1)^2) #2 ((#5)^2)+«39»+16384 #2 ((#3)^2) ((#4)^2) #6 ((#7)^2)+4 ((#1)^2) ((#2)^5) ((#7)^4)+16 ((#1)^2) ((#2)^4) #3 ((#7)^4)-128 #1 ((#2)^4) #3 #4 ((#7)^4)-512 #1 ((#2)^3) ((#3)^2) #4 ((#7)^4)+1024 ((#2)^3) ((#3)^2) ((#4)^2) ((#7)^4)+4096 ((#2)^2) ((#3)^3) ((#4)^2) ((#7)^4)-4 ((#1)^2) ((#2)^3) #5 ((#7)^4)-16 ((#1)^2) ((#2)^2) #3 #5 ((#7)^4)+128 #1 ((#2)^2) #3 #4 #5 ((#7)^4)+512 #1 #2 ((#3)^2) #4 #5 ((#7)^4)-1024 #2 ((#3)^2) ((#4)^2) #5 ((#7)^4)-4096 ((#3)^3) ((#4)^2) #5 ((#7)^4)&},{5,3,«3»,6,1}], ]) ((NumericalApproximation[Root, (RowDefault["1.21", FEExpressionsNumericalApproximationElider]), Root[{70380813052542976-6706695785938944 #1-57086057619456 ((#1)^2)+11464396046272 ((#1)^3)+86904077569 ((#1)^4)-2812789776 ((#1)^5)-3215264 ((#1)^6)+98048 ((#1)^7)+256 ((#1)^8)&,«5»,1024 ((#1)^6) ((#2)^3) ((#5)^2) ((#6)^2)+4096 ((#1)^5) ((#2)^4) ((#5)^2) ((#6)^2)-3072 ((#1)^4) ((#2)^3) #3 ((#5)^2) ((#6)^2)-8192 ((#1)^3) ((#2)^4) #3 ((#5)^2) ((#6)^2)+3072 ((#1)^2) ((#2)^3) ((#3)^2) ((#5)^2) ((#6)^2)+4096 #1 ((#2)^4) ((#3)^2) ((#5)^2) ((#6)^2)-1024 ((#2)^3) ((#3)^3) ((#5)^2) ((#6)^2)-4096 ((#1)^5) ((#2)^2) #4 ((#5)^2) ((#6)^2)+«522»+1610612736 ((#1)^3) ((#2)^2) #3 #4 ((#7)^8)+6442450944 ((#1)^2) ((#2)^3) #3 #4 ((#7)^8)-805306368 #1 ((#2)^2) ((#3)^2) #4 ((#7)^8)+3221225472 ((#1)^4) #2 ((#4)^2) ((#7)^8)+12884901888 ((#1)^3) ((#2)^2) ((#4)^2) ((#7)^8)-3221225472 ((#1)^2) #2 #3 ((#4)^2) ((#7)^8)-4294967296 ((#1)^3) ((#4)^3) ((#7)^8)&},{3,5,4,6,4,5,6}], ])^3)+√((((-24 (NumericalApproximation[Root, (RowDefault["-0.827", FEExpressionsNumericalApproximationElider]), Root[{6019223819124736-710189636845568 #1-13865811276800 ((#1)^2)+2376618535232 ((#1)^3)+12972975489 ((#1)^4)-1361354882 ((#1)^5)+12758020 ((#1)^6)-26632 ((#1)^7)+16 ((#1)^8)&,«11»+98048 ((#2)^7)+256 ((#2)^8)&,«1»&,«1»&,«1»&,4294967296+«7»+4294967296 ((#6)^8)&,4096 ((#1)^4) #2+16384 ((#1)^4) #3+32768 ((#1)^3) #2 #4+131072 ((#1)^3) #3 #4+65536 ((#1)^2) #2 ((#4)^2)+262144 ((#1)^2) #3 ((#4)^2)+128 ((#1)^3) #2 #5+512 ((#1)^3) #3 #5+512 ((#1)^2) #2 #4 #5+2048 ((#1)^2) #3 #4 #5+((#1)^2) #2 ((#5)^2)+«39»+16384 #2 ((#3)^2) ((#4)^2) #6 ((#7)^2)+4 ((#1)^2) ((#2)^5) ((#7)^4)+16 ((#1)^2) ((#2)^4) #3 ((#7)^4)-128 #1 ((#2)^4) #3 #4 ((#7)^4)-512 #1 ((#2)^3) ((#3)^2) #4 ((#7)^4)+1024 ((#2)^3) ((#3)^2) ((#4)^2) ((#7)^4)+4096 ((#2)^2) ((#3)^3) ((#4)^2) ((#7)^4)-4 ((#1)^2) ((#2)^3) #5 ((#7)^4)-16 ((#1)^2) ((#2)^2) #3 #5 ((#7)^4)+128 #1 ((#2)^2) #3 #4 #5 ((#7)^4)+512 #1 #2 ((#3)^2) #4 #5 ((#7)^4)-1024 #2 ((#3)^2) ((#4)^2) #5 ((#7)^4)-4096 ((#3)^3) ((#4)^2) #5 ((#7)^4)&},{5,3,«3»,6,1}], ]) (NumericalApproximation[Root, (RowDefault["1.21", FEExpressionsNumericalApproximationElider]), Root[{70380813052542976-6706695785938944 #1-57086057619456 ((#1)^2)+11464396046272 ((#1)^3)+86904077569 ((#1)^4)-2812789776 ((#1)^5)-3215264 ((#1)^6)+98048 ((#1)^7)+256 ((#1)^8)&,«5»,1024 ((#1)^6) ((#2)^3) ((#5)^2) ((#6)^2)+4096 ((#1)^5) ((#2)^4) ((#5)^2) ((#6)^2)-3072 ((#1)^4) ((#2)^3) #3 ((#5)^2) ((#6)^2)-8192 ((#1)^3) ((#2)^4) #3 ((#5)^2) ((#6)^2)+3072 ((#1)^2) ((#2)^3) ((#3)^2) ((#5)^2) ((#6)^2)+4096 #1 ((#2)^4) ((#3)^2) ((#5)^2) ((#6)^2)-1024 ((#2)^3) ((#3)^3) ((#5)^2) ((#6)^2)-4096 ((#1)^5) ((#2)^2) #4 ((#5)^2) ((#6)^2)+«522»+1610612736 ((#1)^3) ((#2)^2) #3 #4 ((#7)^8)+6442450944 ((#1)^2) ((#2)^3) #3 #4 ((#7)^8)-805306368 #1 ((#2)^2) ((#3)^2) #4 ((#7)^8)+3221225472 ((#1)^4) #2 ((#4)^2) ((#7)^8)+12884901888 ((#1)^3) ((#2)^2) ((#4)^2) ((#7)^8)-3221225472 ((#1)^2) #2 #3 ((#4)^2) ((#7)^8)-4294967296 ((#1)^3) ((#4)^3) ((#7)^8)&},{3,5,4,6,4,5,6}], ])+8 (Sqrt[5]) (NumericalApproximation[Root, (RowDefault["-0.827", FEExpressionsNumericalApproximationElider]), Root[{6019223819124736-710189636845568 #1-13865811276800 ((#1)^2)+2376618535232 ((#1)^3)+12972975489 ((#1)^4)-1361354882 ((#1)^5)+12758020 ((#1)^6)-26632 ((#1)^7)+16 ((#1)^8)&,«11»+98048 ((#2)^7)+256 ((#2)^8)&,«1»&,«1»&,«1»&,4294967296+«7»+4294967296 ((#6)^8)&,4096 ((#1)^4) #2+16384 ((#1)^4) #3+32768 ((#1)^3) #2 #4+131072 ((#1)^3) #3 #4+65536 ((#1)^2) #2 ((#4)^2)+262144 ((#1)^2) #3 ((#4)^2)+128 ((#1)^3) #2 #5+512 ((#1)^3) #3 #5+512 ((#1)^2) #2 #4 #5+2048 ((#1)^2) #3 #4 #5+((#1)^2) #2 ((#5)^2)+«39»+16384 #2 ((#3)^2) ((#4)^2) #6 ((#7)^2)+4 ((#1)^2) ((#2)^5) ((#7)^4)+16 ((#1)^2) ((#2)^4) #3 ((#7)^4)-128 #1 ((#2)^4) #3 #4 ((#7)^4)-512 #1 ((#2)^3) ((#3)^2) #4 ((#7)^4)+1024 ((#2)^3) ((#3)^2) ((#4)^2) ((#7)^4)+4096 ((#2)^2) ((#3)^3) ((#4)^2) ((#7)^4)-4 ((#1)^2) ((#2)^3) #5 ((#7)^4)-16 ((#1)^2) ((#2)^2) #3 #5 ((#7)^4)+128 #1 ((#2)^2) #3 #4 #5 ((#7)^4)+512 #1 #2 ((#3)^2) #4 #5 ((#7)^4)-1024 #2 ((#3)^2) ((#4)^2) #5 ((#7)^4)-4096 ((#3)^3) ((#4)^2) #5 ((#7)^4)&},{5,3,«3»,6,1}], ]) (NumericalApproximation[Root, (RowDefault["1.21", FEExpressionsNumericalApproximationElider]), Root[{70380813052542976-6706695785938944 #1-57086057619456 ((#1)^2)+11464396046272 ((#1)^3)+86904077569 ((#1)^4)-2812789776 ((#1)^5)-3215264 ((#1)^6)+98048 ((#1)^7)+256 ((#1)^8)&,«5»,1024 ((#1)^6) ((#2)^3) ((#5)^2) ((#6)^2)+4096 ((#1)^5) ((#2)^4) ((#5)^2) ((#6)^2)-3072 ((#1)^4) ((#2)^3) #3 ((#5)^2) ((#6)^2)-8192 ((#1)^3) ((#2)^4) #3 ((#5)^2) ((#6)^2)+3072 ((#1)^2) ((#2)^3) ((#3)^2) ((#5)^2) ((#6)^2)+4096 #1 ((#2)^4) ((#3)^2) ((#5)^2) ((#6)^2)-1024 ((#2)^3) ((#3)^3) ((#5)^2) ((#6)^2)-4096 ((#1)^5) ((#2)^2) #4 ((#5)^2) ((#6)^2)+«522»+1610612736 ((#1)^3) ((#2)^2) #3 #4 ((#7)^8)+6442450944 ((#1)^2) ((#2)^3) #3 #4 ((#7)^8)-805306368 #1 ((#2)^2) ((#3)^2) #4 ((#7)^8)+3221225472 ((#1)^4) #2 ((#4)^2) ((#7)^8)+12884901888 ((#1)^3) ((#2)^2) ((#4)^2) ((#7)^8)-3221225472 ((#1)^2) #2 #3 ((#4)^2) ((#7)^8)-4294967296 ((#1)^3) ((#4)^3) ((#7)^8)&},{3,5,4,6,4,5,6}], ])+16 (NumericalApproximation[Root, (RowDefault["-0.827", FEExpressionsNumericalApproximationElider]), Root[{6019223819124736-710189636845568 #1-13865811276800 ((#1)^2)+2376618535232 ((#1)^3)+12972975489 ((#1)^4)-1361354882 ((#1)^5)+12758020 ((#1)^6)-26632 ((#1)^7)+16 ((#1)^8)&,«11»+98048 ((#2)^7)+256 ((#2)^8)&,«1»&,«1»&,«1»&,4294967296+«7»+4294967296 ((#6)^8)&,4096 ((#1)^4) #2+16384 ((#1)^4) #3+32768 ((#1)^3) #2 #4+131072 ((#1)^3) #3 #4+65536 ((#1)^2) #2 ((#4)^2)+262144 ((#1)^2) #3 ((#4)^2)+128 ((#1)^3) #2 #5+512 ((#1)^3) #3 #5+512 ((#1)^2) #2 #4 #5+2048 ((#1)^2) #3 #4 #5+((#1)^2) #2 ((#5)^2)+«39»+16384 #2 ((#3)^2) ((#4)^2) #6 ((#7)^2)+4 ((#1)^2) ((#2)^5) ((#7)^4)+16 ((#1)^2) ((#2)^4) #3 ((#7)^4)-128 #1 ((#2)^4) #3 #4 ((#7)^4)-512 #1 ((#2)^3) ((#3)^2) #4 ((#7)^4)+1024 ((#2)^3) ((#3)^2) ((#4)^2) ((#7)^4)+4096 ((#2)^2) ((#3)^3) ((#4)^2) ((#7)^4)-4 ((#1)^2) ((#2)^3) #5 ((#7)^4)-16 ((#1)^2) ((#2)^2) #3 #5 ((#7)^4)+128 #1 ((#2)^2) #3 #4 #5 ((#7)^4)+512 #1 #2 ((#3)^2) #4 #5 ((#7)^4)-1024 #2 ((#3)^2) ((#4)^2) #5 ((#7)^4)-4096 ((#3)^3) ((#4)^2) #5 ((#7)^4)&},{5,3,«3»,6,1}], ]) ((NumericalApproximation[Root, (RowDefault["1.21", FEExpressionsNumericalApproximationElider]), Root[{70380813052542976-6706695785938944 #1-57086057619456 ((#1)^2)+11464396046272 ((#1)^3)+86904077569 ((#1)^4)-2812789776 ((#1)^5)-3215264 ((#1)^6)+98048 ((#1)^7)+256 ((#1)^8)&,«5»,1024 ((#1)^6) ((#2)^3) ((#5)^2) ((#6)^2)+4096 ((#1)^5) ((#2)^4) ((#5)^2) ((#6)^2)-3072 ((#1)^4) ((#2)^3) #3 ((#5)^2) ((#6)^2)-8192 ((#1)^3) ((#2)^4) #3 ((#5)^2) ((#6)^2)+3072 ((#1)^2) ((#2)^3) ((#3)^2) ((#5)^2) ((#6)^2)+4096 #1 ((#2)^4) ((#3)^2) ((#5)^2) ((#6)^2)-1024 ((#2)^3) ((#3)^3) ((#5)^2) ((#6)^2)-4096 ((#1)^5) ((#2)^2) #4 ((#5)^2) ((#6)^2)+«522»+1610612736 ((#1)^3) ((#2)^2) #3 #4 ((#7)^8)+6442450944 ((#1)^2) ((#2)^3) #3 #4 ((#7)^8)-805306368 #1 ((#2)^2) ((#3)^2) #4 ((#7)^8)+3221225472 ((#1)^4) #2 ((#4)^2) ((#7)^8)+12884901888 ((#1)^3) ((#2)^2) ((#4)^2) ((#7)^8)-3221225472 ((#1)^2) #2 #3 ((#4)^2) ((#7)^8)-4294967296 ((#1)^3) ((#4)^3) ((#7)^8)&},{3,5,4,6,4,5,6}], ])^3)))^2)+64 ((NumericalApproximation[Root, (RowDefault["1.21", FEExpressionsNumericalApproximationElider]), Root[{70380813052542976-6706695785938944 #1-57086057619456 ((#1)^2)+11464396046272 ((#1)^3)+86904077569 ((#1)^4)-2812789776 ((#1)^5)-3215264 ((#1)^6)+98048 ((#1)^7)+256 ((#1)^8)&,«5»,1024 ((#1)^6) ((#2)^3) ((#5)^2) ((#6)^2)+4096 ((#1)^5) ((#2)^4) ((#5)^2) ((#6)^2)-3072 ((#1)^4) ((#2)^3) #3 ((#5)^2) ((#6)^2)-8192 ((#1)^3) ((#2)^4) #3 ((#5)^2) ((#6)^2)+3072 ((#1)^2) ((#2)^3) ((#3)^2) ((#5)^2) ((#6)^2)+4096 #1 ((#2)^4) ((#3)^2) ((#5)^2) ((#6)^2)-1024 ((#2)^3) ((#3)^3) ((#5)^2) ((#6)^2)-4096 ((#1)^5) ((#2)^2) #4 ((#5)^2) ((#6)^2)+«522»+1610612736 ((#1)^3) ((#2)^2) #3 #4 ((#7)^8)+6442450944 ((#1)^2) ((#2)^3) #3 #4 ((#7)^8)-805306368 #1 ((#2)^2) ((#3)^2) #4 ((#7)^8)+3221225472 ((#1)^4) #2 ((#4)^2) ((#7)^8)+12884901888 ((#1)^3) ((#2)^2) ((#4)^2) ((#7)^8)-3221225472 ((#1)^2) #2 #3 ((#4)^2) ((#7)^8)-4294967296 ((#1)^3) ((#4)^3) ((#7)^8)&},{3,5,4,6,4,5,6}], ])^2) (-14+6 (Sqrt[5])+28 ((NumericalApproximation[Root, (RowDefault["1.21", FEExpressionsNumericalApproximationElider]), Root[{70380813052542976-6706695785938944 #1-57086057619456 ((#1)^2)+11464396046272 ((#1)^3)+86904077569 ((#1)^4)-2812789776 ((#1)^5)-3215264 ((#1)^6)+98048 ((#1)^7)+256 ((#1)^8)&,«5»,1024 ((#1)^6) ((#2)^3) ((#5)^2) ((#6)^2)+4096 ((#1)^5) ((#2)^4) ((#5)^2) ((#6)^2)-3072 ((#1)^4) ((#2)^3) #3 ((#5)^2) ((#6)^2)-8192 ((#1)^3) ((#2)^4) #3 ((#5)^2) ((#6)^2)+3072 ((#1)^2) ((#2)^3) ((#3)^2) ((#5)^2) ((#6)^2)+4096 #1 ((#2)^4) ((#3)^2) ((#5)^2) ((#6)^2)-1024 ((#2)^3) ((#3)^3) ((#5)^2) ((#6)^2)-4096 ((#1)^5) ((#2)^2) #4 ((#5)^2) ((#6)^2)+«522»+1610612736 ((#1)^3) ((#2)^2) #3 #4 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((#6)^2)+«522»+1610612736 ((#1)^3) ((#2)^2) #3 #4 ((#7)^8)+6442450944 ((#1)^2) ((#2)^3) #3 #4 ((#7)^8)-805306368 #1 ((#2)^2) ((#3)^2) #4 ((#7)^8)+3221225472 ((#1)^4) #2 ((#4)^2) ((#7)^8)+12884901888 ((#1)^3) ((#2)^2) ((#4)^2) ((#7)^8)-3221225472 ((#1)^2) #2 #3 ((#4)^2) ((#7)^8)-4294967296 ((#1)^3) ((#4)^3) ((#7)^8)&},{3,5,4,6,4,5,6}], ])^2)-4 ((NumericalApproximation[Root, (RowDefault["1.21", FEExpressionsNumericalApproximationElider]), Root[{70380813052542976-6706695785938944 #1-57086057619456 ((#1)^2)+11464396046272 ((#1)^3)+86904077569 ((#1)^4)-2812789776 ((#1)^5)-3215264 ((#1)^6)+98048 ((#1)^7)+256 ((#1)^8)&,«5»,1024 ((#1)^6) ((#2)^3) ((#5)^2) ((#6)^2)+4096 ((#1)^5) ((#2)^4) ((#5)^2) ((#6)^2)-3072 ((#1)^4) ((#2)^3) #3 ((#5)^2) ((#6)^2)-8192 ((#1)^3) ((#2)^4) #3 ((#5)^2) ((#6)^2)+3072 ((#1)^2) ((#2)^3) ((#3)^2) ((#5)^2) ((#6)^2)+4096 #1 ((#2)^4) ((#3)^2) ((#5)^2) ((#6)^2)-1024 ((#2)^3) ((#3)^3) ((#5)^2) ((#6)^2)-4096 ((#1)^5) ((#2)^2) #4 ((#5)^2) ((#6)^2)+«522»+1610612736 ((#1)^3) ((#2)^2) #3 #4 ((#7)^8)+6442450944 ((#1)^2) ((#2)^3) #3 #4 ((#7)^8)-805306368 #1 ((#2)^2) ((#3)^2) #4 ((#7)^8)+3221225472 ((#1)^4) #2 ((#4)^2) ((#7)^8)+12884901888 ((#1)^3) ((#2)^2) ((#4)^2) ((#7)^8)-3221225472 ((#1)^2) #2 #3 ((#4)^2) ((#7)^8)-4294967296 ((#1)^3) ((#4)^3) ((#7)^8)&},{3,5,4,6,4,5,6}], ])^4))))],ψ->-ArcCos[(NumericalApproximation[Root, (RowDefault["-0.827", FEExpressionsNumericalApproximationElider]), Root[{6019223819124736-710189636845568 #1-13865811276800 ((#1)^2)+2376618535232 ((#1)^3)+12972975489 ((#1)^4)-1361354882 ((#1)^5)+12758020 ((#1)^6)-26632 ((#1)^7)+16 ((#1)^8)&,«11»+98048 ((#2)^7)+256 ((#2)^8)&,«1»&,«1»&,«1»&,4294967296+«7»+4294967296 ((#6)^8)&,4096 ((#1)^4) #2+16384 ((#1)^4) #3+32768 ((#1)^3) #2 #4+131072 ((#1)^3) #3 #4+65536 ((#1)^2) #2 ((#4)^2)+262144 ((#1)^2) #3 ((#4)^2)+128 ((#1)^3) #2 #5+512 ((#1)^3) #3 #5+512 ((#1)^2) #2 #4 #5+2048 ((#1)^2) #3 #4 #5+((#1)^2) #2 ((#5)^2)+«39»+16384 #2 ((#3)^2) ((#4)^2) #6 ((#7)^2)+4 ((#1)^2) ((#2)^5) ((#7)^4)+16 ((#1)^2) ((#2)^4) #3 ((#7)^4)-128 #1 ((#2)^4) #3 #4 ((#7)^4)-512 #1 ((#2)^3) ((#3)^2) #4 ((#7)^4)+1024 ((#2)^3) ((#3)^2) ((#4)^2) ((#7)^4)+4096 ((#2)^2) ((#3)^3) ((#4)^2) ((#7)^4)-4 ((#1)^2) ((#2)^3) #5 ((#7)^4)-16 ((#1)^2) ((#2)^2) #3 #5 ((#7)^4)+128 #1 ((#2)^2) #3 #4 #5 ((#7)^4)+512 #1 #2 ((#3)^2) #4 #5 ((#7)^4)-1024 #2 ((#3)^2) ((#4)^2) #5 ((#7)^4)-4096 ((#3)^3) ((#4)^2) #5 ((#7)^4)&},{5,3,«3»,6,1}], ])]}/.soln[[-1]])/.R->(1/128) ((Sqrt[1023]) (-1+(Sqrt[5]))-(Sqrt[2 (5+(Sqrt[5]))])+2 (Sqrt[7681-2559 (Sqrt[5])-(Sqrt[2046 (5-(Sqrt[5]))])]))

