​
periods
1
2
3
4
up factor
1.39
down factor
0.76
initial price
81.3
​
branches
2
3
4
5
6
generations
3
4
5
6
7
angle
length
zoom
thickness
hue
​
position of first branch
number of branchings
show branches
Manipulate[Module[{z=Complex@@pt,t,l},t=Transpose[{Re[#],Im[#]}]&/@NestList[Flatten[Outer[Times,1+#,{z,Conjugate[z]}]]&,{0},n];​​Graphics[{MapIndexed[{PointSize[.02/Sqrt[First[#2]]],Point[#]}&,t],If[branches&&ControlActive[False,True],Map[Line,Flatten[Transpose[MapAt[Join[#,#]&,#,1]]&/@Partition[t,2,1],1]],Opacity[0]],Line[{{-Sqrt[Abs[Re[z]]],0},{0,0}}]},PlotRange->All,AspectRatio->1,ImageSize->{450,300}]],{{pt,{0.5,.5},"position of first branch"},{-1,-1},{1,1}},{{n,10,"number of branchings"},1,10,1},{{branches,False,"show branches"},{True,False}}]
In[]:=
Manipulate[Module[{m1={Reverse[pt1],{-1,1}pt1},m2={Reverse[pt2],{-1,1}pt2}},Graphics[MapIndexed[{Thickness[0.01th^#2[[1]]],Line[#]}&,NestList[Flatten[Map[{{#[[2]],#[[2]]+m1.(#[[2]]-#[[1]])},{#[[2]],#[[2]]+m2.(#[[2]]-#[[1]])}}&,#],1]&,{{{0,-1},{0,0}}},gen]],PlotRange->{{-3,3},{-1,5}},ImageSize->{500,400}]],{{gen,7,"generations"},3,10,1},{{th,0.7,"thickness ratio"},0.01,1.2},{{pt1,{-0.25,0.75}},{-3,-1},{3,3},Locator},{{pt2,{0.25,0.75}},{-3,-1},{3,3},Locator}]
Out[]=
​
generations
thickness ratio

Semigroup Case

Start with identity; apply elements....