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).

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).