pivots[A_]:=Flatten[FirstPosition[1]/@Transpose[A]]

pMatrix[n_]:=Table[Which[i==2&&j==1||i==1&&j==2,-1/2,i==j&&i>2,1,True,0],{i, n},{j,n}]

gramMatrix[W_]:=Simplify[Transpose[W].pMatrix[Length[W]].W]

linearRelationsMatrix[G_]:=RowReduce[G][[1;;MatrixRank[G]]]//Transpose

linearRelations[L_]:=Select[Table[Equal[(Subscript[b, i]),L[[i]].Table[(Subscript[b, i]),{i,pivots[L]}]],{i,Length[L]}],Not[BooleanQ[#]]&]

linearRelationsFromW=linearRelations@*linearRelationsMatrix@*gramMatrix;

inverseLinearRelations[G_]:=Table[If[i==#,1,0],{i,Length[G[[1]]]}]&/@Flatten[FirstPosition[#,1]&/@(RowReduce[G]//Select[Not[AllTrue[#,#==0&]]&])]

quadraticFormMatrix[G_,L_]:=Inverse[Table[G[[i, j]],{i,pivots[L]},{j,pivots[L]}]]

quadraticForm[Q_,L_]:=Module[{v=Table[(Subscript[b, i]),{i,pivots[L]}]},Simplify[v.Q.v==0]]

quadform[W_]:=Module[{G=gramMatrix[W]},Module[{L=linearRelationsMatrix[G]},quadraticForm[quadraticFormMatrix[G,L],L]]]

quadformFromG[G_]:=Module[{L=linearRelationsMatrix[G]},quadraticForm[quadraticFormMatrix[G,L],L]]

graphFromG[G_]:=PlanarGraph[AdjacencyGraph[If[#==-1,1,0]&/@#&/@G,VertexLabels->"Name"]]

nextCandidate[s_,t_,adj_]:=Block[{length,pos},length=Length[adj];
pos=Mod[Position[adj,s][[1,1]]+1,length,1];
{t,adj[[pos]]}];

FindFace[g_?PlanarGraphQ]:=Block[{emb},emb=GraphEmbedding[g,"PlanarEmbedding"];
FindFace[g,emb]];

FindFace[g_?PlanarGraphQ,emb_]:=Block[{m,orderings,pAdj,rightF,s,t,initial,face},m=AdjacencyMatrix[g];
Table[pAdj[v]=SortBy[Pick[VertexList[g],m[[v]],1],ArcTan@@(emb[[v]]-emb[[#]])&],{v,VertexList[g]}];
rightF[_]:=False;
Reap[Table[If[!rightF[e],s=e[[1]];
t=e[[2]];
initial=s;
face={s};
While[t=!=initial,rightF[UndirectedEdge[s,t]]=True;
{s,t}=nextCandidate[s,t,pAdj[t]];
face=Join[face,{s}];];
Sow[face];],{e,EdgeList[g]}]][[2,1]]]

findGenerators[faces_,vertices_,G_]:=Table[Block[{face,otherVertex,verticesOnFace,allVertices,Gp,P,Gnew,L,Q,Qform,sol1,sol2,sols,coeff,coeffTable,small,Li,P2},
face=faces[[i]];
verticesOnFace=Sort[RandomSample[face,3]];
otherVertex=RandomChoice[Select[vertices,Not[MemberQ[face,#]]&]];
allVertices=Append[verticesOnFace,otherVertex];
P=Transpose[IdentityMatrix[Length[vertices]][[#]]&/@DeleteDuplicates[Join[allVertices,vertices]]];
Gnew=Transpose[P].G.P;
L=linearRelationsMatrix[Gnew];
Gp=Table[G[[i,j]],{i,allVertices},{j,allVertices}];
Q=Inverse[Gp];
Qform=(#.Q.#&[Table[(Subscript[b, j]),{j,allVertices}]])==0;
sols=Expand[FullSimplify[Total[#[[1]][[2]]&/@Solve[Qform,(Subscript[b, otherVertex])]]]];
coeff=Table[sols/.Table[(Subscript[b, i])->If[i==curr,1,0],{i,allVertices}],{curr,allVertices}];
small=IdentityMatrix[Length[allVertices]];
small[[-1]]=Table[If[j==4,-1,If[MemberQ[verticesOnFace,allVertices[[j]]],coeff[[j]],0]],{j,Length[allVertices]}];
Li=Table[Table[If[j==k,1,0],{j,Length[vertices]}],{k,allVertices}];
P2=Table[If[j==k,1,0],{j,Length[vertices]},{k,DeleteDuplicates[Join[allVertices,vertices]]}];
P2.L.small.Li
],{i,Length[faces]}]

findGeneratorsN[faces_,vertices_,G_]:=Table[Block[{face,otherVertex,verticesOnFace,allVertices,Gp,P,Gnew,L,Q,Qform,sol1,sol2,sols,coeff,coeffTable,small,Li,P2},
face=faces[[i]];
verticesOnFace=Sort[RandomSample[face,3]];
otherVertex=RandomChoice[Select[vertices,Not[MemberQ[face,#]]&]];
allVertices=Append[verticesOnFace,otherVertex];
P=Transpose[IdentityMatrix[Length[vertices]][[#]]&/@DeleteDuplicates[Join[allVertices,vertices]]];
Gnew=Transpose[P].G.P;
L=linearRelationsMatrix[Gnew];
Gp=Table[G[[i,j]],{i,allVertices},{j,allVertices}];
Q=Inverse[Gp];
Qform=(#.Q.#&[Table[(Subscript[b, j]),{j,allVertices}]])==0;
sols=Expand[FullSimplify[Total[#[[1]][[2]]&/@NSolve[Qform,(Subscript[b, otherVertex])]]]];
coeff=Table[sols/.Table[(Subscript[b, i])->If[i==curr,1,0],{i,allVertices}],{curr,allVertices}];
small=IdentityMatrix[Length[allVertices]];
small[[-1]]=Table[If[j==4,-1,If[MemberQ[verticesOnFace,allVertices[[j]]],coeff[[j]],0]],{j,Length[allVertices]}];
Li=Transpose[Table[If[j==k,1,0],{j,Length[vertices]},{k,allVertices}]];
P2=Table[If[j==k,1,0],{j,Length[vertices]},{k,DeleteDuplicates[Join[allVertices,vertices]]}];
P2.L.small.Li
],{i,Length[faces]}]

findGeneratorsFromG[G_]:=findGeneratorsN[FindFace[PlanarGraph[AdjacencyGraph[If[#==-1,1,0]&/@#&/@G,VertexLabels->"Name"]]],Range[Length[G]],G]

abbctoxyr[{bt_,b_,h1_,h2_}]:={h1/b,h2/b,1/b}

findGeometricGenerators[W_,faces_]:=Block[{circs,pointsOfIntersection,duals},
circs=abbctoxyr/@W;
pointsOfIntersection=Table[If[i==j,{0,0},NSolve[{(((x-circs[[i]][[1]]))^2)+(((y-circs[[i]][[2]]))^2)==((circs[[i]][[3]])^2),(((x-circs[[j]][[1]]))^2)+(((y-circs[[j]][[2]]))^2)==((circs[[j]][[3]])^2)},Reals]/.{{x->a_,y->b_},_}->{a,b}],{i,Length[circs]},{j,Length[circs]}];
duals=(Table[CircleThrough[Table[pointsOfIntersection[[face[[i]],face[[If[i==Length[face],1,i+1]]]]],{i,3}]],{face,faces}]//Chop)/.Circle[{x_,y_},r_]->{x,y,r};
invertAboutCircle@@#&/@duals
]

search[generators_,current_,lastgen_,upperBound_,faces_]:=Block[{totals,invalid},
invalid[σ_,face_]:=σ==lastgen||AllTrue[Delete[σ.current,{#}&/@face],#>upperBound&];
totals[i_]:=If[invalid[generators[[i]],faces[[i]]],0,Length[current]-Length[faces[[i]]]-Length[Select[Delete[generators[[i]].current,{#}&/@faces[[i]]],#>upperBound&]]+search[generators,generators[[i]].current,generators[[i]],upperBound,faces]];
Total[totals/@Range[Length[generators]]]]

