This might do the trick:
Manipulate[
ParametricPlot[
#1 {Cos[#2], Sin[#2]} & @@ {t, Log[i] Floor[t]},
{t, 0, 200}
, Background -> Black
, PlotStyle -> Purple
, Axes -> False
, PerformanceGoal -> "Quality"
, PlotRange -> {{-201, 201}, {-201, 201}}
],
{{i, 3.525}, 3.43, 3.6}
]
Since you enjoyed the animation aspect here is nearly verbatim code I wrote 15 years ago:
Animate[ParametricPlot[#1 {Cos[#2], Sin[#2]} & @@ {t, Log[i] Floor[t]}, {t, 0, 200},
Background -> Black, ImageSize -> 400, PlotPoints -> 150, Axes -> False,
PlotRange -> {{-201, 201}, {-201, 201}}], {i, 1, 12.365}, DefaultDuration -> 200,
AnimationRepetitions -> 1]
The animation is much too long to practically include as a .GIF here, but I hope you enjoy the patterns that emerge from this simple function.