Coxeter Configuration

Author

Eric W. Weisstein
May 22, 2026

This notebook downloaded from https://mathworld.wolfram.com/notebooks/Configurations/CoxeterConfiguration.nb.

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

©2026 Wolfram Research, Inc. except for portions noted otherwise

Construction

g0=GraphData["CoxeterConfigurationGraph","Graph",{"Labeled","Configuration"}][[1]]

v0=GraphEmbedding[g0]

ring1

(Subscript[12, 3]) == "NauruConfiguration"

Configuration

UnlabelGraph@//StyleGraphs

Source

https://arxiv.org/pdf/1301.1067.pdf

Construction

From Ed Pegg, Jr.
Dec 14, 2023

From https://www.researchgate.net/figure/The-Coxeter-12-3-configuration-The-automorphism-group-of-this-configuration-has-order_fig4_319491035

coxeter={{-1,-1},{-1,1},{1,-1},{1,1},{1,(NumericalApproximation[Root, (RowDefault["0.596", FEExpressionsNumericalApproximationElider]), Root[-2+3 #1+((#1)^3)&,1], ])},{-1,(NumericalApproximation[Root, (RowDefault["-0.596", FEExpressionsNumericalApproximationElider]), Root[2+3 #1+((#1)^3)&,1], ])},{(NumericalApproximation[Root, (RowDefault["-0.596", FEExpressionsNumericalApproximationElider]), Root[2+3 #1+((#1)^3)&,1], ]),1},{(NumericalApproximation[Root, (RowDefault["0.596", FEExpressionsNumericalApproximationElider]), Root[-2+3 #1+((#1)^3)&,1], ]),-1},{(NumericalApproximation[Root, (RowDefault["-0.192", FEExpressionsNumericalApproximationElider]), Root[3+15 #1-3 ((#1)^2)+((#1)^3)&,1], ]),(NumericalApproximation[Root, (RowDefault["-0.355", FEExpressionsNumericalApproximationElider]), Root[4+9 #1-6 ((#1)^2)+((#1)^3)&,1], ])},{(NumericalApproximation[Root, (RowDefault["0.355", FEExpressionsNumericalApproximationElider]), Root[-4+9 #1+6 ((#1)^2)+((#1)^3)&,1], ]),(NumericalApproximation[Root, (RowDefault["-0.192", FEExpressionsNumericalApproximationElider]), Root[3+15 #1-3 ((#1)^2)+((#1)^3)&,1], ])},{(NumericalApproximation[Root, (RowDefault["0.192", FEExpressionsNumericalApproximationElider]), Root[-3+15 #1+3 ((#1)^2)+((#1)^3)&,1], ]),(NumericalApproximation[Root, (RowDefault["0.355", FEExpressionsNumericalApproximationElider]), Root[-4+9 #1+6 ((#1)^2)+((#1)^3)&,1], ])},{(NumericalApproximation[Root, (RowDefault["-0.355", FEExpressionsNumericalApproximationElider]), Root[4+9 #1-6 ((#1)^2)+((#1)^3)&,1], ]),(NumericalApproximation[Root, (RowDefault["0.192", FEExpressionsNumericalApproximationElider]), Root[-3+15 #1+3 ((#1)^2)+((#1)^3)&,1], ])}};

lin=ResourceFunction["FindExtraordinaryLines"][coxeter]

Graphics[{EdgeForm[Black],Black,Line[coxeter[[#]]]&/@lin,Blue,Disk[#,.08]&/@coxeter,White, Table[Text[k,coxeter[[k]]],{k,1,12}]}]

IsomorphicGraphQ[Graph[UndirectedEdge@@@Flatten[Table[{lin[[a,b]], lin[[a]]},{a,1,12},{b,1,3}],1]],
GraphData["NauruGraph"]]

I like to see what algebraic space something is in. I don’t think I’ve seen this one pop up.

NumberFieldDiscriminant[(NumericalApproximation[Root, (RowDefault["0.596", FEExpressionsNumericalApproximationElider]), Root[-2+3 #1+((#1)^3)&,1], ])]

points=coxeter

lines=lin

LeviGraph[lines]//RecognizeGraph

coxeter123=ConfigurationGraph[points,lines,VertexLabels->Automatic]

EdgeList[coxeter123]

RecognizeGraph@coxeter123

(Subscript[21, 3]) == configuration graph *is* "CoxeterGraph"

Construction from Ed [(Subscript[21, 3]) whose segments and points directly correspond to the Coxeter graph]

Ed Pegg, Jr
Dec. 14, 2023

coxe=CirclePoints[14];
ResourceFunction["NumberedPointPlot"][coxe]

outer=RootReduce[ResourceFunction["LineIntersection"][{coxe[[{10,6}]],coxe[[{13,5}]]}]]

add1=Table[RootReduce[RotationMatrix[2 k Pi/7].{(NumericalApproximation[Root, (RowDefault["1.62", FEExpressionsNumericalApproximationElider]), Root[-1+12 #1-20 ((#1)^2)+8 ((#1)^3)&,3], ]),(NumericalApproximation[Root, (RowDefault["1.01", FEExpressionsNumericalApproximationElider]), Root[-841+2072 ((#1)^2)-1680 ((#1)^4)+448 ((#1)^6)&,4], ])}],{k,0,6}]

coxe2=Join[coxe,add1];
coxlin=ResourceFunction["FindExtraordinaryLines"][coxe2];

Graphics[{Line[coxe2[[#]]]&/@coxlin,ResourceFunction["NumberedPointPlot"][coxe2][[1]]}]