$RecursionLimit=16384

fractalDimensionFromG[G_,max_,delta_,root_]:=Block[{xs,ys,data,graph,faces,sigmas,nlm,params},
xs=Range[delta,max,delta];
graph=graphFromG[G];
faces=FindFace[graph];
sigmas=findGeneratorsFromG[G];
ys=search[sigmas,root,{},#,faces]+Length[root]&/@xs;
data={xs[[#]],ys[[#]]}&/@Range[Length[ys]];
nlm=NonlinearModelFit[data,a (x^b),{a,b},x];
params=nlm["BestFitParameters"];
Print[nlm["RSquared"]];
b/.params
]

pyramidG[n_]:=Table[Which[i==0&&j==0,1,i==0||j==0,-1,True,1-((2((Sin[((Abs[i-j]π)/n)])^2))/((Sin[(π/n)])^2))],{i,0,n},{j,0,n}]
prismG[n_]:=Table[If[i>n&&j≤n||i≤n&&j>n,-1-((2((Sin[(((Abs[i-j]-n)π)/n)])^2))/((Sin[(π/n)])^2)),1-((2((Sin[((Abs[i-j]π)/n)])^2))/((Sin[(π/n)])^2))],{i,1,2n},{j,1,2n}]

pyramidFractalDimension[n_,max_,delta_]:=fractalDimensionFromG[N[pyramidG[n]],max,delta,N[Table[If[i==n+1,-1,((Sec[(π/n)]+Tan[(π/n)])/(Tan[(π/n)]))],{i,n+1}]]]

pyramidTuple[n_]:=N[Table[If[i==1,-1,((Sec[(π/n)]+Tan[(π/n)])/(Tan[(π/n)]))],{i,n+1}]]

tetrahedron=({{1, -1, -1, -1}, {-1, 1, -1, -1}, {-1, -1, 1, -1}, {-1, -1, -1, 1}});
octahedron=({{1, -1, -1, -1, -1, -3}, {-1, 1, -1, -1, -3, -1}, {-1, -1, 1, -3, -1, -1}, {-1, -1, -3, 1, -1, -1}, {-1, -3, -1, -1, 1, -1}, {-3, -1, -1, -1, -1, 1}});

cube=({{1, -1, -3, -1, -1, -3, -5, -3}, {-1, 1, -1, -3, -3, -1, -3, -5}, {-3, -1, 1, -1, -5, -3, -1, -3}, {-1, -3, -1, 1, -3, -5, -3, -1}, {-1, -3, -5, -3, 1, -1, -3, -1}, {-3, -1, -3, -5, -1, 1, -1, -3}, {-5, -3, -1, -3, -3, -1, 1, -1}, {-3, -5, -3, -1, -1, -3, -1, 1}});

g6v6f1=({{1, -1, -(Sqrt[5]), -1, -1, -1}, {-1, 1, -1, -2-(Sqrt[5]), -3-2(Sqrt[5]), -1}, {-(Sqrt[5]), -1, 1, -1, -2-(Sqrt[5]), -1}, {-1, -2-(Sqrt[5]), -1, 1, -1, -(Sqrt[5])}, {-1, -3-2(Sqrt[5]), -2-(Sqrt[5]), -1, 1, -1}, {-1, -1, -1, -(Sqrt[5]), -1, 1}});