N[%]

g=Graph[Range[20],UndirectedEdge@@@e2,VertexCoordinates->vc,VertexLabels->"Name"]

Redo

v=Join[
(* outer pentagon *)
Table[Csc[(π/5)]Through[{Cos,Sin}[2π (k+1/4)/5+4π/5]],{k,5}],

(* tilted pentagon *)
Table[R Csc[(π/5)]Through[{Cos,Sin}[2π (k+1/4)/5+4π/5+ϕ]],{k,5}],

(* middle pentagon *)
Table[(1/2)r Csc[(π/5)]Through[{Cos,Sin}[2π (k+1/4)/5+4π/5+ψ]],{k,5}],

(* inner pentagon *)
Table[(1/2)Csc[(π/5)]Through[{Cos,Sin}[2π (k+1/4)/5+4π/5+θ]],{k,5}]
]/.{
ϕ->π/100,
θ->-1.70560978792577961038601596146083756671243650575033830256890571236629180161806`50.,ψ->-0.59690260418206071530790224282310554799754850595296608734085729074044126728344`50.,r->1.20786944362003084727646891808969176051871599141625694902742932477722499564001`50.,R->0.95707815686611063222867711536412461546114343529004809765978406494528295603317`50.}

e=Select[Subsets[Range[20],{2}],Abs[Round[d=EuclideanDistance@@v[[#]]]-d]<=10^-2&]

g0=Graph[Range[20],UndirectedEdge@@@e,VertexCoordinates->v,VertexLabels->"Name"]

