# [GIF] Recapitulation (Fourier series with fivefold symmetry)

GROUPS:
 Clayton Shonkwiler 2 Votes RecapitulationInspired by a talk Frank Farris gave at the ICERM Workshop on Illustrating Mathematics (see also @symmetric_curve); the idea is that a Fourier series whose only non-vanishing coefficients are congruent to 1 mod $n$ will produce a parametrized curve having $n$-fold symmetry. In this case, $n=5$; here's the code: ToReal[z_] := {Re[z], Im[z]}; DynamicModule[{a = -1/4, b = -3/4, c = 1/3, d = 1/3, cols}, cols = RGBColor /@ {"#8BDEFF", "#A888FF", "#C2FFFF", "#26466F"}; Manipulate[ ParametricPlot[ ToReal[E^(-I π/10)*(a E^(-4 I θ) + b E^(6 I θ) + c Sin[t] E^(-9 I θ)) + d Cos[t] E^(11 I θ)], {θ, 0, 2 π}, PlotRange -> 2.5, Axes -> None, ImageSize -> 540, PlotStyle -> Thickness[.005], ColorFunction -> Function[{x, y, θ}, Blend[Append[#, First[#]] &[cols[[;; 3]]], Mod[(θ + π/2 + t)/(2 π), 1]]], ColorFunctionScaling -> False, Background -> Last[cols]], {t, -π/2, π/2}] ] 
 Neat as always.ToReal is not defined but I suppose it can be replaced with ReIm.
 Oops, sorry about that. Just added the definition of ToReal to the post.As you guessed, it's the same function as ReIm, which I didn't even know existed, but am very happy to learn about. Thanks!