icos={{1,-1,-4-Sqrt[5],-1,-2-Sqrt[5],-1,-2-Sqrt[5],-2-Sqrt[5],-1,-1,-2-Sqrt[5],-2-Sqrt[5]},{-1,1,-2-Sqrt[5],-2-Sqrt[5],-1,-1,-4-Sqrt[5],-2-Sqrt[5],-2-Sqrt[5],-1,-2-Sqrt[5],-1},{-4-Sqrt[5],-2-Sqrt[5],1,-2-Sqrt[5],-1,-2-Sqrt[5],-1,-1,-2-Sqrt[5],-2-Sqrt[5],-1,-1},{-1,-2-Sqrt[5],-2-Sqrt[5],1,-4-Sqrt[5],-2-Sqrt[5],-1,-1,-1,-1,-2-Sqrt[5],-2-Sqrt[5]},{-2-Sqrt[5],-1,-1,-4-Sqrt[5],1,-1,-2-Sqrt[5],-2-Sqrt[5],-2-Sqrt[5],-2-Sqrt[5],-1,-1},{-1,-1,-2-Sqrt[5],-2-Sqrt[5],-1,1,-2-Sqrt[5],-4-Sqrt[5],-1,-2-Sqrt[5],-1,-2-Sqrt[5]},{-2-Sqrt[5],-4-Sqrt[5],-1,-1,-2-Sqrt[5],-2-Sqrt[5],1,-1,-1,-2-Sqrt[5],-1,-2-Sqrt[5]},{-2-Sqrt[5],-2-Sqrt[5],-1,-1,-2-Sqrt[5],-4-Sqrt[5],-1,1,-2-Sqrt[5],-1,-2-Sqrt[5],-1},{-1,-2-Sqrt[5],-2-Sqrt[5],-1,-2-Sqrt[5],-1,-1,-2-Sqrt[5],1,-2-Sqrt[5],-1,-4-Sqrt[5]},{-1,-1,-2-Sqrt[5],-1,-2-Sqrt[5],-2-Sqrt[5],-2-Sqrt[5],-1,-2-Sqrt[5],1,-4-Sqrt[5],-1},{-2-Sqrt[5],-2-Sqrt[5],-1,-2-Sqrt[5],-1,-1,-1,-2-Sqrt[5],-1,-4-Sqrt[5],1,-2-Sqrt[5]},{-2-Sqrt[5],-1,-1,-2-Sqrt[5],-1,-2-Sqrt[5],-2-Sqrt[5],-1,-4-Sqrt[5],-1,-2-Sqrt[5],1}};

cuboc=({{1, -1, -3, -1, -1, -1, -5, -5, -5, -3, -5, -7}, {-1, 1, -1, -3, -5, -1, -1, -5, -7, -5, -3, -5}, {-3, -1, 1, -1, -5, -5, -1, -1, -5, -7, -5, -3}, {-1, -3, -1, 1, -1, -5, -5, -1, -3, -5, -7, -5}, {-1, -5, -5, -1, 1, -3, -7, -3, -1, -1, -5, -5}, {-1, -1, -5, -5, -3, 1, -3, -7, -5, -1, -1, -5}, {-5, -1, -1, -5, -7, -3, 1, -3, -5, -5, -1, -1}, {-5, -5, -1, -1, -3, -7, -3, 1, -1, -5, -5, -1}, {-5, -7, -5, -3, -1, -5, -5, -1, 1, -1, -3, -1}, {-3, -5, -7, -5, -1, -1, -5, -5, -1, 1, -1, -3}, {-5, -3, -5, -7, -5, -1, -1, -5, -3, -1, 1, -1}, {-7, -5, -3, -5, -5, -5, -1, -1, -1, -3, -1, 1}});

