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The Hyperbolic
It’s getting more “obvious”
Now .... project onto a hyperboloid!!

Some code

argPoint[arg_]:={Cos[arg],Sin[arg]};
PLine[arg1_,arg2_]:=IfNCos
arg2-arg1
2
0,Line[{argPoint[arg1],argPoint[arg2]}],IfN[arg2-arg1>π],PLine[arg2,arg1],IfN[arg2-arg1]<0,PLine[arg1,arg2+2π],CircleSec
arg2-arg1
2
argPoint
arg1+arg2
2
,Tan
arg2-arg1
2
,arg2+
π
2
,arg1-
π
2
+If[N[arg2+π]>arg1,2π,0];
Graphics[{{RGBColor[.33,.26,.78],Circle[]},​​Line[{{0,-1},{0,1}}],​​Table[PLine[#[[1]],#[[2]]]&/@N[Partition[Table[2kPi/(2^n),{k,0,2^n-1}],2,1,1]],{n,1,6+1}]},ImageSize->{600,600},PlotRange{{-1,1},{-1,1}}]

Art from friend Gwen Fisher, 1 hour later

She just coincidentally finished an art piece with the same thing.
Doodle No. 69 “Three Trees”
12” square

Another method

inf=Flatten[Table[Sort/@(Partition[Table[2kPi/(2^n),{k,0,2^n-1}],2,1,1]/Pi),{n,1,6+1}],1];
Graph[UndirectedEdge@@@inf]

Make the dual graph

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