From gosper.org/flowsnakes.pdf (q.v. for 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[149]:= FlowS /@ {7/22, 113/355} Out[149]= {1693/2762 + (543 I Sqrt[3])/2762, 13159704187...887406573947767669/ 21456243153017...26366300007824801961964892568 + (152317993023640247...951911649014 I)/ (268203039412723583...100245245611571 Sqrt[3])} In[150]:= N[%] Out[150]= {0.612962 + 0.340515 I, 0.613328 + 0.327889 I} As usual, it redefines itself twice and has no obvious termination condition: In[119]:= 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, w*((-1)^(1/3) + (-1)^(2/3)* FlowS[u, a*(-1)^(1/3)*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]]]) Toward the end of the pdf are two plots that suggest there may be a typo in this definition. --rwg