Generalized Octagon

Author

Eric W. Weisstein
December 3, 2023

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

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

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

Table

Many of these are listed in van Dam and Haemers 2003

go=Select[GraphData["GeneralizedPolygon"],First[GraphData[#,"NotationRules","GeneralizedPolygon"]]==8&]

TextGrid[Sort[
{
GraphData[#,"NotationRules","GeneralizedPolygon"][[2]],
GraphData[#,"V"],
GraphData[#,"Name"],
GraphData[#,"Notation"],
GraphData[#,"Spectrum"]//SpectrumForm
}&/@go],Dividers->All,Alignment->{{Automatic,Decimal,{Left}}}]

Embeddings

<<MathWorld`Graphs`

{"GeneralizedOctagon", {1, 2}} == "LeviGraph"

GraphData[{"GeneralizedOctagon",{1,2}},"StandardName"]

{"GeneralizedOctagon", {1, 3}} == {"Cage", {4, 8}}

GraphData[{"GeneralizedOctagon",{1,3}},"StandardName"]

{"GeneralizedOctagon", {1, 4}} == {"Cage", {5, 8}}

GraphData[{"GeneralizedOctagon",{1,4}},"StandardName"]

{"GeneralizedOctagon", {1, 8}} == {"Cage", {9, 8}}

GraphData[{"GeneralizedOctagon",{1,8}},"StandardName"]

{"GeneralizedOctagon", {2, 1}}

Names

GraphData[{"GeneralizedOctagon",{2,1}},"StandardName"]

GraphData[{"GeneralizedOctagon",{2,1}},"StandardNames"]

Primary

GraphData[{"GeneralizedOctagon",{2,1}}]//StyleGraphs[#,ImageSize->Small]&

All

GraphData[{"GeneralizedOctagon",{2,1}},"Graph","All"]//StyleGraphs

Inexact

Select[GraphData[{"GeneralizedOctagon",{2,1}},"Graph","All"],(!FreeQ[GraphEmbedding[#],_?InexactNumberQ]&)]//StyleGraphs

Embedding types

GraphDataEmbeddings[GraphData[{"GeneralizedOctagon",{2,1}},"EdgeList"]]

Default embedding

GraphPlot@GraphData[{"GeneralizedOctagon",{2,1}},"Edges"]

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

RecognizeGraph[g]

Degenerate

GraphData[{"GeneralizedOctagon",{2,1}},"Graph","Degenerate"]//StyleGraphs

DegenerateGraphEmbeddingTypes/@GraphData[{"GeneralizedOctagon",{2,1}},"Graph","Degenerate"]

Perspective

GraphData[{"GeneralizedOctagon",{2,1}},"Graph","Perspective"]//StyleGraphs

Perspective construction

GraphData[{"GeneralizedOctagon",{2,1}},"Polyhedron"]

Bilateral

GraphData[{"GeneralizedOctagon",{2,1}},"Graph","Bilateral"]//StyleGraphs

Circular

GraphData[{"GeneralizedOctagon",{2,1}},"Graph","Circular"]//StyleGraphs

Planar

GraphData[{"GeneralizedOctagon",{2,1}},"Planar"]

GraphData[{"GeneralizedOctagon",{2,1}},"Graph","Planar"]//StyleGraphs

MinimalCrossing

GraphData[{"GeneralizedOctagon",{2,1}},{"RCN","CN"}]

GraphData[{"GeneralizedOctagon",{2,1}},"Graph","MinimalCrossing"]//StyleGraphs

MinimalCrossing construction

CrossingNumber[GraphData[{"GeneralizedOctagon",{2,1}}],TimeConstraint->10]//AbsoluteTiming

CrossingNumber[GraphData[{"GeneralizedOctagon",{2,1}}],TimeConstraint->60]//AbsoluteTiming

g=RotateGraph[MinimalCrossingGraph[GraphData[{"GeneralizedOctagon",{2,1}}],"CrossingNumber"->12]]

RecognizeGraph[g]

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

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData[{"GeneralizedOctagon",{2,1}},"Graph","Circulant"]//StyleGraphs

Halin

GraphData[{"GeneralizedOctagon",{2,1}},"Halin"]

GraphData[{"GeneralizedOctagon",{2,1}},"Graph","Halin"]//StyleGraphs

LCF

GraphData[{"GeneralizedOctagon",{2,1}},"LCF"]

GraphData[{"GeneralizedOctagon",{2,1}},"HamiltonianCycleCount"]

GraphData[{"GeneralizedOctagon",{2,1}},"LCFSignature"]

GraphData[{"GeneralizedOctagon",{2,1}},"LCFNotations"]

GraphData[{"GeneralizedOctagon",{2,1}},"Graph","LCF"]//StyleGraphs

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

LCF compute all

Divisors[GraphData[{"GeneralizedOctagon",{2,1}},"V"]]

Length[FindHamiltonianCycle[GraphData[{"GeneralizedOctagon",{2,1}}],All][[All,All,1]]]//Timing

Tally[(lcf=LCF[GraphData[{"GeneralizedOctagon",{2,1}}],All])[[All,2]]]//Timing

MaxMemoryUsed[]/10.^9 "GB"

Tally[(lcf=ReadProperty["LCFNotations",GraphNameToFileName[{"GeneralizedOctagon",{2,1}},".m.gz"]])[[All,2]]]//Timing

DeleteCases[lcf,{_,1}]

LCFGraph@@@DeleteCases[lcf,{_,1}]//StyleGraphs

Tally[LCFNotation[#][[-1]]&/@(gs=NiceLCFGraphs[lcf,2])]//Timing

Length[gs]

gs

BilateralGraphEmbeddingQ/@gs

UnitDistance

GraphData[{"GeneralizedOctagon",{2,1}},"UnitDistance"]

UnitDistanceGraphQ[GraphData[{"GeneralizedOctagon",{2,1}}],Debug->True]//Timing

GraphData[{"GeneralizedOctagon",{2,1}},"Graph","UnitDistance"]//StyleGraphs

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

Select[UnitDistanceEmbeddingRefine/@Complement@@(GraphData[{"GeneralizedOctagon",{2,1}},"Graph",#]&/@{"All","UnitDistance"}),UnitDistanceGraphEmbeddingQ]

FindUnitDistanceEmbedding[GraphData[{"GeneralizedOctagon",{2,1}}],MaxIterations->5000,"ReturnWithMessage"->True]//Timing

UnitDistance exact construction

g0=GraphData[{"GeneralizedOctagon",{2,1}},"Graph","UnitDistance"][[1]]

z=Position[Norm/@GraphEmbedding[g0],_?(#<.01&)]//Flatten

GraphicsGrid[Partition[HighlightGraph[g0,Style[UndirectedEdge[#,_],Directive[Red,AbsoluteThickness[5]]]]&/@z,UpTo[3]],ImageSize->1000]

Annotate[HighlightGraph[g0,Style[UndirectedEdge[#,_],Directive[Red,AbsoluteThickness[5]]]],{VertexSize->1,PlotRange->.05}]&/@z

LabelGraph@g0

eps=Norm[v0[[2]]]

Norm[v0[[#]]]&/@z

a=Norm[v0[[7]]]

Select[Norm/@v0,.1<#<.2&]

ab=Norm[v0[[34]]]

Norm/@v0[[{34,41,1,14,25}]]

Select[Norm/@v0,.98<#<1&]

b=Norm@v0[[30]]

Select[Norm/@v0,.73<#<.75&]

q1=VectorAngle@@v0[[{8,12}]]

c=Norm[v0[[12]]]

R=Norm@v0[[8]]

q2=VectorAngle@@v0[[{8,45}]]

d=Norm[v0[[45]]]

a=0.14269099672768115`;
b=0.7406618491806436`;
ab=0.9978495537852953`;
c=1.0045895447977276`;
R=1.4994692518734554`;
q1=0.41745315183217274`;
q2=0.5603978089016111`;
d=0.8762542134634074`;
eps=0.006889522390429495`;
v=Join[
CirclePoints[{a,-Pi/2},5],CirclePoints[{ab,-Pi/2},5],CirclePoints[{b,-Pi/2},5],
CirclePoints[{c,Pi/2+q1},5],CirclePoints[{c,Pi/2-q1},5],
CirclePoints[{R,Pi/2},5],
CirclePoints[{d,Pi/2+q2},5],CirclePoints[{d,Pi/2-q2},5],
CirclePoints[eps,5]
];
ee=Select[Subsets[Range[Length[v]],{2}],Abs[EuclideanDistance@@v[[#]]-1]<10^-4&]//Quiet;
g=Graph[Range[Length[v]],ee,VertexCoordinates->v,VertexLabels->Automatic]

RecognizeGraph[g]

EdgeLengths[g]

Show[Annotate[g,VertexSize->.2],PlotRange->.02]

Clear[a,b,ab,c,R,d,q1,q2,eps];

v=Join[
CirclePoints[{a,-Pi/2},5],CirclePoints[{ab,-Pi/2},5],CirclePoints[{b,-Pi/2},5],
CirclePoints[{c,Pi/2+q1},5],CirclePoints[{c,Pi/2-q1},5],
CirclePoints[{R,Pi/2},5],
CirclePoints[{d,Pi/2+q2},5],CirclePoints[{d,Pi/2-q2},5],
CirclePoints[eps,5]
];

eqn=DeleteCases[UnitEdgeEquations[v,ee]//FullSimplify//Union,True]

eqn2=Union[Subtract@@@((DeleteCases[TrigExpand[FullSimplify[TrigExpand[eqn]]],True]//Union//FullSimplify//Union//TrigExpand))]

gb=Join[eqn2/.{Cos[q1]->q1c,Sin[q1]->q1s,Cos[q2]->q2c,Sin[q2]->q2s},{q1c^2+q1s^2-1,q2c^2+q2s^2-1}]

gb={-1+(ab^2)-((ab eps)/2)+(1/2) (Sqrt[5]) ab eps+(eps^2),-1+(b^2)-((b R)/2)-(1/2) (Sqrt[5]) b R+(R^2),-1+(a^2)+(d^2)+2 a d q2c,-2+2 (a^2)+2 (c^2)-a c q1c+(Sqrt[5]) a c q1c-(Sqrt[2 (5+(Sqrt[5]))]) a c q1s,-2+2 (ab^2)+2 (c^2)-ab c q1c-(Sqrt[5]) ab c q1c+(Sqrt[10-2 (Sqrt[5])]) ab c q1s,-2+2 (c^2)+2 (eps^2)+c eps q1c-(Sqrt[5]) c eps q1c-(Sqrt[2 (5+(Sqrt[5]))]) c eps q1s,-2+2 (b^2)+2 (d^2)-b d q2c+(Sqrt[5]) b d q2c-(Sqrt[2 (5+(Sqrt[5]))]) b d q2s,-2+2 (d^2)+d q2c R-(Sqrt[5]) d q2c R-(Sqrt[2 (5+(Sqrt[5]))]) d q2s R+2 (R^2),-2+2 (c^2)+2 (d^2)+c d q1c q2c-(Sqrt[5]) c d q1c q2c+(Sqrt[2 (5+(Sqrt[5]))]) c d q1s q2c-(Sqrt[2 (5+(Sqrt[5]))]) c d q1c q2s+c d q1s q2s-(Sqrt[5]) c d q1s q2s,-1+(q1c^2)+(q1s^2),-1+(q2c^2)+(q2s^2)};

Length[gb]

vars=Variables[gb]

Length/@{gb,vars}

Trying to solve using GroebnerBasis fails:

GroebnerBasis[gb,a,Complement[vars,{a}],MonomialOrder->EliminationOrder]

egb=gb/.(Reverse/@{Cos[q1]->q1c,Sin[q1]->q1s,Cos[q2]->q2c,Sin[q2]->q2s})

Length[egb]

rules={a->0.14269099672768115`,b->0.7406618491806436`,ab->0.9978495537852953`,c->1.0045895447977276`,R->1.4994692518734554`,q1->0.41745315183217274`,q2->0.5603978089016111`,d->0.8762542134634074`,eps->0.006889522390429495`};

egb/.rules

Chop[%]

soln=FindRoot[gb,{{a,0.14269099672768115`},{b,0.7406618491806436`},{ab,0.9978495537852953`},{c,1.0045895447977276`},{R,1.4994692518734554`},{q1c,Cos@0.41745315183217274`},{q1s,Sin@0.41745315183217274`},
{q2c,Cos@0.5603978089016111`},{q2s,Sin@0.5603978089016111`},
{d,0.8762542134634074`},{eps,0.006889522390429495`}},WorkingPrecision->1000];