squareantibipyramid=(({{1, -1-2 (Sqrt[2]), -3-4 (Sqrt[2]), -1-2 (Sqrt[2]), -1, -1, -5-2 (Sqrt[2]), -5-2 (Sqrt[2]), -1, -1-(Sqrt[2])}, {-1-2 (Sqrt[2]), 1, -1-2 (Sqrt[2]), -3-4 (Sqrt[2]), -5-2 (Sqrt[2]), -1, -1, -5-2 (Sqrt[2]), -1, -1-(Sqrt[2])}, {-3-4 (Sqrt[2]), -1-2 (Sqrt[2]), 1, -1-2 (Sqrt[2]), -5-2 (Sqrt[2]), -5-2 (Sqrt[2]), -1, -1, -1, -1-(Sqrt[2])}, {-1-2 (Sqrt[2]), -3-4 (Sqrt[2]), -1-2 (Sqrt[2]), 1, -1, -5-2 (Sqrt[2]), -5-2 (Sqrt[2]), -1, -1, -1-(Sqrt[2])}, {-1, -5-2 (Sqrt[2]), -5-2 (Sqrt[2]), -1, 1, -1-2 (Sqrt[2]), -3-4 (Sqrt[2]), -1-2 (Sqrt[2]), -1-(Sqrt[2]), -1}, {-1, -1, -5-2 (Sqrt[2]), -5-2 (Sqrt[2]), -1-2 (Sqrt[2]), 1, -1-2 (Sqrt[2]), -3-4 (Sqrt[2]), -1-(Sqrt[2]), -1}, {-5-2 (Sqrt[2]), -1, -1, -5-2 (Sqrt[2]), -3-4 (Sqrt[2]), -1-2 (Sqrt[2]), 1, -1-2 (Sqrt[2]), -1-(Sqrt[2]), -1}, {-5-2 (Sqrt[2]), -5-2 (Sqrt[2]), -1, -1, -1-2 (Sqrt[2]), -3-4 (Sqrt[2]), -1-2 (Sqrt[2]), 1, -1-(Sqrt[2]), -1}, {-1, -1, -1, -1, -1-(Sqrt[2]), -1-(Sqrt[2]), -1-(Sqrt[2]), -1-(Sqrt[2]), 1, -1-(Sqrt[2])}, {-1-(Sqrt[2]), -1-(Sqrt[2]), -1-(Sqrt[2]), -1-(Sqrt[2]), -1, -1, -1, -1, -1-(Sqrt[2]), 1}}));

rhombdodec=({{1, -5, -11, -5, -1, -1, -7, -7, -11, -5, -11, -1, -17, -7}, {-5, 1, -5, -11, -7, -1, -1, -7, -17, -11, -5, -1, -11, -7}, {-11, -5, 1, -5, -7, -7, -1, -1, -11, -17, -11, -1, -5, -7}, {-5, -11, -5, 1, -1, -7, -7, -1, -5, -11, -17, -1, -11, -7}, {-1, -7, -7, -1, 1, -2, -5, -2, -1, -1, -7, -2, -7, -2}, {-1, -1, -7, -7, -2, 1, -2, -5, -7, -1, -1, -2, -7, -2}, {-7, -1, -1, -7, -5, -2, 1, -2, -7, -7, -1, -2, -1, -2}, {-7, -7, -1, -1, -2, -5, -2, 1, -1, -7, -7, -2, -1, -2}, {-11, -17, -11, -5, -1, -7, -7, -1, 1, -5, -11, -7, -5, -1}, {-5, -11, -17, -11, -1, -1, -7, -7, -5, 1, -5, -7, -11, -1}, {-11, -5, -11, -17, -7, -1, -1, -7, -11, -5, 1, -7, -5, -1}, {-1, -1, -1, -1, -2, -2, -2, -2, -7, -7, -7, 1, -7, -5}, {-17, -11, -5, -11, -7, -7, -1, -1, -5, -11, -5, -7, 1, -1}, {-7, -7, -7, -7, -2, -2, -2, -2, -1, -1, -1, -5, -1, 1}});

