Touching Spirals:
With[{n = 2}, ParametricPlot[{v Sqrt[u] Cos[u], v Sqrt[u] Sin[u]}, {u, 0, n 2 Pi}, {v, Sqrt[(n - 1)/n], 1}, Epilog -> {Red, Point[{Sqrt[ (n - 1) 2 Pi], 0}]}, PlotPoints -> 30 + 10 n, Frame -> False, Axes -> False]]
n=3:
