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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"
  ],
 {{i, 3.525}, 3.43, 3.6}
]