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g6v7f1=({{1, -1, -1, -1, -1, -1}, {-1, 1, -1, -3, -1, -1}, {-1, -1, 1, -1, -3, -5-4(Sqrt[2])}, {-1, -3, -1, 1, -1, -5-4(Sqrt[2])}, {-1, -1, -3, -1, 1, -1}, {-1, -1, -5-4(Sqrt[2]), -5-4(Sqrt[2]), -1, 1}});

dodeca={{1,-8-3 Sqrt[5],-2-Sqrt[5],-2-Sqrt[5],3 (-2-Sqrt[5]),3 (-2-Sqrt[5]),-2-Sqrt[5],-2-Sqrt[5],-2-Sqrt[5],-2-Sqrt[5],-5-2 Sqrt[5],-5-2 Sqrt[5],3 (-2-Sqrt[5]),-1,-1,-1,-5-2 Sqrt[5],-5-2 Sqrt[5],-5-2 Sqrt[5],-5-2 Sqrt[5]},{-8-3 Sqrt[5],1,-5-2 Sqrt[5],-5-2 Sqrt[5],-1,-1,-5-2 Sqrt[5],-5-2 Sqrt[5],-5-2 Sqrt[5],-5-2 Sqrt[5],-2-Sqrt[5],-2-Sqrt[5],-1,3 (-2-Sqrt[5]),3 (-2-Sqrt[5]),3 (-2-Sqrt[5]),-2-Sqrt[5],-2-Sqrt[5],-2-Sqrt[5],-2-Sqrt[5]},{-2-Sqrt[5],-5-2 Sqrt[5],1,3 (-2-Sqrt[5]),-2-Sqrt[5],3 (-2-Sqrt[5]),-1,-5-2 Sqrt[5],-2-Sqrt[5],-5-2 Sqrt[5],-2-Sqrt[5],-5-2 Sqrt[5],-5-2 Sqrt[5],-1,-5-2 Sqrt[5],-2-Sqrt[5],-2-Sqrt[5],3 (-2-Sqrt[5]),-1,-8-3 Sqrt[5]},{-2-Sqrt[5],-5-2 Sqrt[5],3 (-2-Sqrt[5]),1,3 (-2-Sqrt[5]),-2-Sqrt[5],-5-2 Sqrt[5],-1,-5-2 Sqrt[5],-2-Sqrt[5],-5-2 Sqrt[5],-2-Sqrt[5],-5-2 Sqrt[5],-5-2 Sqrt[5],-1,-2-Sqrt[5],3 (-2-Sqrt[5]),-2-Sqrt[5],-8-3 Sqrt[5],-1},{3 (-2-Sqrt[5]),-1,-2-Sqrt[5],3 (-2-Sqrt[5]),1,-2-Sqrt[5],-2-Sqrt[5],-5-2 Sqrt[5],-5-2 Sqrt[5],3 (-2-Sqrt[5]),-1,-2-Sqrt[5],-2-Sqrt[5],-5-2 Sqrt[5],-8-3 Sqrt[5],-5-2 Sqrt[5],-2-Sqrt[5],-5-2 Sqrt[5],-1,-5-2 Sqrt[5]},{3 (-2-Sqrt[5]),-1,3 (-2-Sqrt[5]),-2-Sqrt[5],-2-Sqrt[5],1,-5-2 Sqrt[5],-2-Sqrt[5],3 (-2-Sqrt[5]),-5-2 Sqrt[5],-2-Sqrt[5],-1,-2-Sqrt[5],-8-3 Sqrt[5],-5-2 Sqrt[5],-5-2 Sqrt[5],-5-2 Sqrt[5],-2-Sqrt[5],-5-2 Sqrt[5],-1},{-2-Sqrt[5],-5-2 Sqrt[5],-1,-5-2 Sqrt[5],-2-Sqrt[5],-5-2 Sqrt[5],1,-2-Sqrt[5],-5-2 Sqrt[5],3 (-2-Sqrt[5]),-1,-2-Sqrt[5],3 (-2-Sqrt[5]),-2-Sqrt[5],-5-2 Sqrt[5],-1,-5-2 Sqrt[5],-8-3 