From (corrected) gosper.org/flowsnakes.pdf (q.v. for (improved) pix): FlowS[t] gives the exact value for rational t. E.g.,
In[148]:= FlowS /@ {0, 1/7, 1/3, 1} Out[148]= {0, 5/14 - (I Sqrt[3])/14, 4/7 + (2 I Sqrt[3])/7, 1} It's a continuous function: In[168]:= FlowS /@ {7/22, 113/355} Out[168]= {1693/2762 + (543 I Sqrt[3])/2762, 2345845273398817840092950245183202891348356982612226750061526434654879728078331327658322166322544455096921202357/ 3824784042812146922648846082969066584366282400370622809380296235226226222138134929551565620475339497558916970979 + (2172173105963516197779522804535911788205830019423847609465739564008821336204362398272785130819862477878379140827 I)/(3824784042812146922648846082969066584366282400370622809380296235226226222138134929551565620475339497558916970979 Sqrt[3])} (*Fixing the typo considerably shortened these numbers.*) In[170]:= N[%168] Out[170]= {0.612962 + 0.340515 I, 0.613328 + 0.327889 I} As usual, it redefines itself twice and has no obvious termination condition: Clear[FlowS]; FlowS[t_, a_: 1, b_: 0] := FlowS[t, x_: 0, y_: 0] = (FlowS[t, s1_: 1, s0_: 0] = (b - s0)/(s1 - a); Module[{u = t*7, n}, u -= (n = Floor[u]); ComplexExpand[Switch[n, 0, w*FlowS[u, a*w, b], 1, w*(1 + (-1)^(1/3) + (-1)^(4/3)* FlowS[1 - u, a*(-1)^(4/3)*w, b + a*w*(1 + (-1)^(1/3))]), 2, w*((-1)^(1/3) + FlowS[1 - u, a*w, b + a*w*(-1)^(1/3)]), 3, (*Typo fixed*) w*((-1)^(1/3) + (-1)^(2/3)*FlowS[u, a*(-1)^(2/3)*w, b + a*w*(-1)^(1/3)]), 4, w*(Sqrt[-3] + FlowS[u, a*w, b + a*w*Sqrt[-3]]), 5, w*(2*(-1)^(1/3) + FlowS[u, a*w, b + a*w*2*(-1)^(1/3)]), 6, w*(2 + (-1)^(1/3) + ((-1)^(1/3) - 1)* FlowS[1 - u, a*w*((-1)^(1/3) - 1), b + a*w (2 + (-1)^(1/3))]), 7, 1]]]) --rwg My Firefox pdf viewer renders the first page blank and spods on the 2nd. Judging by Google Images, the name Flowsnake (which Mandelbrot detested) is losing out to "Gosper curve". Or maybe not--a lot of the "hits" are things like glider guns and Foxtrot strips. ?