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}
    ]

[![enter image description here][1]][1]


  [1]: https://i.sstatic.net/wMybs.png