Sqrt[5],-2-Sqrt[5],3 (-2-Sqrt[5])},{-2-Sqrt[5],-5-2 Sqrt[5],-5-2 Sqrt[5],-1,-5-2 Sqrt[5],-2-Sqrt[5],-2-Sqrt[5],1,3 (-2-Sqrt[5]),-5-2 Sqrt[5],-2-Sqrt[5],-1,3 (-2-Sqrt[5]),-5-2 Sqrt[5],-2-Sqrt[5],-1,-8-3 Sqrt[5],-5-2 Sqrt[5],3 (-2-Sqrt[5]),-2-Sqrt[5]},{-2-Sqrt[5],-5-2 Sqrt[5],-2-Sqrt[5],-5-2 Sqrt[5],-5-2 Sqrt[5],3 (-2-Sqrt[5]),-5-2 Sqrt[5],3 (-2-Sqrt[5]),1,-1,3 (-2-Sqrt[5]),-8-3 Sqrt[5],-2-Sqrt[5],-1,-2-Sqrt[5],-5-2 Sqrt[5],-1,-2-Sqrt[5],-2-Sqrt[5],-5-2 Sqrt[5]},{-2-Sqrt[5],-5-2 Sqrt[5],-5-2 Sqrt[5],-2-Sqrt[5],3 (-2-Sqrt[5]),-5-2 Sqrt[5],3 (-2-Sqrt[5]),-5-2 Sqrt[5],-1,1,-8-3 Sqrt[5],3 (-2-Sqrt[5]),-2-Sqrt[5],-2-Sqrt[5],-1,-5-2 Sqrt[5],-2-Sqrt[5],-1,-5-2 Sqrt[5],-2-Sqrt[5]},{-5-2 Sqrt[5],-2-Sqrt[5],-2-Sqrt[5],-5-2 Sqrt[5],-1,-2-Sqrt[5],-1,-2-Sqrt[5],3 (-2-Sqrt[5]),-8-3 Sqrt[5],1,-1,-5-2 Sqrt[5],-5-2 Sqrt[5],3 (-2-Sqrt[5]),-2-Sqrt[5],-5-2 Sqrt[5],3 (-2-Sqrt[5]),-2-Sqrt[5],-5-2 Sqrt[5]},{-5-2 Sqrt[5],-2-Sqrt[5],-5-2 Sqrt[5],-2-Sqrt[5],-2-Sqrt[5],-1,-2-Sqrt[5],-1,-8-3 Sqrt[5],3 (-2-Sqrt[5]),-1,1,-5-2 Sqrt[5],3 (-2-Sqrt[5]),-5-2 Sqrt[5],-2-Sqrt[5],3 (-2-Sqrt[5]),-5-2 Sqrt[5],-5-2 Sqrt[5],-2-Sqrt[5]},{3 (-2-Sqrt[5]),-1,-5-2 Sqrt[5],-5-2 Sqrt[5],-2-Sqrt[5],-2-Sqrt[5],3 (-2-Sqrt[5]),3 (-2-Sqrt[5]),-2-Sqrt[5],-2-Sqrt[5],-5-2 Sqrt[5],-5-2 Sqrt[5],1,-5-2 Sqrt[5],-5-2 Sqrt[5],-8-3 Sqrt[5],-1,-1,-2-Sqrt[5],-2-Sqrt[5]},{-1,3 (-2-Sqrt[5]),-1,-5-2 Sqrt[5],-5-2 Sqrt[5],-8-3 Sqrt[5],-2-Sqrt[5],-5-2 Sqrt[5],-1,-2-Sqrt[5],-5-2 Sqrt[5],3 (-2-Sqrt[5]),-5-2 Sqrt[5],1,-2-Sqrt[5],-2-Sqrt[5],-2-Sqrt[5],-5-2 Sqrt[5],-2-Sqrt[5],3 (-2-Sqrt[5])},{-1,3 (-2-Sqrt[5]),-5-2 Sqrt[5],-1,-8-3 Sqrt[5],-5-2 Sqrt[5],-5-2 Sqrt[5],-2-Sqrt[5],-2-Sqrt[5],-1,3 (-2-Sqrt[5]),-5-2 Sqrt[5],-5-2 Sqrt[5],-2-Sqrt[5],1,-2-Sqrt[5],-5-2 Sqrt[5],-2-Sqrt[5],3 (-2-Sqrt[5]),-2-Sqrt[5]},{-1,3 (-2-Sqrt[5]),-2-Sqrt[5],-2-Sqrt[5],-5-2 Sqrt[5],-5-2 Sqrt[5],-1,-1,-5-2 Sqrt[5],-5-2 Sqrt[5],-2-Sqrt[5],-2-Sqrt[5],-8-3 Sqrt[5],-2-Sqrt[5],-2-Sqrt[5],1,3 (-2-Sqrt[5]),3 (-2-Sqrt[5]),-5-2 Sqrt[5],-5-2 Sqrt[5]},{-5-2 Sqrt[5],-2-Sqrt[5],-2-Sqrt[5],3 (-2-Sqrt[5]),-2-Sqrt[5],-5-2 Sqrt[5],-5-2 Sqrt[5],-8-3 Sqrt[5],-1,-2-Sqrt[5],-5-2 Sqrt[5],3 (-2-Sqrt[5]),-1,-2-Sqrt[5],-5-2 Sqrt[5],3 (-2-Sqrt[5]),1,-2-Sqrt[5],-1,-5-2 Sqrt[5]},{-5-2 Sqrt[5],-2-Sqrt[5],3 (-2-Sqrt[5]),-2-Sqrt[5],-5-2 Sqrt[5],-2-Sqrt[5],-8-3 Sqrt[5],-5-2 Sqrt[5],-2-Sqrt[5],-1,3 (-2-Sqrt[5]),-5-2 Sqrt[5],-1,-5-2 Sqrt[5],-2-Sqrt[5],3 (-2-Sqrt[5]),-2-Sqrt[5],1,-5-2 Sqrt[5],-1},{-5-2 Sqrt[5],-2-Sqrt[5],-1,-8-3 Sqrt[5],-1,-5-2 Sqrt[5],-2-Sqrt[5],3 (-2-Sqrt[5]),-2-Sqrt[5],-5-2 Sqrt[5],-2-Sqrt[5],-5-2 Sqrt[5],-2-Sqrt[5],-2-Sqrt[5],3 (-2-Sqrt[5]),-5-2 Sqrt[5],-1,-5-2 Sqrt[5],1,3 (-2-Sqrt[5])},{-5-2 Sqrt[5],-2-Sqrt[5],-8-3 Sqrt[5],-1,-5-2 Sqrt[5],-1,3 (-2-Sqrt[5]),-2-Sqrt[5],-5-2 Sqrt[5],-2-Sqrt[5],-5-2 Sqrt[5],-2-Sqrt[5],-2-Sqrt[5],3 (-2-Sqrt[5]),-2-Sqrt[5],-5-2 Sqrt[5],-5-2 Sqrt[5],-1,3 (-2-Sqrt[5]),1}};

g=pyramidG[7];

g//graphFromG

root=rootTupleFromG[g,"Face"->{1,2,3},"Invert"->5];
graphAbbc[root]

root

alg=findGeneratorsFromG[N[g]]//Chop

FindFace[graphFromG[g]]-1

root=rootTupleFromG[octahedron,"Exact"->True,"Invert"->5]

alg=RootApproximant[findGeneratorsFromG[octahedron]];

geom=Transpose[root].Transpose[#].PseudoInverse[Transpose[root]]&/@alg//Simplify;

c=root[[-1]]

geom[[8]].c

root//graphAbbc

root//N

{{10.`,8.`,-5.656854249492381`,7.`},{24.`,24.`,-16.970562748477143`,17.`},{8.`,10.`,-5.656854249492381`,7.`},{12.`,10.`,-8.485281374238571`,7.`},{-4.`,-4.`,2.8284271247461903`,-3.`},{10.`,12.`,-8.485281374238571`,7.`}}