Recapitulation

Inspired 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 ?/10)*(a E^(-4 I ?) + b  E^(6 I ?) +
             c Sin[t] E^(-9 I ?)) + 
         d  Cos[t] E^(11 I ?)], {?, 0, 2 ?}, 
       PlotRange -> 2.5, Axes -> None, ImageSize -> 540, 
       PlotStyle -> Thickness[.005], 
       ColorFunction -> 
        Function[{x, y, ?}, 
         Blend[Append[#, First[#]] &[cols[[;; 3]]], 
          Mod[(? + ?/2 + t)/(2 ?), 1]]], 
       ColorFunctionScaling -> False, 
       Background -> Last[cols]], {t, -?/2, ?/2}]
     ]