Golden, silver and metallic spirals and, more generally, plane curves consisting of successive circle arcs, are nice examples of curves which are $C_1$ but not $C_2$, since they show continuous tangent lines, but piecewise constant, whence non-continuous, curvature. On the contrary logarithmic spirals, where the radial distance r is an exponential function of the polar angle, are $C_{\infty}$ curves.
Writing that "Similarly, one can construct a spiral for $\lambda _ n$ for any $n$. These are logarithmic spirals ..." may cause some confusion. Better were to write that metallic spirals are very good approximations to logarithmic spirals (and vice versa).