Length[soln]

esoln=soln/.x_?InexactNumberQ:>RootApproximant[N[x,1000],64]

esoln=soln/.x_?InexactNumberQ:>RootApproximant[N[x,1000],72]

esoln1=Drop[esoln,{6,9}]

(gb/.esoln1)//Column

Solve[-1+((NumericalApproximation[Root, (RowDefault["0.876", FEExpressionsNumericalApproximationElider]), Root[19504021-1021550925 ((#1)^2)+16826333925 ((#1)^4)-144026777475 ((#1)^6)+767601900150 ((#1)^8)-2786101493175 ((#1)^10)+7248691249875 ((#1)^12)-13951699884000 ((#1)^14)+20279794395375 ((#1)^16)-22595324827500 ((#1)^18)+19541839275000 ((#1)^20)-13276625962500 ((#1)^22)+7140777412500 ((#1)^24)-3010836825000 ((#1)^26)+945857250000 ((#1)^28)-195230250000 ((#1)^30)+19460250000 ((#1)^32)&,22], ])^2)+2 q2c (NumericalApproximation[Root, (RowDefault["0.876", FEExpressionsNumericalApproximationElider]), Root[19504021-1021550925 ((#1)^2)+16826333925 ((#1)^4)-144026777475 ((#1)^6)+767601900150 ((#1)^8)-2786101493175 ((#1)^10)+7248691249875 ((#1)^12)-13951699884000 ((#1)^14)+20279794395375 ((#1)^16)-22595324827500 ((#1)^18)+19541839275000 ((#1)^20)-13276625962500 ((#1)^22)+7140777412500 ((#1)^24)-3010836825000 ((#1)^26)+945857250000 ((#1)^28)-195230250000 ((#1)^30)+19460250000 ((#1)^32)&,22], ]) (NumericalApproximation[Root, (RowDefault["0.143", FEExpressionsNumericalApproximationElider]), Root[1-450 ((#1)^2)+56175 ((#1)^4)-3183900 ((#1)^6)+95178000 ((#1)^8)-1593148050 ((#1)^10)+15069672000 ((#1)^12)-77965348500 ((#1)^14)+200904161625 ((#1)^16)-175240530000 ((#1)^18)-244969953750 ((#1)^20)+623406712500 ((#1)^22)-308513475000 ((#1)^24)-260502075000 ((#1)^26)+351135000000 ((#1)^28)-140616000000 ((#1)^30)+19460250000 ((#1)^32)&,16], ])+((NumericalApproximation[Root, (RowDefault["0.143", FEExpressionsNumericalApproximationElider]), Root[1-450 ((#1)^2)+56175 ((#1)^4)-3183900 ((#1)^6)+95178000 ((#1)^8)-1593148050 ((#1)^10)+15069672000 ((#1)^12)-77965348500 ((#1)^14)+200904161625 ((#1)^16)-175240530000 ((#1)^18)-244969953750 ((#1)^20)+623406712500 ((#1)^22)-308513475000 ((#1)^24)-260502075000 ((#1)^26)+351135000000 ((#1)^28)-140616000000 ((#1)^30)+19460250000 ((#1)^32)&,16], ])^2)==0,q2c]

q2ce=RootApproximant[N[((1-((NumericalApproximation[Root, (RowDefault["0.876", FEExpressionsNumericalApproximationElider]), Root[19504021-1021550925 ((#1)^2)+16826333925 ((#1)^4)-144026777475 ((#1)^6)+767601900150 ((#1)^8)-2786101493175 ((#1)^10)+7248691249875 ((#1)^12)-13951699884000 ((#1)^14)+20279794395375 ((#1)^16)-22595324827500 ((#1)^18)+19541839275000 ((#1)^20)-13276625962500 ((#1)^22)+7140777412500 ((#1)^24)-3010836825000 ((#1)^26)+945857250000 ((#1)^28)-195230250000 ((#1)^30)+19460250000 ((#1)^32)&,22], ])^2)-((NumericalApproximation[Root, (RowDefault["0.143", FEExpressionsNumericalApproximationElider]), Root[1-450 ((#1)^2)+56175 ((#1)^4)-3183900 ((#1)^6)+95178000 ((#1)^8)-1593148050 ((#1)^10)+15069672000 ((#1)^12)-77965348500 ((#1)^14)+200904161625 ((#1)^16)-175240530000 ((#1)^18)-244969953750 ((#1)^20)+623406712500 ((#1)^22)-308513475000 ((#1)^24)-260502075000 ((#1)^26)+351135000000 ((#1)^28)-140616000000 ((#1)^30)+19460250000 ((#1)^32)&,16], ])^2))/(2 (NumericalApproximation[Root, (RowDefault["0.876", FEExpressionsNumericalApproximationElider]), Root[19504021-1021550925 ((#1)^2)+16826333925 ((#1)^4)-144026777475 ((#1)^6)+767601900150 ((#1)^8)-2786101493175 ((#1)^10)+7248691249875 ((#1)^12)-13951699884000 ((#1)^14)+20279794395375 ((#1)^16)-22595324827500 ((#1)^18)+19541839275000 ((#1)^20)-13276625962500 ((#1)^22)+7140777412500 ((#1)^24)-3010836825000 ((#1)^26)+945857250000 ((#1)^28)-195230250000 ((#1)^30)+19460250000 ((#1)^32)&,22], ]) (NumericalApproximation[Root, (RowDefault["0.143", FEExpressionsNumericalApproximationElider]), Root[1-450 ((#1)^2)+56175 ((#1)^4)-3183900 ((#1)^6)+95178000 ((#1)^8)-1593148050 ((#1)^10)+15069672000 ((#1)^12)-77965348500 ((#1)^14)+200904161625 ((#1)^16)-175240530000 ((#1)^18)-244969953750 ((#1)^20)+623406712500 ((#1)^22)-308513475000 ((#1)^24)-260502075000 ((#1)^26)+351135000000 ((#1)^28)-140616000000 ((#1)^30)+19460250000 ((#1)^32)&,16], ]))),2000],64]

q1soln=Solve[{-2+2 ((NumericalApproximation[Root, (RowDefault["1.00", FEExpressionsNumericalApproximationElider]), Root[2474611-63748665 ((#1)^2)+626509275 ((#1)^4)-2733606225 ((#1)^6)+2323439700 ((#1)^8)+28637733825 ((#1)^10)-130615403625 ((#1)^12)+202845804750 ((#1)^14)+191641136625 ((#1)^16)-1527099345000 ((#1)^18)+3444302317500 ((#1)^20)-4598543812500 ((#1)^22)+4041882450000 ((#1)^24)-2371767075000 ((#1)^26)+897469875000 ((#1)^28)-198369000000 ((#1)^30)+19460250000 ((#1)^32)&,25], ])^2)-q1c (NumericalApproximation[Root, (RowDefault["1.00", FEExpressionsNumericalApproximationElider]), Root[2474611-63748665 ((#1)^2)+626509275 ((#1)^4)-2733606225 ((#1)^6)+2323439700 ((#1)^8)+28637733825 ((#1)^10)-130615403625 ((#1)^12)+202845804750 ((#1)^14)+191641136625 ((#1)^16)-1527099345000 ((#1)^18)+3444302317500 ((#1)^20)-4598543812500 ((#1)^22)+4041882450000 ((#1)^24)-2371767075000 ((#1)^26)+897469875000 ((#1)^28)-198369000000 ((#1)^30)+19460250000 ((#1)^32)&,25], ]) (NumericalApproximation[Root, (RowDefault["0.143", FEExpressionsNumericalApproximationElider]), Root[1-450 ((#1)^2)+56175 ((#1)^4)-3183900 ((#1)^6)+95178000 ((#1)^8)-1593148050 ((#1)^10)+15069672000 ((#1)^12)-77965348500 ((#1)^14)+200904161625 ((#1)^16)-175240530000 ((#1)^18)-244969953750 ((#1)^20)+623406712500 ((#1)^22)-308513475000 ((#1)^24)-260502075000 ((#1)^26)+351135000000 ((#1)^28)-140616000000 ((#1)^30)+19460250000 ((#1)^32)&,16], ])+(Sqrt[5]) q1c (NumericalApproximation[Root, (RowDefault["1.00", FEExpressionsNumericalApproximationElider]), Root[2474611-63748665 ((#1)^2)+626509275 ((#1)^4)-2733606225 ((#1)^6)+2323439700 ((#1)^8)+28637733825 ((#1)^10)-130615403625 ((#1)^12)+202845804750 ((#1)^14)+191641136625 ((#1)^16)-1527099345000 ((#1)^18)+3444302317500 ((#1)^20)-4598543812500 ((#1)^22)+4041882450000 ((#1)^24)-2371767075000 ((#1)^26)+897469875000 ((#1)^28)-198369000000 ((#1)^30)+19460250000 ((#1)^32)&,25], ]) (NumericalApproximation[Root, (RowDefault["0.143", FEExpressionsNumericalApproximationElider]), Root[1-450 ((#1)^2)+56175 ((#1)^4)-3183900 ((#1)^6)+95178000 ((#1)^8)-1593148050 ((#1)^10)+15069672000 ((#1)^12)-77965348500 ((#1)^14)+200904161625 ((#1)^16)-175240530000 ((#1)^18)-244969953750 ((#1)^20)+623406712500 ((#1)^22)-308513475000 ((#1)^24)-260502075000 ((#1)^26)+351135000000 ((#1)^28)-140616000000 ((#1)^30)+19460250000 ((#1)^32)&,16], ])-(Sqrt[2 (5+(Sqrt[5]))]) q1s (NumericalApproximation[Root, (RowDefault["1.00", FEExpressionsNumericalApproximationElider]), Root[2474611-63748665 ((#1)^2)+626509275 ((#1)^4)-2733606225 ((#1)^6)+2323439700 ((#1)^8)+28637733825 ((#1)^10)-130615403625 ((#1)^12)+202845804750 ((#1)^14)+191641136625 ((#1)^16)-1527099345000 ((#1)^18)+3444302317500 ((#1)^20)-4598543812500 ((#1)^22)+4041882450000 ((#1)^24)-2371767075000 ((#1)^26)+897469875000 ((#1)^28)-198369000000 ((#1)^30)+19460250000 ((#1)^32)&,25], ]) (NumericalApproximation[Root, (RowDefault["0.143", FEExpressionsNumericalApproximationElider]), Root[1-450 ((#1)^2)+56175 ((#1)^4)-3183900 ((#1)^6)+95178000 ((#1)^8)-1593148050 ((#1)^10)+15069672000 ((#1)^12)-77965348500 ((#1)^14)+200904161625 ((#1)^16)-175240530000 ((#1)^18)-244969953750 ((#1)^20)+623406712500 ((#1)^22)-308513475000 ((#1)^24)-260502075000 ((#1)^26)+351135000000 ((#1)^28)-140616000000 ((#1)^30)+19460250000 ((#1)^32)&,16], ])+2 ((NumericalApproximation[Root, (RowDefault["0.143", FEExpressionsNumericalApproximationElider]), Root[1-450 ((#1)^2)+56175 ((#1)^4)-3183900 ((#1)^6)+95178000 ((#1)^8)-1593148050 ((#1)^10)+15069672000 ((#1)^12)-77965348500 ((#1)^14)+200904161625 ((#1)^16)-175240530000 ((#1)^18)-244969953750 ((#1)^20)+623406712500 ((#1)^22)-308513475000 ((#1)^24)-260502075000 ((#1)^26)+351135000000 ((#1)^28)-140616000000 ((#1)^30)+19460250000 ((#1)^32)&,16], ])^2)==0,q1c^2+q1s^2==1},{q1c,q1s}]

N[q1soln]

q1solns=q1c/.q1soln

ArcCos/@q1solns//N

Subtract@@%

4Pi/5.

1st solution:

q1c/.q1soln[[1]]

q1ce=RootApproximant[N[(-2 (NumericalApproximation[Root, (RowDefault["1.00", FEExpressionsNumericalApproximationElider]), Root[2474611-63748665 ((#1)^2)+626509275 ((#1)^4)-2733606225 ((#1)^6)+2323439700 ((#1)^8)+28637733825 ((#1)^10)-130615403625 ((#1)^12)+202845804750 ((#1)^14)+191641136625 ((#1)^16)-1527099345000 ((#1)^18)+3444302317500 ((#1)^20)-4598543812500 ((#1)^22)+4041882450000 ((#1)^24)-2371767075000 ((#1)^26)+897469875000 ((#1)^28)-198369000000 ((#1)^30)+19460250000 ((#1)^32)&,25], ]) (NumericalApproximation[Root, (RowDefault["0.143", FEExpressionsNumericalApproximationElider]), Root[1-450 ((#1)^2)+56175 ((#1)^4)-3183900 ((#1)^6)+95178000 ((#1)^8)-1593148050 ((#1)^10)+15069672000 ((#1)^12)-77965348500 ((#1)^14)+200904161625 ((#1)^16)-175240530000 ((#1)^18)-244969953750 ((#1)^20)+623406712500 ((#1)^22)-308513475000 ((#1)^24)-260502075000 ((#1)^26)+351135000000 ((#1)^28)-140616000000 ((#1)^30)+19460250000 ((#1)^32)&,16], ])+2 (Sqrt[5]) (NumericalApproximation[Root, (RowDefault["1.00", FEExpressionsNumericalApproximationElider]), Root[2474611-63748665 ((#1)^2)+626509275 ((#1)^4)-2733606225 ((#1)^6)+2323439700 ((#1)^8)+28637733825 ((#1)^10)-130615403625 ((#1)^12)+202845804750 ((#1)^14)+191641136625 ((#1)^16)-1527099345000 ((#1)^18)+3444302317500 ((#1)^20)-4598543812500 ((#1)^22)+4041882450000 ((#1)^24)-2371767075000 ((#1)^26)+897469875000 ((#1)^28)-198369000000 ((#1)^30)+19460250000 ((#1)^32)&,25], ]) (NumericalApproximation[Root, (RowDefault["0.143", FEExpressionsNumericalApproximationElider]), Root[1-450 ((#1)^2)+56175 ((#1)^4)-3183900 ((#1)^6)+95178000 ((#1)^8)-1593148050 ((#1)^10)+15069672000 ((#1)^12)-77965348500 ((#1)^14)+200904161625 ((#1)^16)-175240530000 ((#1)^18)-244969953750 ((#1)^20)+623406712500 ((#1)^22)-308513475000 ((#1)^24)-260502075000 ((#1)^26)+351135000000 ((#1)^28)-140616000000 ((#1)^30)+19460250000 ((#1)^32)&,16], ])+2 ((NumericalApproximation[Root, (RowDefault["1.00", FEExpressionsNumericalApproximationElider]), Root[2474611-63748665 ((#1)^2)+626509275 ((#1)^4)-2733606225 ((#1)^6)+2323439700 ((#1)^8)+28637733825 ((#1)^10)-130615403625 ((#1)^12)+202845804750 ((#1)^14)+191641136625 ((#1)^16)-1527099345000 ((#1)^18)+3444302317500 ((#1)^20)-4598543812500 ((#1)^22)+4041882450000 ((#1)^24)-2371767075000 ((#1)^26)+897469875000 ((#1)^28)-198369000000 ((#1)^30)+19460250000 ((#1)^32)&,25], ])^3) (NumericalApproximation[Root, (RowDefault["0.143", FEExpressionsNumericalApproximationElider]), Root[1-450 ((#1)^2)+56175 ((#1)^4)-3183900 ((#1)^6)+95178000 ((#1)^8)-1593148050 ((#1)^10)+15069672000 ((#1)^12)-77965348500 ((#1)^14)+200904161625 ((#1)^16)-175240530000 ((#1)^18)-244969953750 ((#1)^20)+623406712500 ((#1)^22)-308513475000 ((#1)^24)-260502075000 ((#1)^26)+351135000000 ((#1)^28)-140616000000 ((#1)^30)+19460250000 ((#1)^32)&,16], ])-2 (Sqrt[5]) ((NumericalApproximation[Root, (RowDefault["1.00", FEExpressionsNumericalApproximationElider]), Root[2474611-63748665 ((#1)^2)+626509275 ((#1)^4)-2733606225 ((#1)^6)+2323439700 ((#1)^8)+28637733825 ((#1)^10)-130615403625 ((#1)^12)+202845804750 ((#1)^14)+191641136625 ((#1)^16)-1527099345000 ((#1)^18)+3444302317500 ((#1)^20)-4598543812500 ((#1)^22)+4041882450000 ((#1)^24)-2371767075000 ((#1)^26)+897469875000 ((#1)^28)-198369000000 ((#1)^30)+19460250000 ((#1)^32)&,25], ])^3) (NumericalApproximation[Root, (RowDefault["0.143", FEExpressionsNumericalApproximationElider]), Root[1-450 ((#1)^2)+56175 ((#1)^4)-3183900 ((#1)^6)+95178000 ((#1)^8)-1593148050 ((#1)^10)+15069672000 ((#1)^12)-77965348500 ((#1)^14)+200904161625 ((#1)^16)-175240530000 ((#1)^18)-244969953750 ((#1)^20)+623406712500 ((#1)^22)-308513475000 ((#1)^24)-260502075000 ((#1)^26)+351135000000 ((#1)^28)-140616000000 ((#1)^30)+19460250000 ((#1)^32)&,16], ])+2 (NumericalApproximation[Root, (RowDefault["1.00", FEExpressionsNumericalApproximationElider]), Root[2474611-63748665 ((#1)^2)+626509275 ((#1)^4)-2733606225 ((#1)^6)+2323439700 ((#1)^8)+28637733825 ((#1)^10)-130615403625 ((#1)^12)+202845804750 ((#1)^14)+191641136625 ((#1)^16)-1527099345000 ((#1)^18)+3444302317500 ((#1)^20)-4598543812500 ((#1)^22)+4041882450000 ((#1)^24)-2371767075000 ((#1)^26)+897469875000 ((#1)^28)-198369000000 ((#1)^30)+19460250000 ((#1)^32)&,25], ]) ((NumericalApproximation[Root, (RowDefault["0.143", FEExpressionsNumericalApproximationElider]), Root[1-450 ((#1)^2)+56175 ((#1)^4)-3183900 ((#1)^6)+95178000 ((#1)^8)-1593148050 ((#1)^10)+15069672000 ((#1)^12)-77965348500 ((#1)^14)+200904161625 ((#1)^16)-175240530000 ((#1)^18)-244969953750 ((#1)^20)+623406712500 ((#1)^22)-308513475000 ((#1)^24)-260502075000 ((#1)^26)+351135000000 ((#1)^28)-140616000000 ((#1)^30)+19460250000 ((#1)^32)&,16], ])^3)-2 (Sqrt[5]) (NumericalApproximation[Root, (RowDefault["1.00", FEExpressionsNumericalApproximationElider]), Root[2474611-63748665 ((#1)^2)+626509275 ((#1)^4)-2733606225 ((#1)^6)+2323439700 ((#1)^8)+28637733825 ((#1)^10)-130615403625 ((#1)^12)+202845804750 ((#1)^14)+191641136625 ((#1)^16)-1527099345000 ((#1)^18)+3444302317500 ((#1)^20)-4598543812500 ((#1)^22)+4041882450000 ((#1)^24)-2371767075000 ((#1)^26)+897469875000 ((#1)^28)-198369000000 ((#1)^30)+19460250000 ((#1)^32)&,25], ]) ((NumericalApproximation[Root, (RowDefault["0.143", FEExpressionsNumericalApproximationElider]), Root[1-450 ((#1)^2)+56175 ((#1)^4)-3183900 ((#1)^6)+95178000 ((#1)^8)-1593148050 ((#1)^10)+15069672000 ((#1)^12)-77965348500 ((#1)^14)+200904161625 ((#1)^16)-175240530000 ((#1)^18)-244969953750 ((#1)^20)+623406712500 ((#1)^22)-308513475000 ((#1)^24)-260502075000 ((#1)^26)+351135000000 ((#1)^28)-140616000000 ((#1)^30)+19460250000 ((#1)^32)&,16], ])^3)-√((((2 (NumericalApproximation[Root, (RowDefault["1.00", FEExpressionsNumericalApproximationElider]), Root[2474611-63748665 ((#1)^2)+626509275 ((#1)^4)-2733606225 ((#1)^6)+2323439700 ((#1)^8)+28637733825 ((#1)^10)-130615403625 ((#1)^12)+202845804750 ((#1)^14)+191641136625 ((#1)^16)-1527099345000 ((#1)^18)+3444302317500 ((#1)^20)-4598543812500 ((#1)^22)+4041882450000 ((#1)^24)-2371767075000 ((#1)^26)+897469875000 ((#1)^28)-198369000000 ((#1)^30)+19460250000 ((#1)^32)&,25], ]) (NumericalApproximation[Root, (RowDefault["0.143", FEExpressionsNumericalApproximationElider]), Root[1-450 ((#1)^2)+56175 ((#1)^4)-3183900 ((#1)^6)+95178000 ((#1)^8)-1593148050 ((#1)^10)+15069672000 ((#1)^12)-77965348500 ((#1)^14)+200904161625 ((#1)^16)-175240530000 ((#1)^18)-244969953750 ((#1)^20)+623406712500 ((#1)^22)-308513475000 ((#1)^24)-260502075000 ((#1)^26)+351135000000 ((#1)^28)-140616000000 ((#1)^30)+19460250000 ((#1)^32)&,16], ])-2 (Sqrt[5]) (NumericalApproximation[Root, (RowDefault["1.00", FEExpressionsNumericalApproximationElider]), Root[2474611-63748665 ((#1)^2)+626509275 ((#1)^4)-2733606225 ((#1)^6)+2323439700 ((#1)^8)+28637733825 ((#1)^10)-130615403625 ((#1)^12)+202845804750 ((#1)^14)+191641136625 ((#1)^16)-1527099345000 ((#1)^18)+3444302317500 ((#1)^20)-4598543812500 ((#1)^22)+4041882450000 ((#1)^24)-2371767075000 ((#1)^26)+897469875000 ((#1)^28)-198369000000 ((#1)^30)+19460250000 ((#1)^32)&,25], ]) (NumericalApproximation[Root, (RowDefault["0.143", FEExpressionsNumericalApproximationElider]), Root[1-450 ((#1)^2)+56175 ((#1)^4)-3183900 ((#1)^6)+95178000 ((#1)^8)-1593148050 ((#1)^10)+15069672000 ((#1)^12)-77965348500 ((#1)^14)+200904161625 ((#1)^16)-175240530000 ((#1)^18)-244969953750 ((#1)^20)+623406712500 ((#1)^22)-308513475000 ((#1)^24)-260502075000 ((#1)^26)+351135000000 ((#1)^28)-140616000000 ((#1)^30)+19460250000 ((#1)^32)&,16], ])-2 ((NumericalApproximation[Root, (RowDefault["1.00", FEExpressionsNumericalApproximationElider]), Root[2474611-63748665 ((#1)^2)+626509275 ((#1)^4)-2733606225 ((#1)^6)+2323439700 ((#1)^8)+28637733825 ((#1)^10)-130615403625 ((#1)^12)+202845804750 ((#1)^14)+191641136625 ((#1)^16)-1527099345000 ((#1)^18)+3444302317500 ((#1)^20)-4598543812500 ((#1)^22)+4041882450000 ((#1)^24)-2371767075000 ((#1)^26)+897469875000 ((#1)^28)-198369000000 ((#1)^30)+19460250000 ((#1)^32)&,25], ])^3) (NumericalApproximation[Root, (RowDefault["0.143", FEExpressionsNumericalApproximationElider]), Root[1-450 ((#1)^2)+56175 ((#1)^4)-3183900 ((#1)^6)+95178000 ((#1)^8)-1593148050 ((#1)^10)+15069672000 ((#1)^12)-77965348500 ((#1)^14)+200904161625 ((#1)^16)-175240530000 ((#1)^18)-244969953750 ((#1)^20)+623406712500 ((#1)^22)-308513475000 ((#1)^24)-260502075000 ((#1)^26)+351135000000 ((#1)^28)-140616000000 ((#1)^30)+19460250000 ((#1)^32)&,16], ])+2 (Sqrt[5]) ((NumericalApproximation[Root, (RowDefault["1.00", FEExpressionsNumericalApproximationElider]), Root[2474611-63748665 ((#1)^2)+626509275 ((#1)^4)-2733606225 ((#1)^6)+2323439700 ((#1)^8)+28637733825 ((#1)^10)-130615403625 ((#1)^12)+202845804750 ((#1)^14)+191641136625 ((#1)^16)-1527099345000 ((#1)^18)+3444302317500 ((#1)^20)-4598543812500 ((#1)^22)+4041882450000 ((#1)^24)-2371767075000 ((#1)^26)+897469875000 ((#1)^28)-198369000000 ((#1)^30)+19460250000 ((#1)^32)&,25], ])^3) (NumericalApproximation[Root, (RowDefault["0.143", FEExpressionsNumericalApproximationElider]), Root[1-450 ((#1)^2)+56175 ((#1)^4)-3183900 ((#1)^6)+95178000 ((#1)^8)-1593148050 ((#1)^10)+15069672000 ((#1)^12)-77965348500 ((#1)^14)+200904161625 ((#1)^16)-175240530000 ((#1)^18)-244969953750 ((#1)^20)+623406712500 ((#1)^22)-308513475000 ((#1)^24)-260502075000 ((#1)^26)+351135000000 ((#1)^28)-140616000000 ((#1)^30)+19460250000 ((#1)^32)&,16], ])-2 (NumericalApproximation[Root, (RowDefault["1.00", FEExpressionsNumericalApproximationElider]), Root[2474611-63748665 ((#1)^2)+626509275 ((#1)^4)-2733606225 ((#1)^6)+2323439700 ((#1)^8)+28637733825 ((#1)^10)-130615403625 ((#1)^12)+202845804750 ((#1)^14)+191641136625 ((#1)^16)-1527099345000 ((#1)^18)+3444302317500 ((#1)^20)-4598543812500 ((#1)^22)+4041882450000 ((#1)^24)-2371767075000 ((#1)^26)+897469875000 ((#1)^28)-198369000000 ((#1)^30)+19460250000 ((#1)^32)&,25], ]) ((NumericalApproximation[Root, (RowDefault["0.143", FEExpressionsNumericalApproximationElider]), Root[1-450 ((#1)^2)+56175 ((#1)^4)-3183900 ((#1)^6)+95178000 ((#1)^8)-1593148050 ((#1)^10)+15069672000 ((#1)^12)-77965348500 ((#1)^14)+200904161625 ((#1)^16)-175240530000 ((#1)^18)-244969953750 ((#1)^20)+623406712500 ((#1)^22)-308513475000 ((#1)^24)-260502075000 ((#1)^26)+351135000000 ((#1)^28)-140616000000 ((#1)^30)+19460250000 ((#1)^32)&,16], ])^3)+2 (Sqrt[5]) (NumericalApproximation[Root, (RowDefault["1.00", FEExpressionsNumericalApproximationElider]), Root[2474611-63748665 ((#1)^2)+626509275 ((#1)^4)-2733606225 ((#1)^6)+2323439700 ((#1)^8)+28637733825 ((#1)^10)-130615403625 ((#1)^12)+202845804750 ((#1)^14)+191641136625 ((#1)^16)-1527099345000 ((#1)^18)+3444302317500 ((#1)^20)-4598543812500 ((#1)^22)+4041882450000 ((#1)^24)-2371767075000 ((#1)^26)+897469875000 ((#1)^28)-198369000000 ((#1)^30)+19460250000 ((#1)^32)&,25], ]) ((NumericalApproximation[Root, (RowDefault["0.143", FEExpressionsNumericalApproximationElider]), Root[1-450 ((#1)^2)+56175 ((#1)^4)-3183900 ((#1)^6)+95178000 ((#1)^8)-1593148050 ((#1)^10)+15069672000 ((#1)^12)-77965348500 ((#1)^14)+200904161625 ((#1)^16)-175240530000 ((#1)^18)-244969953750 ((#1)^20)+623406712500 ((#1)^22)-308513475000 ((#1)^24)-260502075000 ((#1)^26)+351135000000 ((#1)^28)-140616000000 ((#1)^30)+19460250000 ((#1)^32)&,16], ])^3)))^2)-32 ((NumericalApproximation[Root, (RowDefault["1.00", FEExpressionsNumericalApproximationElider]), Root[2474611-63748665 ((#1)^2)+626509275 ((#1)^4)-2733606225 ((#1)^6)+2323439700 ((#1)^8)+28637733825 ((#1)^10)-130615403625 ((#1)^12)+202845804750 ((#1)^14)+191641136625 ((#1)^16)-1527099345000 ((#1)^18)+3444302317500 ((#1)^20)-4598543812500 ((#1)^22)+4041882450000 ((#1)^24)-2371767075000 ((#1)^26)+897469875000 ((#1)^28)-198369000000 ((#1)^30)+19460250000 ((#1)^32)&,25], ])^2) ((NumericalApproximation[Root, (RowDefault["0.143", FEExpressionsNumericalApproximationElider]), Root[1-450 ((#1)^2)+56175 ((#1)^4)-3183900 ((#1)^6)+95178000 ((#1)^8)-1593148050 ((#1)^10)+15069672000 ((#1)^12)-77965348500 ((#1)^14)+200904161625 ((#1)^16)-175240530000 ((#1)^18)-244969953750 ((#1)^20)+623406712500 ((#1)^22)-308513475000 ((#1)^24)-260502075000 ((#1)^26)+351135000000 ((#1)^28)-140616000000 ((#1)^30)+19460250000 ((#1)^32)&,16], ])^2) (2-4 ((NumericalApproximation[Root, (RowDefault["1.00", FEExpressionsNumericalApproximationElider]), Root[2474611-63748665 ((#1)^2)+626509275 ((#1)^4)-2733606225 ((#1)^6)+2323439700 ((#1)^8)+28637733825 ((#1)^10)-130615403625 ((#1)^12)+202845804750 ((#1)^14)+191641136625 ((#1)^16)-1527099345000 ((#1)^18)+3444302317500 ((#1)^20)-4598543812500 ((#1)^22)+4041882450000 ((#1)^24)-2371767075000 ((#1)^26)+897469875000 ((#1)^28)-198369000000 ((#1)^30)+19460250000 ((#1)^32)&,25], ])^2)+2 ((NumericalApproximation[Root, (RowDefault["1.00", FEExpressionsNumericalApproximationElider]), Root[2474611-63748665 ((#1)^2)+626509275 ((#1)^4)-2733606225 ((#1)^6)+2323439700 ((#1)^8)+28637733825 ((#1)^10)-130615403625 ((#1)^12)+202845804750 ((#1)^14)+191641136625 ((#1)^16)-1527099345000 ((#1)^18)+3444302317500 ((#1)^20)-4598543812500 ((#1)^22)+4041882450000 ((#1)^24)-2371767075000 ((#1)^26)+897469875000 ((#1)^28)-198369000000 ((#1)^30)+19460250000 ((#1)^32)&,25], ])^4)-4 ((NumericalApproximation[Root, (RowDefault["0.143", FEExpressionsNumericalApproximationElider]), Root[1-450 ((#1)^2)+56175 ((#1)^4)-3183900 ((#1)^6)+95178000 ((#1)^8)-1593148050 ((#1)^10)+15069672000 ((#1)^12)-77965348500 ((#1)^14)+200904161625 ((#1)^16)-175240530000 ((#1)^18)-244969953750 ((#1)^20)+623406712500 ((#1)^22)-308513475000 ((#1)^24)-260502075000 ((#1)^26)+351135000000 ((#1)^28)-140616000000 ((#1)^30)+19460250000 ((#1)^32)&,16], ])^2)-((NumericalApproximation[Root, (RowDefault["1.00", FEExpressionsNumericalApproximationElider]), Root[2474611-63748665 ((#1)^2)+626509275 ((#1)^4)-2733606225 ((#1)^6)+2323439700 ((#1)^8)+28637733825 ((#1)^10)-130615403625 ((#1)^12)+202845804750 ((#1)^14)+191641136625 ((#1)^16)-1527099345000 ((#1)^18)+3444302317500 ((#1)^20)-4598543812500 ((#1)^22)+4041882450000 ((#1)^24)-2371767075000 ((#1)^26)+897469875000 ((#1)^28)-198369000000 ((#1)^30)+19460250000 ((#1)^32)&,25], ])^2) ((NumericalApproximation[Root, (RowDefault["0.143", FEExpressionsNumericalApproximationElider]), Root[1-450 ((#1)^2)+56175 ((#1)^4)-3183900 ((#1)^6)+95178000 ((#1)^8)-1593148050 ((#1)^10)+15069672000 ((#1)^12)-77965348500 ((#1)^14)+200904161625 ((#1)^16)-175240530000 ((#1)^18)-244969953750 ((#1)^20)+623406712500 ((#1)^22)-308513475000 ((#1)^24)-260502075000 ((#1)^26)+351135000000 ((#1)^28)-140616000000 ((#1)^30)+19460250000 ((#1)^32)&,16], ])^2)-(Sqrt[5]) ((NumericalApproximation[Root, (RowDefault["1.00", FEExpressionsNumericalApproximationElider]), Root[2474611-63748665 ((#1)^2)+626509275 ((#1)^4)-2733606225 ((#1)^6)+2323439700 ((#1)^8)+28637733825 ((#1)^10)-130615403625 ((#1)^12)+202845804750 ((#1)^14)+191641136625 ((#1)^16)-1527099345000 ((#1)^18)+3444302317500 ((#1)^20)-4598543812500 ((#1)^22)+4041882450000 ((#1)^24)-2371767075000 ((#1)^26)+897469875000 ((#1)^28)-198369000000 ((#1)^30)+19460250000 ((#1)^32)&,25], ])^2) ((NumericalApproximation[Root, (RowDefault["0.143", FEExpressionsNumericalApproximationElider]), Root[1-450 ((#1)^2)+56175 ((#1)^4)-3183900 ((#1)^6)+95178000 ((#1)^8)-1593148050 ((#1)^10)+15069672000 ((#1)^12)-77965348500 ((#1)^14)+200904161625 ((#1)^16)-175240530000 ((#1)^18)-244969953750 ((#1)^20)+623406712500 ((#1)^22)-308513475000 ((#1)^24)-260502075000 ((#1)^26)+351135000000 ((#1)^28)-140616000000 ((#1)^30)+19460250000 ((#1)^32)&,16], ])^2)+2 ((NumericalApproximation[Root, (RowDefault["0.143", FEExpressionsNumericalApproximationElider]), Root[1-450 ((#1)^2)+56175 ((#1)^4)-3183900 ((#1)^6)+95178000 ((#1)^8)-1593148050 ((#1)^10)+15069672000 ((#1)^12)-77965348500 ((#1)^14)+200904161625 ((#1)^16)-175240530000 ((#1)^18)-244969953750 ((#1)^20)+623406712500 ((#1)^22)-308513475000 ((#1)^24)-260502075000 ((#1)^26)+351135000000 ((#1)^28)-140616000000 ((#1)^30)+19460250000 ((#1)^32)&,16], ])^4))))/(16 ((NumericalApproximation[Root, (RowDefault["1.00", FEExpressionsNumericalApproximationElider]), Root[2474611-63748665 ((#1)^2)+626509275 ((#1)^4)-2733606225 ((#1)^6)+2323439700 ((#1)^8)+28637733825 ((#1)^10)-130615403625 ((#1)^12)+202845804750 ((#1)^14)+191641136625 ((#1)^16)-1527099345000 ((#1)^18)+3444302317500 ((#1)^20)-4598543812500 ((#1)^22)+4041882450000 ((#1)^24)-2371767075000 ((#1)^26)+897469875000 ((#1)^28)-198369000000 ((#1)^30)+19460250000 ((#1)^32)&,25], ])^2) ((NumericalApproximation[Root, (RowDefault["0.143", FEExpressionsNumericalApproximationElider]), Root[1-450 ((#1)^2)+56175 ((#1)^4)-3183900 ((#1)^6)+95178000 ((#1)^8)-1593148050 ((#1)^10)+15069672000 ((#1)^12)-77965348500 ((#1)^14)+200904161625 ((#1)^16)-175240530000 ((#1)^18)-244969953750 ((#1)^20)+623406712500 ((#1)^22)-308513475000 ((#1)^24)-260502075000 ((#1)^26)+351135000000 ((#1)^28)-140616000000 ((#1)^30)+19460250000 ((#1)^32)&,16], ])^2)),2000],64]

finalsoln1={a->(NumericalApproximation[Root, (RowDefault["0.143", FEExpressionsNumericalApproximationElider]), Root[1-450 ((#1)^2)+56175 ((#1)^4)-3183900 ((#1)^6)+95178000 ((#1)^8)-1593148050 ((#1)^10)+15069672000 ((#1)^12)-77965348500 ((#1)^14)+200904161625 ((#1)^16)-175240530000 ((#1)^18)-244969953750 ((#1)^20)+623406712500 ((#1)^22)-308513475000 ((#1)^24)-260502075000 ((#1)^26)+351135000000 ((#1)^28)-140616000000 ((#1)^30)+19460250000 ((#1)^32)&,16], ]),b->(NumericalApproximation[Root, (RowDefault["0.741", FEExpressionsNumericalApproximationElider]), Root[1-133545 ((#1)^2)+15392955 ((#1)^4)-658758525 ((#1)^6)+13385251050 ((#1)^8)-140651762175 ((#1)^10)+852442151625 ((#1)^12)-3256025184000 ((#1)^14)+8320298902875 ((#1)^16)-14768681703750 ((#1)^18)+18601688981250 ((#1)^20)-16757836875000 ((#1)^22)+10739082937500 ((#1)^24)-4786478325000 ((#1)^26)+1411293375000 ((#1)^28)-247333500000 ((#1)^30)+19460250000 ((#1)^32)&,21], ]),ab->(NumericalApproximation[Root, (RowDefault["0.998", FEExpressionsNumericalApproximationElider]), Root[73441-528217755 ((#1)^2)+23460408885 ((#1)^4)-379618995675 ((#1)^6)+3255086120100 ((#1)^8)-17156053557675 ((#1)^10)+59993606215125 ((#1)^12)-145401765113250 ((#1)^14)+250375803161625 ((#1)^16)-309939662115000 ((#1)^18)+276042485857500 ((#1)^20)-175100781037500 ((#1)^22)+77310013275000 ((#1)^24)-22821340950000 ((#1)^26)+4214237625000 ((#1)^28)-434403000000 ((#1)^30)+19460250000 ((#1)^32)&,25], ]),c->(NumericalApproximation[Root, (RowDefault["1.00", FEExpressionsNumericalApproximationElider]), Root[2474611-63748665 ((#1)^2)+626509275 ((#1)^4)-2733606225 ((#1)^6)+2323439700 ((#1)^8)+28637733825 ((#1)^10)-130615403625 ((#1)^12)+202845804750 ((#1)^14)+191641136625 ((#1)^16)-1527099345000 ((#1)^18)+3444302317500 ((#1)^20)-4598543812500 ((#1)^22)+4041882450000 ((#1)^24)-2371767075000 ((#1)^26)+897469875000 ((#1)^28)-198369000000 ((#1)^30)+19460250000 ((#1)^32)&,25], ]),R->(NumericalApproximation[Root, (RowDefault["1.50", FEExpressionsNumericalApproximationElider]), Root[73441-13991070 ((#1)^2)+940040580 ((#1)^4)-26938210200 ((#1)^6)+315440426475 ((#1)^8)-1301745023550 ((#1)^10)+2222016066000 ((#1)^12)-780590052000 ((#1)^14)-2073848628375 ((#1)^16)+484548041250 ((#1)^18)+6867324011250 ((#1)^20)-12410051850000 ((#1)^22)+10768629037500 ((#1)^24)-5456935575000 ((#1)^26)+1646092125000 ((#1)^28)-274326750000 ((#1)^30)+19460250000 ((#1)^32)&,26], ]),q1->ArcCos[(NumericalApproximation[Root, (RowDefault["-0.978", FEExpressionsNumericalApproximationElider]), Root[961-1657861 ((#1)^2)+573178150 ((#1)^4)-50490728030 ((#1)^6)+1716129484325 ((#1)^8)-28064457368368 ((#1)^10)+252975507764048 ((#1)^12)-1379795708384000 ((#1)^14)+4852683847778560 ((#1)^16)-11438736264243200 ((#1)^18)+18417044453040128 ((#1)^20)-20298765578764288 ((#1)^22)+15106483222937600 ((#1)^24)-7364941544161280 ((#1)^26)+2249369190400000 ((#1)^28)-411435947524096 ((#1)^30)+41517083262976 ((#1)^32)&,3], ])],q2->ArcCos[(NumericalApproximation[Root, (RowDefault["0.847", FEExpressionsNumericalApproximationElider]), Root[73441-34197271 ((#1)^2)+4525338125 ((#1)^4)-231217822005 ((#1)^6)+4849448134775 ((#1)^8)-55324230938408 ((#1)^10)+394636911570928 ((#1)^12)-1897986578659200 ((#1)^14)+6429316497813760 ((#1)^16)-15727635924633600 ((#1)^18)+28114830613024768 ((#1)^20)-36748859931885568 ((#1)^22)+34727007941427200 ((#1)^24)-23089990800506880 ((#1)^26)+10242838967091200 ((#1)^28)-2720679304626176 ((#1)^30)+327223173185536 ((#1)^32)&,26], ])],d->(NumericalApproximation[Root, (RowDefault["0.876", FEExpressionsNumericalApproximationElider]), Root[19504021-1021550925 ((#1)^2)+16826333925 ((#1)^4)-144026777475 ((#1)^6)+767601900150 ((#1)^8)-2786101493175 ((#1)^10)+7248691249875 ((#1)^12)-13951699884000 ((#1)^14)+20279794395375 ((#1)^16)-22595324827500 ((#1)^18)+19541839275000 ((#1)^20)-13276625962500 ((#1)^22)+7140777412500 ((#1)^24)-3010836825000 ((#1)^26)+945857250000 ((#1)^28)-195230250000 ((#1)^30)+19460250000 ((#1)^32)&,22], ]),eps->(NumericalApproximation[Root, (RowDefault["6.89", FEExpressionsNumericalApproximationElider, " × ", (Superscript["10", "-3"])]), Root[1-21330 ((#1)^2)+5543010 ((#1)^4)-456619500 ((#1)^6)+15778625475 ((#1)^8)-258239033550 ((#1)^10)+2174982842250 ((#1)^12)-10516099014000 ((#1)^14)+31582836559125 ((#1)^16)-61799968608750 ((#1)^18)+80698222687500 ((#1)^20)-70620534112500 ((#1)^22)+40806591412500 ((#1)^24)-15004020825000 ((#1)^26)+3286878750000 ((#1)^28)-385438500000 ((#1)^30)+19460250000 ((#1)^32)&,15], ])};