RootReduce[RotationMatrix[2 6 Pi/7].(coxe2[[17]]+{x,0})]

RootReduce[Solve[(NumericalApproximation[Root, (RowDefault["1.36", FEExpressionsNumericalApproximationElider]), Root[-169+840 ((#1)^2)-1232 ((#1)^4)+448 ((#1)^6)&,6], ])==(x+(NumericalApproximation[Root, (RowDefault["-1.35", FEExpressionsNumericalApproximationElider]), Root[-13-8 #1+12 ((#1)^2)+8 ((#1)^3)&,1], ])) (NumericalApproximation[Root, (RowDefault["-0.782", FEExpressionsNumericalApproximationElider]), Root[-7+56 ((#1)^2)-112 ((#1)^4)+64 ((#1)^6)&,2], ])+(NumericalApproximation[Root, (RowDefault["0.847", FEExpressionsNumericalApproximationElider]), Root[-169+8624 ((#1)^2)-32256 ((#1)^4)+28672 ((#1)^6)&,6], ])]]

more=Table[RootReduce[RotationMatrix[2 k Pi/7].(coxe2[[17]] +{ (NumericalApproximation[Root, (RowDefault["0.692", FEExpressionsNumericalApproximationElider]), Root[1-4 #1+3 ((#1)^2)+((#1)^3)&,3], ]),0})],{k,0,6}];

coxe3=Join[coxe,add1,more];

coxline=ResourceFunction["FindExtraordinaryLines"][coxe3]

coxe3=;coxline={{1,7,16},{1,9,20},{2,6,20},{2,12,18},{3,9,17},{3,11,21},{4,8,21},{4,14,19},{5,11,18},{5,13,15},{6,10,15},{7,13,19},{8,12,16},{10,14,17},{15,26,27},{16,27,28},{17,22,28},{18,22,23},{19,23,24},{20,24,25},{21,25,26}};

Graphics[{Line[coxe3[[#]]]&/@coxline,ResourceFunction["NumberedPointPlot"][coxe3][[1]]}]

Iconize[coxe3]

points=;lines={{1,7,16},{1,9,20},{2,6,20},{2,12,18},{3,9,17},{3,11,21},{4,8,21},{4,14,19},{5,11,18},{5,13,15},{6,10,15},{7,13,19},{8,12,16},{10,14,17},{15,26,27},{16,27,28},{17,22,28},{18,22,23},{19,23,24},{20,24,25},{21,25,26}};

Length[lines]

ConfigurationDiagram[points,lines,.3,VertexLabels->None]//Quiet

coxeter283=ConfigurationGraph[points,lines]//Quiet

RecognizeGraph[coxeter283]

LeviGraph[lines]

What to call this?  213LeviGraph?

(Subscript[28, 3])

Source

https://arxiv.org/pdf/1301.1067.pdf

Construction from Fig. (5b) [Levi graph is Foster056C]

16/.Import["https://arxiv.org/pdf/1301.1067.pdf","EmbeddedImages"]

v0={{436.70834090358534`,306.2678052325583`},{587.528804808624`,1128.7148134689924`},{734.5794977390183`,990.6923146802328`},{333.87`,935.53`},{136.74862726098195`,910.171784156977`},{618.5164703447999`,838.427635961725`},{488.63`,816.18`},{386.3607906371125`,754.5965207122096`},{734.2832485465117`,797.4267477188309`},{1039.74`,909.6`},{1018.3417867813309`,713.8918816618219`},{831.8010764898256`,703.2417231912146`},{317.12735434835275`,640.2369261749033`},{872.1872476582689`,587.9045063186371`},{859.2781890947999`,455.1552431564925`},{302.6555812944122`,517.2342614462211`},{127.59452721253231`,594.6960190568477`},{796.0141740350452`,351.28287003391506`},{350.95`,388.22`},{21.974283854166657`,423.67136022286843`},{1151.565048651486`,423.19736151485813`},{984.6952847424097`,313.2444737160854`},{686.2760669008399`,279.15359788840465`},{561.2218765140506`,265.2150733809756`},{650.4`,94.95`},{264.8912154796512`,213.91211946866952`},{840.1775224079458`,30.393150940730266`},{338.9757322957042`,29.526622052648918`}};

Length[v0]

v=(#-Mean[v0])/578&/@v0

v={{-0.2623121108935409`,-0.42246887454555027`},{-0.001377052234304372`,1.000449824825443`},{0.2530359113130905`,0.7616565743258589`},{-0.44023311591704495`,0.6662200437372208`},{-0.781273553181782`,0.622347697988046`},{0.052234825855965325`,0.498222874120828`},{-0.17248225086514182`,0.45973215446386434`},{-0.34941859924384`,0.35318634254727194`},{0.2525233694575428`,0.4272870813130529`},{0.7809952577853773`,0.6213584520417191`},{0.7439741276492716`,0.2827630915950442`},{0.4212393347573939`,0.2643372118881112`},{-0.46919963088529276`,0.15533237275954478`},{0.4911116032149082`,0.06479185397707742`},{0.4687775918594255`,-0.16487815149376106`},{-0.4942373005287886`,-0.05747500566378071`},{-0.7971110965182002`,0.07654187601550407`},{0.3593242786072548`,-0.3445881395951061`},{-0.41068294290666435`,-0.28068307044963037`},{-0.9798450815672757`,-0.21934853719207256`},{0.9744642000889863`,-0.22016860416094852`},{0.6857621864054633`,-0.41039851384740655`},{0.169465961766069`,-0.4693792678745359`},{-0.04689076900692285`,-0.4934943621780464`},{0.10739664186842215`,-0.7880706137022256`},{-0.5595735735647073`,-0.5822537980124859`},{0.4357314557229998`,-0.8997606639777788`},{-0.431399669038664`,-0.9012598489052551`}};

