Animate[Show[
ParametricPlot3D[
RotationTransform[
t, {0, #[[1]], 0}][{Sin[\[Theta]], Cos[\[Theta]],
0} (E^Cos[\[Theta]] - 2 Cos[4 \[Theta]] +
Sin[\[Theta]/12]^5)], {\[Theta], 0, 2 \[Pi]}, Axes -> False,
Boxed -> False,
ColorFunction -> Function[{x, y, z, \[Theta]}, Hue[\[Theta]]],
RegionFunction ->
Function[{x, y, z, \[Theta]}, #[[2]][x, 0]]] & /@ {{-1,
Less}, {1, Greater}}], {t, -(\[Pi]/2), \[Pi]/2},
RefreshRate -> 10, DisplayAllSteps -> True, AnimationRate -> 1,
AnimationRunning -> False, AnimationDirection -> ForwardBackward]

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