ee=Select[Subsets[Range[Length[v]],{2}],Abs[EuclideanDistance@@v[[#]]-1]<10^-4&]//Quiet

g1=Graph[Range[Length[v]],ee,VertexCoordinates->(v/.finalsoln1)]

RecognizeGraph[g1]

2nd solution:

q1ce=RootApproximant[N[(1/(16 ((NumericalApproximation[Root, (RowDefault["1.00", FEExpressionsNumericalApproximationElider]), Root[2474611-63748665 ((#1)^2)+626509275 ((#1)^4)-2733606225 ((#1)^6)+2323439700 ((#1)^8)+28637733825 ((#1)^10)-130615403625 ((#1)^12)+202845804750 ((#1)^14)+191641136625 ((#1)^16)-1527099345000 ((#1)^18)+3444302317500 ((#1)^20)-4598543812500 ((#1)^22)+4041882450000 ((#1)^24)-2371767075000 ((#1)^26)+897469875000 ((#1)^28)-198369000000 ((#1)^30)+19460250000 ((#1)^32)&,25], ])^2) ((NumericalApproximation[Root, (RowDefault["0.143", FEExpressionsNumericalApproximationElider]), Root[1-450 ((#1)^2)+56175 ((#1)^4)-3183900 ((#1)^6)+95178000 ((#1)^8)-1593148050 ((#1)^10)+15069672000 ((#1)^12)-77965348500 ((#1)^14)+200904161625 ((#1)^16)-175240530000 ((#1)^18)-244969953750 ((#1)^20)+623406712500 ((#1)^22)-308513475000 ((#1)^24)-260502075000 ((#1)^26)+351135000000 ((#1)^28)-140616000000 ((#1)^30)+19460250000 ((#1)^32)&,16], ])^2)))(-2 (NumericalApproximation[Root, (RowDefault["1.00", FEExpressionsNumericalApproximationElider]), Root[2474611-63748665 ((#1)^2)+626509275 ((#1)^4)-2733606225 ((#1)^6)+2323439700 ((#1)^8)+28637733825 ((#1)^10)-130615403625 ((#1)^12)+202845804750 ((#1)^14)+191641136625 ((#1)^16)-1527099345000 ((#1)^18)+3444302317500 ((#1)^20)-4598543812500 ((#1)^22)+4041882450000 ((#1)^24)-2371767075000 ((#1)^26)+897469875000 ((#1)^28)-198369000000 ((#1)^30)+19460250000 ((#1)^32)&,25], ]) (NumericalApproximation[Root, (RowDefault["0.143", FEExpressionsNumericalApproximationElider]), Root[1-450 ((#1)^2)+56175 ((#1)^4)-3183900 ((#1)^6)+95178000 ((#1)^8)-1593148050 ((#1)^10)+15069672000 ((#1)^12)-77965348500 ((#1)^14)+200904161625 ((#1)^16)-175240530000 ((#1)^18)-244969953750 ((#1)^20)+623406712500 ((#1)^22)-308513475000 ((#1)^24)-260502075000 ((#1)^26)+351135000000 ((#1)^28)-140616000000 ((#1)^30)+19460250000 ((#1)^32)&,16], ])+2 (Sqrt[5]) (NumericalApproximation[Root, (RowDefault["1.00", FEExpressionsNumericalApproximationElider]), Root[2474611-63748665 ((#1)^2)+626509275 ((#1)^4)-2733606225 ((#1)^6)+2323439700 ((#1)^8)+28637733825 ((#1)^10)-130615403625 ((#1)^12)+202845804750 ((#1)^14)+191641136625 ((#1)^16)-1527099345000 ((#1)^18)+3444302317500 ((#1)^20)-4598543812500 ((#1)^22)+4041882450000 ((#1)^24)-2371767075000 ((#1)^26)+897469875000 ((#1)^28)-198369000000 ((#1)^30)+19460250000 ((#1)^32)&,25], ]) (NumericalApproximation[Root, (RowDefault["0.143", FEExpressionsNumericalApproximationElider]), Root[1-450 ((#1)^2)+56175 ((#1)^4)-3183900 ((#1)^6)+95178000 ((#1)^8)-1593148050 ((#1)^10)+15069672000 ((#1)^12)-77965348500 ((#1)^14)+200904161625 ((#1)^16)-175240530000 ((#1)^18)-244969953750 ((#1)^20)+623406712500 ((#1)^22)-308513475000 ((#1)^24)-260502075000 ((#1)^26)+351135000000 ((#1)^28)-140616000000 ((#1)^30)+19460250000 ((#1)^32)&,16], ])+2 ((NumericalApproximation[Root, (RowDefault["1.00", FEExpressionsNumericalApproximationElider]), Root[2474611-63748665 ((#1)^2)+626509275 ((#1)^4)-2733606225 ((#1)^6)+2323439700 ((#1)^8)+28637733825 ((#1)^10)-130615403625 ((#1)^12)+202845804750 ((#1)^14)+191641136625 ((#1)^16)-1527099345000 ((#1)^18)+3444302317500 ((#1)^20)-4598543812500 ((#1)^22)+4041882450000 ((#1)^24)-2371767075000 ((#1)^26)+897469875000 ((#1)^28)-198369000000 ((#1)^30)+19460250000 ((#1)^32)&,25], ])^3) (NumericalApproximation[Root, (RowDefault["0.143", FEExpressionsNumericalApproximationElider]), Root[1-450 ((#1)^2)+56175 ((#1)^4)-3183900 ((#1)^6)+95178000 ((#1)^8)-1593148050 ((#1)^10)+15069672000 ((#1)^12)-77965348500 ((#1)^14)+200904161625 ((#1)^16)-175240530000 ((#1)^18)-244969953750 ((#1)^20)+623406712500 ((#1)^22)-308513475000 ((#1)^24)-260502075000 ((#1)^26)+351135000000 ((#1)^28)-140616000000 ((#1)^30)+19460250000 ((#1)^32)&,16], ])-2 (Sqrt[5]) ((NumericalApproximation[Root, (RowDefault["1.00", FEExpressionsNumericalApproximationElider]), Root[2474611-63748665 ((#1)^2)+626509275 ((#1)^4)-2733606225 ((#1)^6)+2323439700 ((#1)^8)+28637733825 ((#1)^10)-130615403625 ((#1)^12)+202845804750 ((#1)^14)+191641136625 ((#1)^16)-1527099345000 ((#1)^18)+3444302317500 ((#1)^20)-4598543812500 ((#1)^22)+4041882450000 ((#1)^24)-2371767075000 ((#1)^26)+897469875000 ((#1)^28)-198369000000 ((#1)^30)+19460250000 ((#1)^32)&,25], ])^3) (NumericalApproximation[Root, (RowDefault["0.143", FEExpressionsNumericalApproximationElider]), Root[1-450 ((#1)^2)+56175 ((#1)^4)-3183900 ((#1)^6)+95178000 ((#1)^8)-1593148050 ((#1)^10)+15069672000 ((#1)^12)-77965348500 ((#1)^14)+200904161625 ((#1)^16)-175240530000 ((#1)^18)-244969953750 ((#1)^20)+623406712500 ((#1)^22)-308513475000 ((#1)^24)-260502075000 ((#1)^26)+351135000000 ((#1)^28)-140616000000 ((#1)^30)+19460250000 ((#1)^32)&,16], ])+2 (NumericalApproximation[Root, (RowDefault["1.00", FEExpressionsNumericalApproximationElider]), Root[2474611-63748665 ((#1)^2)+626509275 ((#1)^4)-2733606225 ((#1)^6)+2323439700 ((#1)^8)+28637733825 ((#1)^10)-130615403625 ((#1)^12)+202845804750 ((#1)^14)+191641136625 ((#1)^16)-1527099345000 ((#1)^18)+3444302317500 ((#1)^20)-4598543812500 ((#1)^22)+4041882450000 ((#1)^24)-2371767075000 ((#1)^26)+897469875000 ((#1)^28)-198369000000 ((#1)^30)+19460250000 ((#1)^32)&,25], ]) ((NumericalApproximation[Root, (RowDefault["0.143", FEExpressionsNumericalApproximationElider]), Root[1-450 ((#1)^2)+56175 ((#1)^4)-3183900 ((#1)^6)+95178000 ((#1)^8)-1593148050 ((#1)^10)+15069672000 ((#1)^12)-77965348500 ((#1)^14)+200904161625 ((#1)^16)-175240530000 ((#1)^18)-244969953750 ((#1)^20)+623406712500 ((#1)^22)-308513475000 ((#1)^24)-260502075000 ((#1)^26)+351135000000 ((#1)^28)-140616000000 ((#1)^30)+19460250000 ((#1)^32)&,16], ])^3)-2 (Sqrt[5]) (NumericalApproximation[Root, (RowDefault["1.00", FEExpressionsNumericalApproximationElider]), Root[2474611-63748665 ((#1)^2)+626509275 ((#1)^4)-2733606225 ((#1)^6)+2323439700 ((#1)^8)+28637733825 ((#1)^10)-130615403625 ((#1)^12)+202845804750 ((#1)^14)+191641136625 ((#1)^16)-1527099345000 ((#1)^18)+3444302317500 ((#1)^20)-4598543812500 ((#1)^22)+4041882450000 ((#1)^24)-2371767075000 ((#1)^26)+897469875000 ((#1)^28)-198369000000 ((#1)^30)+19460250000 ((#1)^32)&,25], ]) ((NumericalApproximation[Root, (RowDefault["0.143", FEExpressionsNumericalApproximationElider]), Root[1-450 ((#1)^2)+56175 ((#1)^4)-3183900 ((#1)^6)+95178000 ((#1)^8)-1593148050 ((#1)^10)+15069672000 ((#1)^12)-77965348500 ((#1)^14)+200904161625 ((#1)^16)-175240530000 ((#1)^18)-244969953750 ((#1)^20)+623406712500 ((#1)^22)-308513475000 ((#1)^24)-260502075000 ((#1)^26)+351135000000 ((#1)^28)-140616000000 ((#1)^30)+19460250000 ((#1)^32)&,16], ])^3)+√((((2 (NumericalApproximation[Root, (RowDefault["1.00", FEExpressionsNumericalApproximationElider]), Root[2474611-63748665 ((#1)^2)+626509275 ((#1)^4)-2733606225 ((#1)^6)+2323439700 ((#1)^8)+28637733825 ((#1)^10)-130615403625 ((#1)^12)+202845804750 ((#1)^14)+191641136625 ((#1)^16)-1527099345000 ((#1)^18)+3444302317500 ((#1)^20)-4598543812500 ((#1)^22)+4041882450000 ((#1)^24)-2371767075000 ((#1)^26)+897469875000 ((#1)^28)-198369000000 ((#1)^30)+19460250000 ((#1)^32)&,25], ]) (NumericalApproximation[Root, (RowDefault["0.143", FEExpressionsNumericalApproximationElider]), Root[1-450 ((#1)^2)+56175 ((#1)^4)-3183900 ((#1)^6)+95178000 ((#1)^8)-1593148050 ((#1)^10)+15069672000 ((#1)^12)-77965348500 ((#1)^14)+200904161625 ((#1)^16)-175240530000 ((#1)^18)-244969953750 ((#1)^20)+623406712500 ((#1)^22)-308513475000 ((#1)^24)-260502075000 ((#1)^26)+351135000000 ((#1)^28)-140616000000 ((#1)^30)+19460250000 ((#1)^32)&,16], ])-2 (Sqrt[5]) (NumericalApproximation[Root, (RowDefault["1.00", FEExpressionsNumericalApproximationElider]), Root[2474611-63748665 ((#1)^2)+626509275 ((#1)^4)-2733606225 ((#1)^6)+2323439700 ((#1)^8)+28637733825 ((#1)^10)-130615403625 ((#1)^12)+202845804750 ((#1)^14)+191641136625 ((#1)^16)-1527099345000 ((#1)^18)+3444302317500 ((#1)^20)-4598543812500 ((#1)^22)+4041882450000 ((#1)^24)-2371767075000 ((#1)^26)+897469875000 ((#1)^28)-198369000000 ((#1)^30)+19460250000 ((#1)^32)&,25], ]) (NumericalApproximation[Root, (RowDefault["0.143", FEExpressionsNumericalApproximationElider]), Root[1-450 ((#1)^2)+56175 ((#1)^4)-3183900 ((#1)^6)+95178000 ((#1)^8)-1593148050 ((#1)^10)+15069672000 ((#1)^12)-77965348500 ((#1)^14)+200904161625 ((#1)^16)-175240530000 ((#1)^18)-244969953750 ((#1)^20)+623406712500 ((#1)^22)-308513475000 ((#1)^24)-260502075000 ((#1)^26)+351135000000 ((#1)^28)-140616000000 ((#1)^30)+19460250000 ((#1)^32)&,16], ])-2 ((NumericalApproximation[Root, (RowDefault["1.00", FEExpressionsNumericalApproximationElider]), Root[2474611-63748665 ((#1)^2)+626509275 ((#1)^4)-2733606225 ((#1)^6)+2323439700 ((#1)^8)+28637733825 ((#1)^10)-130615403625 ((#1)^12)+202845804750 ((#1)^14)+191641136625 ((#1)^16)-1527099345000 ((#1)^18)+3444302317500 ((#1)^20)-4598543812500 ((#1)^22)+4041882450000 ((#1)^24)-2371767075000 ((#1)^26)+897469875000 ((#1)^28)-198369000000 ((#1)^30)+19460250000 ((#1)^32)&,25], ])^3) (NumericalApproximation[Root, (RowDefault["0.143", FEExpressionsNumericalApproximationElider]), Root[1-450 ((#1)^2)+56175 ((#1)^4)-3183900 ((#1)^6)+95178000 ((#1)^8)-1593148050 ((#1)^10)+15069672000 ((#1)^12)-77965348500 ((#1)^14)+200904161625 ((#1)^16)-175240530000 ((#1)^18)-244969953750 ((#1)^20)+623406712500 ((#1)^22)-308513475000 ((#1)^24)-260502075000 ((#1)^26)+351135000000 ((#1)^28)-140616000000 ((#1)^30)+19460250000 ((#1)^32)&,16], ])+2 (Sqrt[5]) ((NumericalApproximation[Root, (RowDefault["1.00", FEExpressionsNumericalApproximationElider]), Root[2474611-63748665 ((#1)^2)+626509275 ((#1)^4)-2733606225 ((#1)^6)+2323439700 ((#1)^8)+28637733825 ((#1)^10)-130615403625 ((#1)^12)+202845804750 ((#1)^14)+191641136625 ((#1)^16)-1527099345000 ((#1)^18)+3444302317500 ((#1)^20)-4598543812500 ((#1)^22)+4041882450000 ((#1)^24)-2371767075000 ((#1)^26)+897469875000 ((#1)^28)-198369000000 ((#1)^30)+19460250000 ((#1)^32)&,25], ])^3) (NumericalApproximation[Root, (RowDefault["0.143", FEExpressionsNumericalApproximationElider]), Root[1-450 ((#1)^2)+56175 ((#1)^4)-3183900 ((#1)^6)+95178000 ((#1)^8)-1593148050 ((#1)^10)+15069672000 ((#1)^12)-77965348500 ((#1)^14)+200904161625 ((#1)^16)-175240530000 ((#1)^18)-244969953750 ((#1)^20)+623406712500 ((#1)^22)-308513475000 ((#1)^24)-260502075000 ((#1)^26)+351135000000 ((#1)^28)-140616000000 ((#1)^30)+19460250000 ((#1)^32)&,16], ])-2 (NumericalApproximation[Root, (RowDefault["1.00", FEExpressionsNumericalApproximationElider]), Root[2474611-63748665 ((#1)^2)+626509275 ((#1)^4)-2733606225 ((#1)^6)+2323439700 ((#1)^8)+28637733825 ((#1)^10)-130615403625 ((#1)^12)+202845804750 ((#1)^14)+191641136625 ((#1)^16)-1527099345000 ((#1)^18)+3444302317500 ((#1)^20)-4598543812500 ((#1)^22)+4041882450000 ((#1)^24)-2371767075000 ((#1)^26)+897469875000 ((#1)^28)-198369000000 ((#1)^30)+19460250000 ((#1)^32)&,25], ]) ((NumericalApproximation[Root, (RowDefault["0.143", FEExpressionsNumericalApproximationElider]), Root[1-450 ((#1)^2)+56175 ((#1)^4)-3183900 ((#1)^6)+95178000 ((#1)^8)-1593148050 ((#1)^10)+15069672000 ((#1)^12)-77965348500 ((#1)^14)+200904161625 ((#1)^16)-175240530000 ((#1)^18)-244969953750 ((#1)^20)+623406712500 ((#1)^22)-308513475000 ((#1)^24)-260502075000 ((#1)^26)+351135000000 ((#1)^28)-140616000000 ((#1)^30)+19460250000 ((#1)^32)&,16], ])^3)+2 (Sqrt[5]) (NumericalApproximation[Root, (RowDefault["1.00", FEExpressionsNumericalApproximationElider]), Root[2474611-63748665 ((#1)^2)+626509275 ((#1)^4)-2733606225 ((#1)^6)+2323439700 ((#1)^8)+28637733825 ((#1)^10)-130615403625 ((#1)^12)+202845804750 ((#1)^14)+191641136625 ((#1)^16)-1527099345000 ((#1)^18)+3444302317500 ((#1)^20)-4598543812500 ((#1)^22)+4041882450000 ((#1)^24)-2371767075000 ((#1)^26)+897469875000 ((#1)^28)-198369000000 ((#1)^30)+19460250000 ((#1)^32)&,25], ]) ((NumericalApproximation[Root, (RowDefault["0.143", FEExpressionsNumericalApproximationElider]), Root[1-450 ((#1)^2)+56175 ((#1)^4)-3183900 ((#1)^6)+95178000 ((#1)^8)-1593148050 ((#1)^10)+15069672000 ((#1)^12)-77965348500 ((#1)^14)+200904161625 ((#1)^16)-175240530000 ((#1)^18)-244969953750 ((#1)^20)+623406712500 ((#1)^22)-308513475000 ((#1)^24)-260502075000 ((#1)^26)+351135000000 ((#1)^28)-140616000000 ((#1)^30)+19460250000 ((#1)^32)&,16], ])^3)))^2)-32 ((NumericalApproximation[Root, (RowDefault["1.00", FEExpressionsNumericalApproximationElider]), Root[2474611-63748665 ((#1)^2)+626509275 ((#1)^4)-2733606225 ((#1)^6)+2323439700 ((#1)^8)+28637733825 ((#1)^10)-130615403625 ((#1)^12)+202845804750 ((#1)^14)+191641136625 ((#1)^16)-1527099345000 ((#1)^18)+3444302317500 ((#1)^20)-4598543812500 ((#1)^22)+4041882450000 ((#1)^24)-2371767075000 ((#1)^26)+897469875000 ((#1)^28)-198369000000 ((#1)^30)+19460250000 ((#1)^32)&,25], ])^2) ((NumericalApproximation[Root, (RowDefault["0.143", FEExpressionsNumericalApproximationElider]), Root[1-450 ((#1)^2)+56175 ((#1)^4)-3183900 ((#1)^6)+95178000 ((#1)^8)-1593148050 ((#1)^10)+15069672000 ((#1)^12)-77965348500 ((#1)^14)+200904161625 ((#1)^16)-175240530000 ((#1)^18)-244969953750 ((#1)^20)+623406712500 ((#1)^22)-308513475000 ((#1)^24)-260502075000 ((#1)^26)+351135000000 ((#1)^28)-140616000000 ((#1)^30)+19460250000 ((#1)^32)&,16], ])^2) (2-4 ((NumericalApproximation[Root, (RowDefault["1.00", FEExpressionsNumericalApproximationElider]), Root[2474611-63748665 ((#1)^2)+626509275 ((#1)^4)-2733606225 ((#1)^6)+2323439700 ((#1)^8)+28637733825 ((#1)^10)-130615403625 ((#1)^12)+202845804750 ((#1)^14)+191641136625 ((#1)^16)-1527099345000 ((#1)^18)+3444302317500 ((#1)^20)-4598543812500 ((#1)^22)+4041882450000 ((#1)^24)-2371767075000 ((#1)^26)+897469875000 ((#1)^28)-198369000000 ((#1)^30)+19460250000 ((#1)^32)&,25], ])^2)+2 ((NumericalApproximation[Root, (RowDefault["1.00", FEExpressionsNumericalApproximationElider]), Root[2474611-63748665 ((#1)^2)+626509275 ((#1)^4)-2733606225 ((#1)^6)+2323439700 ((#1)^8)+28637733825 ((#1)^10)-130615403625 ((#1)^12)+202845804750 ((#1)^14)+191641136625 ((#1)^16)-1527099345000 ((#1)^18)+3444302317500 ((#1)^20)-4598543812500 ((#1)^22)+4041882450000 ((#1)^24)-2371767075000 ((#1)^26)+897469875000 ((#1)^28)-198369000000 ((#1)^30)+19460250000 ((#1)^32)&,25], ])^4)-4 ((NumericalApproximation[Root, (RowDefault["0.143", FEExpressionsNumericalApproximationElider]), Root[1-450 ((#1)^2)+56175 ((#1)^4)-3183900 ((#1)^6)+95178000 ((#1)^8)-1593148050 ((#1)^10)+15069672000 ((#1)^12)-77965348500 ((#1)^14)+200904161625 ((#1)^16)-175240530000 ((#1)^18)-244969953750 ((#1)^20)+623406712500 ((#1)^22)-308513475000 ((#1)^24)-260502075000 ((#1)^26)+351135000000 ((#1)^28)-140616000000 ((#1)^30)+19460250000 ((#1)^32)&,16], ])^2)-((NumericalApproximation[Root, (RowDefault["1.00", FEExpressionsNumericalApproximationElider]), Root[2474611-63748665 ((#1)^2)+626509275 ((#1)^4)-2733606225 ((#1)^6)+2323439700 ((#1)^8)+28637733825 ((#1)^10)-130615403625 ((#1)^12)+202845804750 ((#1)^14)+191641136625 ((#1)^16)-1527099345000 ((#1)^18)+3444302317500 ((#1)^20)-4598543812500 ((#1)^22)+4041882450000 ((#1)^24)-2371767075000 ((#1)^26)+897469875000 ((#1)^28)-198369000000 ((#1)^30)+19460250000 ((#1)^32)&,25], ])^2) ((NumericalApproximation[Root, (RowDefault["0.143", FEExpressionsNumericalApproximationElider]), Root[1-450 ((#1)^2)+56175 ((#1)^4)-3183900 ((#1)^6)+95178000 ((#1)^8)-1593148050 ((#1)^10)+15069672000 ((#1)^12)-77965348500 ((#1)^14)+200904161625 ((#1)^16)-175240530000 ((#1)^18)-244969953750 ((#1)^20)+623406712500 ((#1)^22)-308513475000 ((#1)^24)-260502075000 ((#1)^26)+351135000000 ((#1)^28)-140616000000 ((#1)^30)+19460250000 ((#1)^32)&,16], ])^2)-(Sqrt[5]) ((NumericalApproximation[Root, (RowDefault["1.00", FEExpressionsNumericalApproximationElider]), Root[2474611-63748665 ((#1)^2)+626509275 ((#1)^4)-2733606225 ((#1)^6)+2323439700 ((#1)^8)+28637733825 ((#1)^10)-130615403625 ((#1)^12)+202845804750 ((#1)^14)+191641136625 ((#1)^16)-1527099345000 ((#1)^18)+3444302317500 ((#1)^20)-4598543812500 ((#1)^22)+4041882450000 ((#1)^24)-2371767075000 ((#1)^26)+897469875000 ((#1)^28)-198369000000 ((#1)^30)+19460250000 ((#1)^32)&,25], ])^2) ((NumericalApproximation[Root, (RowDefault["0.143", FEExpressionsNumericalApproximationElider]), Root[1-450 ((#1)^2)+56175 ((#1)^4)-3183900 ((#1)^6)+95178000 ((#1)^8)-1593148050 ((#1)^10)+15069672000 ((#1)^12)-77965348500 ((#1)^14)+200904161625 ((#1)^16)-175240530000 ((#1)^18)-244969953750 ((#1)^20)+623406712500 ((#1)^22)-308513475000 ((#1)^24)-260502075000 ((#1)^26)+351135000000 ((#1)^28)-140616000000 ((#1)^30)+19460250000 ((#1)^32)&,16], ])^2)+2 ((NumericalApproximation[Root, (RowDefault["0.143", FEExpressionsNumericalApproximationElider]), Root[1-450 ((#1)^2)+56175 ((#1)^4)-3183900 ((#1)^6)+95178000 ((#1)^8)-1593148050 ((#1)^10)+15069672000 ((#1)^12)-77965348500 ((#1)^14)+200904161625 ((#1)^16)-175240530000 ((#1)^18)-244969953750 ((#1)^20)+623406712500 ((#1)^22)-308513475000 ((#1)^24)-260502075000 ((#1)^26)+351135000000 ((#1)^28)-140616000000 ((#1)^30)+19460250000 ((#1)^32)&,16], ])^4)))),2000],64]

esoln

finalsoln={a->(NumericalApproximation[Root, (RowDefault["0.143", FEExpressionsNumericalApproximationElider]), Root[1-450 ((#1)^2)+56175 ((#1)^4)-3183900 ((#1)^6)+95178000 ((#1)^8)-1593148050 ((#1)^10)+15069672000 ((#1)^12)-77965348500 ((#1)^14)+200904161625 ((#1)^16)-175240530000 ((#1)^18)-244969953750 ((#1)^20)+623406712500 ((#1)^22)-308513475000 ((#1)^24)-260502075000 ((#1)^26)+351135000000 ((#1)^28)-140616000000 ((#1)^30)+19460250000 ((#1)^32)&,16], ]),b->(NumericalApproximation[Root, (RowDefault["0.741", FEExpressionsNumericalApproximationElider]), Root[1-133545 ((#1)^2)+15392955 ((#1)^4)-658758525 ((#1)^6)+13385251050 ((#1)^8)-140651762175 ((#1)^10)+852442151625 ((#1)^12)-3256025184000 ((#1)^14)+8320298902875 ((#1)^16)-14768681703750 ((#1)^18)+18601688981250 ((#1)^20)-16757836875000 ((#1)^22)+10739082937500 ((#1)^24)-4786478325000 ((#1)^26)+1411293375000 ((#1)^28)-247333500000 ((#1)^30)+19460250000 ((#1)^32)&,21], ]),ab->(NumericalApproximation[Root, (RowDefault["0.998", FEExpressionsNumericalApproximationElider]), Root[73441-528217755 ((#1)^2)+23460408885 ((#1)^4)-379618995675 ((#1)^6)+3255086120100 ((#1)^8)-17156053557675 ((#1)^10)+59993606215125 ((#1)^12)-145401765113250 ((#1)^14)+250375803161625 ((#1)^16)-309939662115000 ((#1)^18)+276042485857500 ((#1)^20)-175100781037500 ((#1)^22)+77310013275000 ((#1)^24)-22821340950000 ((#1)^26)+4214237625000 ((#1)^28)-434403000000 ((#1)^30)+19460250000 ((#1)^32)&,25], ]),c->(NumericalApproximation[Root, (RowDefault["1.00", FEExpressionsNumericalApproximationElider]), Root[2474611-63748665 ((#1)^2)+626509275 ((#1)^4)-2733606225 ((#1)^6)+2323439700 ((#1)^8)+28637733825 ((#1)^10)-130615403625 ((#1)^12)+202845804750 ((#1)^14)+191641136625 ((#1)^16)-1527099345000 ((#1)^18)+3444302317500 ((#1)^20)-4598543812500 ((#1)^22)+4041882450000 ((#1)^24)-2371767075000 ((#1)^26)+897469875000 ((#1)^28)-198369000000 ((#1)^30)+19460250000 ((#1)^32)&,25], ]),R->(NumericalApproximation[Root, (RowDefault["1.50", FEExpressionsNumericalApproximationElider]), Root[73441-13991070 ((#1)^2)+940040580 ((#1)^4)-26938210200 ((#1)^6)+315440426475 ((#1)^8)-1301745023550 ((#1)^10)+2222016066000 ((#1)^12)-780590052000 ((#1)^14)-2073848628375 ((#1)^16)+484548041250 ((#1)^18)+6867324011250 ((#1)^20)-12410051850000 ((#1)^22)+10768629037500 ((#1)^24)-5456935575000 ((#1)^26)+1646092125000 ((#1)^28)-274326750000 ((#1)^30)+19460250000 ((#1)^32)&,26], ]),q1->ArcCos[(NumericalApproximation[Root, (RowDefault["0.914", FEExpressionsNumericalApproximationElider]), Root[13134470470801-762391850100136 ((#1)^2)+16543868123531680 ((#1)^4)-177402682696553600 ((#1)^6)+1123112575590068480 ((#1)^8)-4653714413381521408 ((#1)^10)+13417625517914697728 ((#1)^12)-27928253847844290560 ((#1)^14)+42863736327238451200 ((#1)^16)-48989291872200949760 ((#1)^18)+41725139048163442688 ((#1)^20)-26287196539294056448 ((#1)^22)+12065494962934906880 ((#1)^24)-3945709373594009600 ((#1)^26)+892795390032609280 ((#1)^28)-132758076922003456 ((#1)^30)+10628373315321856 ((#1)^32)&,21], ])],q2->ArcCos[(NumericalApproximation[Root, (RowDefault["0.847", FEExpressionsNumericalApproximationElider]), Root[73441-34197271 ((#1)^2)+4525338125 ((#1)^4)-231217822005 ((#1)^6)+4849448134775 ((#1)^8)-55324230938408 ((#1)^10)+394636911570928 ((#1)^12)-1897986578659200 ((#1)^14)+6429316497813760 ((#1)^16)-15727635924633600 ((#1)^18)+28114830613024768 ((#1)^20)-36748859931885568 ((#1)^22)+34727007941427200 ((#1)^24)-23089990800506880 ((#1)^26)+10242838967091200 ((#1)^28)-2720679304626176 ((#1)^30)+327223173185536 ((#1)^32)&,26], ])],d->(NumericalApproximation[Root, (RowDefault["0.876", FEExpressionsNumericalApproximationElider]), Root[19504021-1021550925 ((#1)^2)+16826333925 ((#1)^4)-144026777475 ((#1)^6)+767601900150 ((#1)^8)-2786101493175 ((#1)^10)+7248691249875 ((#1)^12)-13951699884000 ((#1)^14)+20279794395375 ((#1)^16)-22595324827500 ((#1)^18)+19541839275000 ((#1)^20)-13276625962500 ((#1)^22)+7140777412500 ((#1)^24)-3010836825000 ((#1)^26)+945857250000 ((#1)^28)-195230250000 ((#1)^30)+19460250000 ((#1)^32)&,22], ]),eps->(NumericalApproximation[Root, (RowDefault["6.89", FEExpressionsNumericalApproximationElider, " × ", (Superscript["10", "-3"])]), Root[1-21330 ((#1)^2)+5543010 ((#1)^4)-456619500 ((#1)^6)+15778625475 ((#1)^8)-258239033550 ((#1)^10)+2174982842250 ((#1)^12)-10516099014000 ((#1)^14)+31582836559125 ((#1)^16)-61799968608750 ((#1)^18)+80698222687500 ((#1)^20)-70620534112500 ((#1)^22)+40806591412500 ((#1)^24)-15004020825000 ((#1)^26)+3286878750000 ((#1)^28)-385438500000 ((#1)^30)+19460250000 ((#1)^32)&,15], ])};

g2=Graph[Range[Length[v]],ee,VertexCoordinates->(v/.finalsoln)]