Mean[v]

{Mean[#],Length[#]}&/@Split[Sort[Norm/@v],Abs[#-#2]<.01&]

Sort[ArcTan@@@Select[v,Abs[Norm[#]-.5]<.1&]]

2Pi-Mod[Subtract@@@Partition[%,2,1,1],2Pi]

lines={{5,4,3},{2,3,11},{10,11,22},{21,22,25},{27,25,26},{28,26,17},{20,17,4},{20,19,18},{28,1,6},{25,1,19},{16,15,21},{21,14,7},{10,12,1},{10,9,13},{19,7,2},{2,6,23},{3,12,14},{5,8,15},{5,13,24},{9,18,27},{27,23,8},{28,24,14},{20,16,12},{4,6,9},{17,8,7},{26,16,13},{18,15,11},{24,23,22}};

Length[lines]

ConfigurationDiagram[v,lines,VertexLabels->None]

cg=Show[ConfigurationGraph[v,lines,VertexLabels->Automatic],Axes->True]

RecognizeGraph@cg

Norm/@v[[{4,17,25,26,22,11,3}]]//Mean

2π-Mod[Subtract@@@Partition[ArcTan@@@v[[{11,3,4,17,26,25,22}]],2,1],2π]

{Norm[v[[2]]],ArcTan@@v[[2]]}

Pi/2.

{Norm[v[[4]]],ArcTan@@v[[4]]}

{Norm[v[[3]]],ArcTan@@v[[3]]}

{Norm[v[[6]]],ArcTan@@v[[6]]}

{Norm[v[[9]]],ArcTan@@v[[9]]}

RecognizeGraph@LeviGraph[lines]

GraphData["Foster056C","Incidence"]

GraphData["Foster056C","IncidenceLines"]

Exact

vnew=Join[CirclePoints[{1,Pi/2},7],CirclePoints[{0.7985326186369939`,2.154721070198766`},7],CirclePoints[{1/2,1.4663356643009722`},7],CirclePoints[{1/2,1.0370264728028047`},7]];
Graph[Range@Length@vnew,UndirectedEdge@@@{e[1,14,13],e[25,23,1],e[19,15,1],e[22,15,8]},VertexCoordinates->vnew,VertexLabels->Automatic]

vsym=Join[CirclePoints[{1,Pi/2},7],CirclePoints[{a1,q1},7],CirclePoints[{1/2,q2},7],CirclePoints[{1/2,q3},7]];

(eqn=FullSimplify[Join[TrigExpand[CollinearPoints[vsym[[#]]]&/@{{8,15,22},{25,23,1},{19,15,1},{13,14,1}}],{s1^2+c1^2==1,s2^2+c2^2==1,s3^2+c3^2==1}]/.{Sin[q1]->s1,Cos[q1]->c1,Sin[q2]->s2,Cos[q2]->c2,Sin[q3]->s3,Cos[q3]->c3},a1>0])//Column

Length[eqn]

var=Variables[Subtract@@@eqn]

soln=FindRoot[eqn,{
{a1,0.7985326186369939`},
{c1,Cos[2.154721070198766`]},{s1,Sin[2.154721070198766`]},
{c2,Cos[1.4663356643009722]},{s2,Sin[1.4663356643009722]},
{c3,Cos[1.0370264728028047`]},{s3,Sin[1.0370264728028047`]}},WorkingPrecision->10000];

N[soln]

ArcCos@-0.5663435356131596`

ArcCos@0.11266842903562059`

ArcCos@0.5153605947352065`

vnew=Join[CirclePoints[{1,Pi/2},7],CirclePoints[{a1,ArcCos[c1]},7],CirclePoints[{1/2,ArcCos[c2]},7],CirclePoints[{1/2,ArcCos[c3]},7]]/.soln;

lines=ResourceFunction["FindExtraordinaryLines"][vnew]

LeviGraph[lines]//RecognizeGraph

Graph[Range@Length@vnew,UndirectedEdge@@@{e[1,14,13],e[25,23,1],e[19,15,1],e[22,15,8]},VertexCoordinates->vnew,VertexLabels->Automatic]

CollinearPoints[vnew[[#]]]&/@{{1,14,13},{25,23,1},{19,15,1},{22,15,8}}

Print[Map[ToAlgebraicRoot[#,"MaxDegree"->64,WorkingPrecision->2500]&,#,{1}]//Timing]&/@Drop[vnew,7]

