w = 1/2 + I/(2 Sqrt[3]); Trid[t_, a_: 1, b_: 0] := Trid[t, x_: 0, y_: 0] = (Trid[t, s1_: 1, s0_: 0] = (b - s0)/(s1 - a); Module[{u = (t + 1)*3/2, n}, n = Floor[u]; u = 2*(u - n) - 1; ComplexExpand[ Switch[n, 0, w*(Trid[u, Expand[a*w], Expand[b + a*w - a]] + 1) - 1, 1, Trid[u, Expand[a/Sqrt[3]/I], b]/Sqrt[3]/I, 2, w*(Trid[u, Expand[a*w], Expand[b - a*w + a]] - 1) + 1, 3, 1]]]) Exactly maps rational -1 <= t <= 1 onto the tridaic dragon with the same endpoints, which is an complete definition since a continuous function is determined by its values on a dense set. Try, e.g., ListLinePlot[{Re[#], Im[#]} & /@ Table[Trid[t], {t, 1/81/3 - 1, 1, 2/81/3}], Axes -> None, AspectRatio -> Automatic] for self-avoidance, and ListLinePlot[{Re[#], Im[#]} & /@ Table[Trid[t], {t, -1, 1, 1/3^n/2}], Axes -> None] n>2 for interesting texture. Puzzles: What is the bounding box of the dragon's range? Give inverse images for a point on a hrizontal edge and a vertical edge. In how many points does this range "curve" intersect this box? Unsolved, but probably easy: What is Trid[theta], the radix 3 stutter-free Morse number formed by pairwise grouping the bits of the Thue number 0.110100110010110... and mapping 11->2, 10->1, 01->1, and 00->0 (ceiling[n/2] = digit sum)? Bonus: Self-avoiding Z-zerpent: ListLinePlot[{Re[#], Im[#]} & /@ Table[Trid[t], {t, -2/3, 0, 2/81/3}], Axes -> None, AspectRatio -> Automatic] --rwg
Omigod, I've been using Vaseline five years past expiration! To make matters worse, the jar fails to list the percentage of inert ingredients! Answers: Trid[13/40] = (-1 - i 3 rt 3)/8, Trid[39/40] = (9 - i rt3)/8, e.g. # intersections = oo! (Subpuzzle: countable?)