RecognizeGraph[g2]

N[EdgeLengths[g2]-1,20000]

vfinal=(v/.finalsoln)

vfinale=Map[RootApproximant[N[#,1000],32]&,vfinal,{2}]

N[vfinal-vfinale,20000]

ee=Select[Subsets[Range[Length[v]],{2}],Abs[EuclideanDistance@@v[[#]]-1]<10^-4&]//Quiet

g2=Graph[Range[Length[vfinale]],ee,VertexCoordinates->vfinale]

RecognizeGraph[g2]

N[EdgeLengths[g2]-1,20000]

Show[{Annotate[g1,{VertexSize->.5,EdgeStyle->Directive[Thick,Red]}],Annotate[g2,{VertexSize->.5,EdgeStyle->Green}]}]

Show[%,PlotRange->.04]

UnitDistance FindSubgraphUnitDistanceEmbedding

{{"GeneralizedOctagon",{2,1}},l=FindSubgraphUnitDistanceEmbedding[GraphData[{"GeneralizedOctagon",{2,1}}],TimeConstraint->300,"Results"->All]//Timing}

gs=UniqueEmbeddings[l]

Matchstick

GraphData[{"GeneralizedOctagon",{2,1}},"Matchstick"]

GraphData[{"GeneralizedOctagon",{2,1}},"Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData[{"GeneralizedOctagon",{2,1}},"Edges"]]

3D All

GraphData[{"GeneralizedOctagon",{2,1}},"Graph",{"3D","All"}]

3D UnitDistance

GraphData[{"GeneralizedOctagon",{2,1}},"Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData[{"GeneralizedOctagon",{2,1}},"Polyhedral"]

GraphData[{"GeneralizedOctagon",{2,1}},"Graph",{"3D","Polyhedron"}]

{"GeneralizedOctagon", {3, 1}}

GraphData[{"GeneralizedOctagon",{3,1}},"StandardName"]

Primary

GraphData[{"GeneralizedOctagon",{3,1}}]//StyleGraphs

Default embedding

GraphPlot@GraphData[{"GeneralizedOctagon",{3,1}},"Edges"]

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

RecognizeGraph[g]

All

GraphData[{"GeneralizedOctagon",{3,1}},"Graph","All"]//StyleGraphs

Planar

GraphData[{"GeneralizedOctagon",{3,1}},"Planar"]

GraphData[{"GeneralizedOctagon",{3,1}},"Graph","Planar"]//StyleGraphs

MinimalCrossing

GraphData[{"GeneralizedOctagon",{3,1}},{"RCN","CN"}]

GraphData[{"GeneralizedOctagon",{3,1}},"Graph","MinimalCrossing"]//StyleGraphs

MinimalCrossing construction

CrossingNumber[GraphData[{"GeneralizedOctagon",{3,1}}],TimeConstraint->10]//AbsoluteTiming

CrossingNumber[GraphData[{"GeneralizedOctagon",{3,1}}],TimeConstraint->60]//AbsoluteTiming

g=RotateGraph[MinimalCrossingGraph[GraphData[{"GeneralizedOctagon",{3,1}}],"CrossingNumber"->-1]]

RecognizeGraph[g]

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

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData[{"GeneralizedOctagon",{3,1}},"Graph","Circulant"]//StyleGraphs

Halin

GraphData[{"GeneralizedOctagon",{3,1}},"Halin"]

GraphData[{"GeneralizedOctagon",{3,1}},"Graph","Halin"]//StyleGraphs

LCF

GraphData[{"GeneralizedOctagon",{3,1}},"LCF"]

GraphData[{"GeneralizedOctagon",{3,1}},"LCFSignature"]

GraphData[{"GeneralizedOctagon",{3,1}},"Graph","LCF"]//StyleGraphs

LCF compute

Length[FindHamiltonianCycle[GraphData[{"GeneralizedOctagon",{3,1}}],All][[All,All,1]]]//Timing

Tally[(lcf=LCF[GraphData[{"GeneralizedOctagon",{3,1}}],All])[[All,2]]]

gs=LCFGraph@@@DeleteCases[lcf,{_,1|2}]

UnitDistance

GraphData[{"GeneralizedOctagon",{3,1}},"UnitDistance"]

UnitDistanceGraphQ[GraphData[{"GeneralizedOctagon",{3,1}}],Debug->True]//Timing

GraphData[{"GeneralizedOctagon",{3,1}},"Graph","UnitDistance"]//StyleGraphs

3D default

Graph3D[GraphData[{"GeneralizedOctagon",{3,1}},"Edges"]]

3D All

GraphData[{"GeneralizedOctagon",{3,1}},"Graph",{"3D","All"}]

3D UnitDistance

GraphData[{"GeneralizedOctagon",{3,1}},"Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData[{"GeneralizedOctagon",{3,1}},"Polyhedral"]

GraphData[{"GeneralizedOctagon",{3,1}},"Graph",{"3D","Polyhedron"}]

{"GeneralizedOctagon", {4, 1}}

GraphData[{"GeneralizedOctagon",{4,1}},"StandardName"]

Primary

GraphData[{"GeneralizedOctagon",{4,1}}]//StyleGraphs

Default embedding

GraphPlot@GraphData[{"GeneralizedOctagon",{4,1}},"Edges"]

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

RecognizeGraph[g]

All

GraphData[{"GeneralizedOctagon",{4,1}},"Graph","All"]//StyleGraphs

Planar

GraphData[{"GeneralizedOctagon",{4,1}},"Planar"]

GraphData[{"GeneralizedOctagon",{4,1}},"Graph","Planar"]//StyleGraphs

MinimalCrossing

GraphData[{"GeneralizedOctagon",{4,1}},{"RCN","CN"}]

GraphData[{"GeneralizedOctagon",{4,1}},"Graph","MinimalCrossing"]//StyleGraphs

MinimalCrossing construction

CrossingNumber[GraphData[{"GeneralizedOctagon",{4,1}}],TimeConstraint->10]//AbsoluteTiming

CrossingNumber[GraphData[{"GeneralizedOctagon",{4,1}}],TimeConstraint->60]//AbsoluteTiming

g=RotateGraph[MinimalCrossingGraph[GraphData[{"GeneralizedOctagon",{4,1}}],"CrossingNumber"->-1]]

RecognizeGraph[g]

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

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData[{"GeneralizedOctagon",{4,1}},"Graph","Circulant"]//StyleGraphs

Halin

GraphData[{"GeneralizedOctagon",{4,1}},"Halin"]

GraphData[{"GeneralizedOctagon",{4,1}},"Graph","Halin"]//StyleGraphs

LCF

GraphData[{"GeneralizedOctagon",{4,1}},"LCF"]

GraphData[{"GeneralizedOctagon",{4,1}},"LCFSignature"]

GraphData[{"GeneralizedOctagon",{4,1}},"Graph","LCF"]//StyleGraphs

LCF compute

Length[FindHamiltonianCycle[GraphData[{"GeneralizedOctagon",{4,1}}],All][[All,All,1]]]//Timing

Tally[(lcf=LCF[GraphData[{"GeneralizedOctagon",{4,1}}],All])[[All,2]]]

gs=LCFGraph@@@DeleteCases[lcf,{_,1|2}]

UnitDistance

GraphData[{"GeneralizedOctagon",{4,1}},"UnitDistance"]

UnitDistanceGraphQ[GraphData[{"GeneralizedOctagon",{4,1}}],Debug->True]//Timing

GraphData[{"GeneralizedOctagon",{4,1}},"Graph","UnitDistance"]//StyleGraphs

Matchstick

GraphData[{"GeneralizedOctagon",{4,1}},"Matchstick"]

GraphData[{"GeneralizedOctagon",{4,1}},"Graph","Matchstick"]//StyleGraphs

3D default

Graph3D[GraphData[{"GeneralizedOctagon",{4,1}},"Edges"]]

3D All

GraphData[{"GeneralizedOctagon",{4,1}},"Graph",{"3D","All"}]

3D UnitDistance

GraphData[{"GeneralizedOctagon",{4,1}},"Graph",{"3D","UnitDistance"}]

3D Polyhedron

GraphData[{"GeneralizedOctagon",{4,1}},"Polyhedral"]

GraphData[{"GeneralizedOctagon",{4,1}},"Graph",{"3D","Polyhedron"}]

{"GeneralizedOctagon", {4, 2}}

distance-transitive on 2925 nodes

Not on DistanceRegular.org

GraphData

GraphClassString

GraphClassString[go]

GraphDataString

GO(2,1)

GraphData[{"Cage",{3,8}},"StandardNames"]

lg=LineGraph[Graph[{"Cage",{3,8}}]]

RecognizeGraph[lg]

Print[GraphDataString[lg,
"Name"->"generalized octagon (2,1)",
"AlternateStandardNames"->{"GO2,1"},
"StandardName"->{"GeneralizedOctagon",{2,1}},
"Information"->"GeneralizedOctagon",
"Classes"->{"DeterminedBySpectrum","GeneralizedPolygon","LineGraph"},
"NotationRules"->{"GeneralizedPolygon"->{8,{2,1}}},
"Skip"->{"ChromaticPolynomial","ClawFree","Perfect"},
Debug->True,
TimeConstraint->120
]]

GO(3,1)

GraphData[{"Cage",{4,8}},"StandardNames"]

lg=LineGraph[Graph[{"Cage",{4,8}}]]

RecognizeGraph[lg]

file=OpenWrite["GO3,1.m"]

WriteString[file,GraphDataString[lg,
"Name"->"generalized octagon (3,1)",
"AlternateStandardNames"->{"GO3,1"},
"StandardName"->{"GeneralizedOctagon",{3,1}},
"Information"->"GeneralizedOctagon",
"Classes"->{"DeterminedBySpectrum","GeneralizedPolygon","LineGraph"},
"NotationRules"->{"GeneralizedPolygon"->{8,{3,1}}},
"Skip"->{"ChromaticPolynomial","ClawFree","Perfect"},
Debug->True,
TimeConstraint->60
]]

GO(4,1)

GraphData[{"Cage",{5,8}},"StandardNames"]

lg=LineGraph[Graph[{"Cage",{5,8}}]]

RecognizeGraph[lg]

file=OpenWrite["GO4,1.m"]

WriteString[file,GraphDataString[lg,
"Name"->"generalized octagon (4,1)",
"AlternateStandardNames"->{"GO4,1"},
"StandardName"->{"GeneralizedOctagon",{4,1}},
"Information"->"GeneralizedOctagon",
"Classes"->{"DeterminedBySpectrum","GeneralizedPolygon","LineGraph"},
"NotationRules"->{"GeneralizedPolygon"->{8,{4,1}}},
"Skip"->{"ChromaticPolynomial","ClawFree","Perfect"},
Debug->True,
TimeConstraint->60
]]

Close[file]