exact=Join[CirclePoints[{1,Pi/2},7],{{(NumericalApproximation[Root, (RowDefault["-0.462", FEExpressionsNumericalApproximationElider]), Root[118503493736761-133121730701680000 ((#1)^2)+47219065521559934464 ((#1)^4)-2588136195862976131072 ((#1)^6)-1530924857863017765076992 ((#1)^8)+171994750628750611350290432 ((#1)^10)+«21»+2202021147838892244965948170314654416896 ((#1)^40)-1393073238547820250361858941166358102016 ((#1)^42)+387945380452412572106197081641864658944 ((#1)^44)-39814532311245058255571845852459696128 ((#1)^46)+1329227995784915872903807060280344576 ((#1)^48)&,9], ]),(NumericalApproximation[Root, (RowDefault["0.673", FEExpressionsNumericalApproximationElider]), Root[47372053-2196774000 #1-5101966656 ((#1)^2)+405021104320 ((#1)^3)-576643805440 ((#1)^4)-227438881792 ((#1)^5)+10289082318848 ((#1)^6)+80899815063552 ((#1)^7)+76123983577088 ((#1)^8)-507121446682624 ((#1)^9)+«9»+339060455106412544 ((#1)^16)+19593034973445095424 ((#1)^17)+11471574248493940736 ((#1)^18)-32723119708101935104 ((#1)^19)-31924891858522603520 ((#1)^20)+10826653504198672384 ((#1)^21)+22752185317475745792 ((#1)^22)+9079256848778919936 ((#1)^23)+1152921504606846976 ((#1)^24)&,15], ])},{(NumericalApproximation[Root, (RowDefault["-0.814", FEExpressionsNumericalApproximationElider]), Root[84150467569-228033780536016 ((#1)^2)+91000584349529344 ((#1)^4)-14914589289722257408 ((#1)^6)+1247756305999490187264 ((#1)^8)-62922317357119123226624 ((#1)^10)+«22»+865320783559116662480957929453147652096 ((#1)^40)-439647073019587409580094386409348005888 ((#1)^42)+130601518364172860913700791012852498432 ((#1)^44)-20623803122100335340523131419662221312 ((#1)^46)+1329227995784915872903807060280344576 ((#1)^48)&,6], ]),(NumericalApproximation[Root, (RowDefault["0.0580", FEExpressionsNumericalApproximationElider]), Root[-7073933+49234380 #1+6237839696 ((#1)^2)-50298248448 ((#1)^3)-1116626905600 ((#1)^4)+9389616339968 ((#1)^5)-5137576591360 ((#1)^6)-157436740026368 ((#1)^7)+585108805058560 ((#1)^8)-2973163450007552 ((#1)^9)+«8»+1573778684152643584 ((#1)^15)-430094039291789312 ((#1)^16)+2155687104254836736 ((#1)^17)+13613655995866677248 ((#1)^18)-21249179310585020416 ((#1)^19)-31696334177433550848 ((#1)^20)+22790465914308395008 ((#1)^21)+14429533206095069184 ((#1)^22)-9079256848778919936 ((#1)^23)+1152921504606846976 ((#1)^24)&,9], ])},{(NumericalApproximation[Root, (RowDefault["-0.553", FEExpressionsNumericalApproximationElider]), Root[353778481-61039779808 ((#1)^2)+2284112845056 ((#1)^4)+3449102573568 ((#1)^6)+571546326138880 ((#1)^8)-7119812744445952 ((#1)^10)+15717987155705856 ((#1)^12)+43698453263417344 ((#1)^14)+76150938389184512 ((#1)^16)-1778156592718413824 ((#1)^18)+4932848966854246400 ((#1)^20)-4719772409484279808 ((#1)^22)+1152921504606846976 ((#1)^24)&,4], ]),(NumericalApproximation[Root, (RowDefault["-0.600", FEExpressionsNumericalApproximationElider]), Root[1+528 #1+7344 ((#1)^2)+33408 ((#1)^3)+4057088 ((#1)^4)+49364992 ((#1)^5)+186683392 ((#1)^6)-532283392 ((#1)^7)-3698851840 ((#1)^8)-4206886912 ((#1)^9)+2902458368 ((#1)^10)+5637144576 ((#1)^11)+1073741824 ((#1)^12)&,3], ])},{(NumericalApproximation[Root, (RowDefault["0.125", FEExpressionsNumericalApproximationElider]), Root[84150467569-228033780536016 ((#1)^2)+91000584349529344 ((#1)^4)-14914589289722257408 ((#1)^6)+1247756305999490187264 ((#1)^8)-62922317357119123226624 ((#1)^10)+«22»+865320783559116662480957929453147652096 ((#1)^40)-439647073019587409580094386409348005888 ((#1)^42)+130601518364172860913700791012852498432 ((#1)^44)-20623803122100335340523131419662221312 ((#1)^46)+1329227995784915872903807060280344576 ((#1)^48)&,21], ]),(NumericalApproximation[Root, (RowDefault["-0.807", FEExpressionsNumericalApproximationElider]), Root[-7073933+49234380 #1+6237839696 ((#1)^2)-50298248448 ((#1)^3)-1116626905600 ((#1)^4)+9389616339968 ((#1)^5)-5137576591360 ((#1)^6)-157436740026368 ((#1)^7)+585108805058560 ((#1)^8)-2973163450007552 ((#1)^9)+«8»+1573778684152643584 ((#1)^15)-430094039291789312 ((#1)^16)+2155687104254836736 ((#1)^17)+13613655995866677248 ((#1)^18)-21249179310585020416 ((#1)^19)-31696334177433550848 ((#1)^20)+22790465914308395008 ((#1)^21)+14429533206095069184 ((#1)^22)-9079256848778919936 ((#1)^23)+1152921504606846976 ((#1)^24)&,2], ])},{(NumericalApproximation[Root, (RowDefault["0.709", FEExpressionsNumericalApproximationElider]), Root[118503493736761-133121730701680000 ((#1)^2)+47219065521559934464 ((#1)^4)-2588136195862976131072 ((#1)^6)-1530924857863017765076992 ((#1)^8)+171994750628750611350290432 ((#1)^10)+«21»+2202021147838892244965948170314654416896 ((#1)^40)-1393073238547820250361858941166358102016 ((#1)^42)+387945380452412572106197081641864658944 ((#1)^44)-39814532311245058255571845852459696128 ((#1)^46)+1329227995784915872903807060280344576 ((#1)^48)&,28], ]),(NumericalApproximation[Root, (RowDefault["-0.406", FEExpressionsNumericalApproximationElider]), Root[47372053-2196774000 #1-5101966656 ((#1)^2)+405021104320 ((#1)^3)-576643805440 ((#1)^4)-227438881792 ((#1)^5)+10289082318848 ((#1)^6)+80899815063552 ((#1)^7)+76123983577088 ((#1)^8)-507121446682624 ((#1)^9)+«9»+339060455106412544 ((#1)^16)+19593034973445095424 ((#1)^17)+11471574248493940736 ((#1)^18)-32723119708101935104 ((#1)^19)-31924891858522603520 ((#1)^20)+10826653504198672384 ((#1)^21)+22752185317475745792 ((#1)^22)+9079256848778919936 ((#1)^23)+1152921504606846976 ((#1)^24)&,8], ])},{(NumericalApproximation[Root, (RowDefault["0.759", FEExpressionsNumericalApproximationElider]), Root[514595326609-2648639350360528 ((#1)^2)+3808179302827341568 ((#1)^4)-1222645685484718444544 ((#1)^6)+90449027529137530208256 ((#1)^8)-2529960203406774506618880 ((#1)^10)+«21»+3031177928024679957492738036783248310272 ((#1)^40)-2243740664907341079062748413849650593792 ((#1)^42)+698702075805846396686659804853543370752 ((#1)^44)-56242959571649252872242336238112079872 ((#1)^46)+1329227995784915872903807060280344576 ((#1)^48)&,28], ]),(NumericalApproximation[Root, (RowDefault["0.301", FEExpressionsNumericalApproximationElider]), Root[-2613533-76252044 #1+274418064 ((#1)^2)+12683703872 ((#1)^3)+8634302464 ((#1)^4)-499118561280 ((#1)^5)+3219824816128 ((#1)^6)-8892177809408 ((#1)^7)-457041812258816 ((#1)^8)+2291671033184256 ((#1)^9)+«9»+1322904352993574912 ((#1)^16)+1278057472719847424 ((#1)^17)-12961562037711798272 ((#1)^18)+25815231198413193216 ((#1)^19)-20298004995535536128 ((#1)^20)+218424581927469056 ((#1)^21)+10016005571271983104 ((#1)^22)-6052837899185946624 ((#1)^23)+1152921504606846976 ((#1)^24)&,8], ])},{(NumericalApproximation[Root, (RowDefault["0.238", FEExpressionsNumericalApproximationElider]), Root[514595326609-2648639350360528 ((#1)^2)+3808179302827341568 ((#1)^4)-1222645685484718444544 ((#1)^6)+90449027529137530208256 ((#1)^8)-2529960203406774506618880 ((#1)^10)+«21»+3031177928024679957492738036783248310272 ((#1)^40)-2243740664907341079062748413849650593792 ((#1)^42)+698702075805846396686659804853543370752 ((#1)^44)-56242959571649252872242336238112079872 ((#1)^46)+1329227995784915872903807060280344576 ((#1)^48)&,23], ]),(NumericalApproximation[Root, (RowDefault["0.781", FEExpressionsNumericalApproximationElider]), Root[-2613533-76252044 #1+274418064 ((#1)^2)+12683703872 ((#1)^3)+8634302464 ((#1)^4)-499118561280 ((#1)^5)+3219824816128 ((#1)^6)-8892177809408 ((#1)^7)-457041812258816 ((#1)^8)+2291671033184256 ((#1)^9)+«9»+1322904352993574912 ((#1)^16)+1278057472719847424 ((#1)^17)-12961562037711798272 ((#1)^18)+25815231198413193216 ((#1)^19)-20298004995535536128 ((#1)^20)+218424581927469056 ((#1)^21)+10016005571271983104 ((#1)^22)-6052837899185946624 ((#1)^23)+1152921504606846976 ((#1)^24)&,12], ])},{(NumericalApproximation[Root, (RowDefault["0.0563", FEExpressionsNumericalApproximationElider]), Root[729-295488 ((#1)^2)+22745088 ((#1)^4)-660865024 ((#1)^6)+7935623168 ((#1)^8)-39728447488 ((#1)^10)+68719476736 ((#1)^12)&,7], ]),(NumericalApproximation[Root, (RowDefault["0.497", FEExpressionsNumericalApproximationElider]), Root[29-3160 #1+2368 ((#1)^2)+50688 ((#1)^3)-69632 ((#1)^4)-163840 ((#1)^5)+262144 ((#1)^6)&,6], ])},{(NumericalApproximation[Root, (RowDefault["-0.353", FEExpressionsNumericalApproximationElider]), Root[28561-3223168 ((#1)^2)+107384832 ((#1)^4)-1627914240 ((#1)^6)+12515803136 ((#1)^8)-47244640256 ((#1)^10)+68719476736 ((#1)^12)&,3], ]),(NumericalApproximation[Root, (RowDefault["0.354", FEExpressionsNumericalApproximationElider]), Root[-167+2048 #1+9536 ((#1)^2)-24576 ((#1)^3)-98304 ((#1)^4)+65536 ((#1)^5)+262144 ((#1)^6)&,5], ])},{(NumericalApproximation[Root, (RowDefault["-0.497", FEExpressionsNumericalApproximationElider]), Root[-2911+39296 ((#1)^2)-176128 ((#1)^4)+262144 ((#1)^6)&,1], ]),(NumericalApproximation[Root, (RowDefault["-0.0556", FEExpressionsNumericalApproximationElider]), Root[-1-16 #1+64 ((#1)^2)+512 ((#1)^3)&,2], ])},{(NumericalApproximation[Root, (RowDefault["-0.266", FEExpressionsNumericalApproximationElider]), Root[28561-3223168 ((#1)^2)+107384832 ((#1)^4)-1627914240 ((#1)^6)+12515803136 ((#1)^8)-47244640256 ((#1)^10)+68719476736 ((#1)^12)&,5], ]),(NumericalApproximation[Root, (RowDefault["-0.423", FEExpressionsNumericalApproximationElider]), Root[-167+2048 #1+9536 ((#1)^2)-24576 ((#1)^3)-98304 ((#1)^4)+65536 ((#1)^5)+262144 ((#1)^6)&,1], ])},{(NumericalApproximation[Root, (RowDefault["0.165", FEExpressionsNumericalApproximationElider]), Root[729-295488 ((#1)^2)+22745088 ((#1)^4)-660865024 ((#1)^6)+7935623168 ((#1)^8)-39728447488 ((#1)^10)+68719476736 ((#1)^12)&,8], ]),(NumericalApproximation[Root, (RowDefault["-0.472", FEExpressionsNumericalApproximationElider]), Root[29-3160 #1+2368 ((#1)^2)+50688 ((#1)^3)-69632 ((#1)^4)-163840 ((#1)^5)+262144 ((#1)^6)&,1], ])},{(NumericalApproximation[Root, (RowDefault["0.472", FEExpressionsNumericalApproximationElider]), Root[841-498880 ((#1)^2)+52076544 ((#1)^4)-1270087680 ((#1)^6)+11811160064 ((#1)^8)-47244640256 ((#1)^10)+68719476736 ((#1)^12)&,12], ]),(NumericalApproximation[Root, (RowDefault["-0.165", FEExpressionsNumericalApproximationElider]), Root[-251+200 #1+10880 ((#1)^2)-13824 ((#1)^3)-98304 ((#1)^4)+65536 ((#1)^5)+262144 ((#1)^6)&,3], ])},{(NumericalApproximation[Root, (RowDefault["0.424", FEExpressionsNumericalApproximationElider]), Root[841-498880 ((#1)^2)+52076544 ((#1)^4)-1270087680 ((#1)^6)+11811160064 ((#1)^8)-47244640256 ((#1)^10)+68719476736 ((#1)^12)&,10], ]),(NumericalApproximation[Root, (RowDefault["0.266", FEExpressionsNumericalApproximationElider]), Root[-251+200 #1+10880 ((#1)^2)-13824 ((#1)^3)-98304 ((#1)^4)+65536 ((#1)^5)+262144 ((#1)^6)&,5], ])},{(NumericalApproximation[Root, (RowDefault["0.258", FEExpressionsNumericalApproximationElider]), Root[841-498880 ((#1)^2)+52076544 ((#1)^4)-1270087680 ((#1)^6)+11811160064 ((#1)^8)-47244640256 ((#1)^10)+68719476736 ((#1)^12)&,9], ]),(NumericalApproximation[Root, (RowDefault["0.428", FEExpressionsNumericalApproximationElider]), Root[-251+200 #1+10880 ((#1)^2)-13824 ((#1)^3)-98304 ((#1)^4)+65536 ((#1)^5)+262144 ((#1)^6)&,6], ])},{(NumericalApproximation[Root, (RowDefault["-0.174", FEExpressionsNumericalApproximationElider]), Root[729-295488 ((#1)^2)+22745088 ((#1)^4)-660865024 ((#1)^6)+7935623168 ((#1)^8)-39728447488 ((#1)^10)+68719476736 ((#1)^12)&,4], ]),(NumericalApproximation[Root, (RowDefault["0.469", FEExpressionsNumericalApproximationElider]), Root[29-3160 #1+2368 ((#1)^2)+50688 ((#1)^3)-69632 ((#1)^4)-163840 ((#1)^5)+262144 ((#1)^6)&,5], ])},{(NumericalApproximation[Root, (RowDefault["-0.475", FEExpressionsNumericalApproximationElider]), Root[-2911+39296 ((#1)^2)-176128 ((#1)^4)+262144 ((#1)^6)&,2], ]),(NumericalApproximation[Root, (RowDefault["0.156", FEExpressionsNumericalApproximationElider]), Root[-1-16 #1+64 ((#1)^2)+512 ((#1)^3)&,3], ])},{(NumericalApproximation[Root, (RowDefault["-0.418", FEExpressionsNumericalApproximationElider]), Root[729-295488 ((#1)^2)+22745088 ((#1)^4)-660865024 ((#1)^6)+7935623168 ((#1)^8)-39728447488 ((#1)^10)+68719476736 ((#1)^12)&,2], ]),(NumericalApproximation[Root, (RowDefault["-0.274", FEExpressionsNumericalApproximationElider]), Root[29-3160 #1+2368 ((#1)^2)+50688 ((#1)^3)-69632 ((#1)^4)-163840 ((#1)^5)+262144 ((#1)^6)&,2], ])},{(NumericalApproximation[Root, (RowDefault["-0.0462", FEExpressionsNumericalApproximationElider]), Root[841-498880 ((#1)^2)+52076544 ((#1)^4)-1270087680 ((#1)^6)+11811160064 ((#1)^8)-47244640256 ((#1)^10)+68719476736 ((#1)^12)&,6], ]),(NumericalApproximation[Root, (RowDefault["-0.498", FEExpressionsNumericalApproximationElider]), Root[-251+200 #1+10880 ((#1)^2)-13824 ((#1)^3)-98304 ((#1)^4)+65536 ((#1)^5)+262144 ((#1)^6)&,1], ])},{(NumericalApproximation[Root, (RowDefault["0.360", FEExpressionsNumericalApproximationElider]), Root[28561-3223168 ((#1)^2)+107384832 ((#1)^4)-1627914240 ((#1)^6)+12515803136 ((#1)^8)-47244640256 ((#1)^10)+68719476736 ((#1)^12)&,11], ]),(NumericalApproximation[Root, (RowDefault["-0.347", FEExpressionsNumericalApproximationElider]), Root[-167+2048 #1+9536 ((#1)^2)-24576 ((#1)^3)-98304 ((#1)^4)+65536 ((#1)^5)+262144 ((#1)^6)&,3], ])},{(NumericalApproximation[Root, (RowDefault["0.496", FEExpressionsNumericalApproximationElider]), Root[28561-3223168 ((#1)^2)+107384832 ((#1)^4)-1627914240 ((#1)^6)+12515803136 ((#1)^8)-47244640256 ((#1)^10)+68719476736 ((#1)^12)&,12], ]),(NumericalApproximation[Root, (RowDefault["0.0657", FEExpressionsNumericalApproximationElider]), Root[-167+2048 #1+9536 ((#1)^2)-24576 ((#1)^3)-98304 ((#1)^4)+65536 ((#1)^5)+262144 ((#1)^6)&,4], ])}}];