RecognizeGraph[g0]

VertexDegree[g0]

e2=Complement[e,{{1,16},{1,17},{2,17},{2,18},{3,18},{3,19},{4,19},{4,20},{5,16},{5,20}}]

EuclideanDistance@@v[[{4,5}]]//FullSimplify

g=Graph[Range[20],UndirectedEdge@@@e2,VertexCoordinates->v,VertexLabels->"Name"]

r=RecognizeGraph[g]

GraphEdgeLengths[g]

l={{r,{0,ReplacePart[GraphData[r,"Embeddings"],10->ToCommonEdges[g,GraphData[r]]]}}}

PropertyReplace[l,"Embeddings","12.1"]

RenameGraphFiles[]

UnitDistance

r=FullSimplify[((Sin[((n+1-2k)/(2(n+1)))π])/(Sin[((2k)/(n+1))π]))/.{n->9,k->2}];
R=FullSimplify[((Sin[((n+1-2k)/(2(n+1)))π])/(Sin[((2k)/(n+1))π]))/.{n->9,k->1}];

ShowLabeledGraph[g=Graph[List/@Join[
Partition[Range[1,9,2],2,1,1],
Partition[Range[2,10,2],2,1,1]
],List/@Join[
Table[r Through[{Cos,Sin}[((2π i)/10)]],{i,10}],
Table[R Through[{Cos,Sin}[((2π i)/10)]],{i,10}]
]]]

Norm[Subtract@@v[[{1,3}]]]//FullSimplify

Norm[Subtract@@v[[{12,13}]]]//FullSimplify

v=Combinatorica`Vertices[g];

p=Intersections[Circle[{0,0},R],Line[v[[{3,5}]]]][[1,1]]//FullSimplify

p//N

Solve[Norm[R{Cos[a],Sin[a]}-v[[3]]]==1,a]//FullSimplify

N[%]

ang=3(2π/10)-ArcSec[Root[80-160 #1+100 ((#1)^2)-20 ((#1)^3)+((#1)^4)&,3]];

ShowLabeledGraph[g=Graph[List/@Join[
Partition[Range[1,9,2],2,1,1],
Partition[Range[2,10,2],2,1,1],
{#,#+10}&/@Range[10],
Partition[Range[11,20],2,1,1]
],List/@Join[
Table[r Through[{Cos,Sin}[((2π i)/10)]],{i,10}],
Table[R Through[{Cos,Sin}[((2π i)/10)-ang]],{i,10}]
]]]

v=Combinatorica`Vertices[g];
Norm[Subtract@@v[[{3,13}]]]//FullSimplify

RecognizeGraph[g]

tilt=ArcTan@@Subtract@@v[[{19,18}]]//FullSimplify

ShowLabeledGraph[g=Graph[List/@Join[
Partition[Range[1,9,2],2,1,1],
Partition[Range[2,10,2],2,1,1],
{#,#+10}&/@Range[10],
Partition[Range[11,20],2,1,1]
],List/@Join[
Table[r Through[{Cos,Sin}[((2π i)/10)-tilt]],{i,10}],
Table[R Through[{Cos,Sin}[((2π i)/10)-ang-tilt]],{i,10}]
]]]

GraphPlot[g,Method->None,ImageSize->200]

v=Combinatorica`Vertices[g]//FullSimplify//ToRadicals

<<Utilities`Simplify`

SymbolicUnion[Norm/@Subtract@@@(Edges[g]/.Thread[Range[20]->v])]//FullSimplify

Insert[GraphData["DodecahedralGraph","Embeddings"],ToCommonEdges[ChangeVertices[g,v],"DodecahedralGraph"],4]

GraphData

GraphClassString

GraphClassString["DodecahedralGraph"]