Astroids ad Infinitum

d\[Theta] = 3;
r[n_] := (1/d\[Theta])^n;
p[0, t_] := {0, 0};
p[n_, t_] :=
  p[n, t] =
   p[n - 1, t] +
    RotationMatrix[(-d\[Theta])^(n - 1) t].{0, r[n - 1] - r[n]};
frame[\[Tau]_] :=
  Show[
   Graphics[
    {
     Table[Circle[p[n, \[Tau]], r[n]], {n, 0, 10}]
     }
    ],
   If[\[Tau] < 2 Pi,
    Table[
     ParametricPlot[p[n, t], {t, -0.01, \[Tau]},
      PlotStyle -> ColorData["RedBlueTones"][n/10]], {n, 0, 10}],
    Table[
     ParametricPlot[p[n, t], {t, \[Tau], 4 Pi + 0.01},
      PlotStyle -> ColorData["RedBlueTones"][n/10]], {n, 0, 10}]
    ], ImageSize -> 300
   ];
Manipulate[frame[t], {t, 0, 4 Pi}]