vnew-exact

g=ConfigurationGraph[exact,lines]

RecognizeGraph@g

Embeddings

<<MathWorld`Graphs`

Renamed from "Foster056CConfigurationGraph" to "CoxeterConfigurationGraph" in V14.2

Names

GraphData["CoxeterConfigurationGraph","StandardName"]

GraphData["CoxeterConfigurationGraph","StandardNames"]

HouseOfGraphs

hogid=GraphData["CoxeterConfigurationGraph","NotationRules","HouseOfGraphs"]

HouseOfGraphsLookup[hogid]

HouseOfGraphsGraph[hogid]//StyleGraphs[#,ImageSize->Small]&

Primary

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

All

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

CurvedEdge

GraphData["CoxeterConfigurationGraph","Graph","CurvedEdge"]//StyleGraphs

Inexact

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

(UD refinement local minimum)

Degenerate

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

DegenerateGraphEmbeddingTypes/@GraphData["CoxeterConfigurationGraph","Graph","Degenerate"]

Embedding types

GraphDataEmbeddings[GraphData["CoxeterConfigurationGraph","EdgeList"]]

Default embedding

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

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

RecognizeGraph[g]

Configuration

GraphData["CoxeterConfigurationGraph","Configuration"]

GraphData["CoxeterConfigurationGraph","Graph","Configuration"]//StyleGraphs

config=%[[1]]

(alledges=AddAllConfigurationEdges[config])//Timing

RecognizeGraph@alledges

ConfigurationDiagram[config,.2,"Simplification"->(N[#,20]&)]

lines=ConfigurationLines[config,"Simplification"->(N[#,20]&)]

GraphData["CoxeterConfigurationGraph","IncidenceLines"]

r=LeviGraph[lines]//RecognizeGraph

GraphData["CoxeterConfigurationGraph","LeviGraph","Name"]

FirstCase[GraphData[r,"StandardNames"],{"Incidence",_}]

Cases[GraphData["CoxeterConfigurationGraph","StandardNames"],_String?(StringMatchQ[#,"*Configuration*"]&)]

GraphData["CoxeterConfigurationGraph","OrdinaryLineGraphName"]

(ordinarylinegraph=OrdinaryLineGraph[config,"Simplification"->(N[#,20]&)])//Timing

RecognizeGraph@ordinarylinegraph

Bilateral

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

Circular

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

Circular construction

GraphData["CoxeterConfigurationGraph","LCF"]

GraphData["CoxeterConfigurationGraph","HamiltonianCycleCount"]

Length[hc=FindHamiltonianCycle[GraphData["CoxeterConfigurationGraph"],All][[All,All,1]]]//Timing

(circ=CircularEmbeddingFromHamiltonianCycle[GraphData["CoxeterConfigurationGraph"],#]&/@hc);//Timing

graphswithsymmetrygroups={#,GraphSymmetryGroup[#]}&/@circ;//Timing

Tally[graphswithsymmetrygroups[[All,-1]]]//Column

(BilateralOrRotationalGraphs={BilateralGraph[#,"ReturnNonbilateralGraph"->True],#2}&@@@SortBy[
DeleteCases[graphswithsymmetrygroups,{_,TransformationGroup[{}]}],
{MatchQ[#[[2]],TransformationGroup[{_Rotation,r_Reflection}|{r_Reflection|_Rotation}]],FirstCase[#[[2]],Rotation[{0,0},r_]:>N[r/π],{},{0,∞}]}&
])//Timing

Length[SymmetricGraphs=DeleteDuplicatesBy[BilateralOrRotationalGraphs,CircularCanonicalGraphEmbedding[#[[1]]]&]]//Timing

SymmetricGraphs

Length[BilateralGraphs=Select[SymmetricGraphs,!FreeQ[#,Reflection]&][[All,1]]]

BilateralGraphs

Length[BilateralAndRotationalGraphs=Select[SymmetricGraphs,And[!FreeQ[#,Rotation],!FreeQ[#,Reflection]]&][[All,1]]]

BilateralAndRotationalGraphs

Planar

GraphData["CoxeterConfigurationGraph","Planar"]

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

MinimalCrossing

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

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

MinimalCrossing construction

(cn=CrossingNumber[GraphData["CoxeterConfigurationGraph"],TimeConstraint->1])//AbsoluteTiming

(cn=CrossingNumber[GraphData["CoxeterConfigurationGraph"],TimeConstraint->10])//AbsoluteTiming

Union[Select[Table[RotateGraph[MinimalCrossingGraph[GraphData["CoxeterConfigurationGraph"],"CrossingNumber"->cn,RandomSeed->n]],{n,500}],Max[OuterFace[#]]<=GraphData["CoxeterConfigurationGraph","V"]&]]//Timing

Table[MinimalCrossingGraph[GraphData["CoxeterConfigurationGraph"],"CrossingNumber"->cn,RandomSeed->i],{i,100}]//Union//Timing

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

RecognizeGraph[g]

RectilinearCrossingCount[g]

Circulant

GraphData["CoxeterConfigurationGraph","Circulant"]

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

Halin

GraphData["CoxeterConfigurationGraph","Halin"]

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

Rigid

GraphData["CoxeterConfigurationGraph","Rigid"]

GraphData["CoxeterConfigurationGraph","Graph","Rigid"]//StyleGraphs

Flexible

GraphData["CoxeterConfigurationGraph","Flexible"]

GraphData["CoxeterConfigurationGraph","Graph","Flexible"]//StyleGraphs

LCF

GraphData["CoxeterConfigurationGraph","LCF"]

GraphData["CoxeterConfigurationGraph","HamiltonianCycleCount"]

GraphData["CoxeterConfigurationGraph","LCFSignature"]

GraphData["CoxeterConfigurationGraph","LCFNotations"]

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

UnitDistance

GraphData["CoxeterConfigurationGraph","UnitDistance"]

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

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

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

UnitDistance by refinement of default embeddings

Graph[edge list]

(g=UnitDistanceEmbeddingRefine[Graph[GraphData["CoxeterConfigurationGraph","EdgeList"]],MaxIterations->500])//Timing

UnitDistanceGraphEmbeddingQ[g]

Graph[vertex list, edge list]

(g=UnitDistanceEmbeddingRefine[Graph[GraphData["CoxeterConfigurationGraph","VertexList"],GraphData["CoxeterConfigurationGraph","EdgeList"]],MaxIterations->500])//Timing

UnitDistanceGraphEmbeddingQ[g]

UnitDistance by refinement of existing embeddings

gs=UnitDistanceEmbeddingRefine[Annotate[#,VertexCoordinates->N[GraphEmbedding[#]]],MaxIterations->500]&/@Complement@@(GraphData["CoxeterConfigurationGraph","Graph",#]&/@{"All","UnitDistance"})

Select[gs,UnitDistanceGraphEmbeddingQ]

GraphsWithSymmetryGroups=DeleteCases[{BilateralGraph[CenterGraph@#,"ReturnNonbilateralGraph"->True],GraphSymmetryGroup[#]}&/@gs,{_,TransformationGroup[{}]}]

NondegenrateSymmetricGraphs=Select[GraphsWithSymmetryGroups[[All,1]],!DegenerateGraphEmbeddingQ[#]&]

Max[Abs[EdgeLengths[#]-1]]&/@NondegenrateSymmetricGraphs

g=RotateGraph[,3.17]

g=

DegenerateGraphEmbeddingQ@g

UnitDistance by FindUnitDistanceEmbedding  (random embeddings)

(g=FindUnitDistanceEmbedding[GraphData["CoxeterConfigurationGraph"],MaxIterations->5000,"ReturnWithMessage"->True])//Timing

UnitDistanceGraphEmbeddingQ[g]

(gs=FindUnitDistanceEmbedding[GraphData["CoxeterConfigurationGraph"],50,MaxIterations->5000,"ReturnWithMessage"->True])//Timing

Select[gs,UnitDistanceGraphEmbeddingQ]

Select[gs,GraphSymmetryGroup[#]=!=TransformationGroup[{}]&]

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

UnitDistance by FindSubgraphUnitDistanceEmbedding

{$Version,DateString[]}

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

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

Select[IndexGraph/@gs,!DegenerateGraphEmbeddingQ[#]&&GraphSymmetryGroup[#]=!=TransformationGroup[{}]&]//Timing

Matchstick

GraphData["CoxeterConfigurationGraph","Matchstick"]

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

3D default

Graph3D[GraphData["CoxeterConfigurationGraph","Edges"]]

3D All

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

3D UnitDistance

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

3D Polyhedron

GraphData["CoxeterConfigurationGraph","Polyhedral"]

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

GraphData

GraphDataString

file=OpenWrite["~/coxeter.m"]

WriteString[file,GraphDataString[coxeter123,
"StandardName"->"NauruConfigurationGraph",
"Name"->"Nauru configuration graph",
"Information"->"NauruConfiguration",
"Skip"->{"ConstructionNumber","EdgeCovers","EdgeCuts","GracefulLabeling","LinklesslyEmbeddable","Minors","ProjectivePlanar","QuadraticEmbeddingConstant"}
]]

Close[file]