This is a gif of a helicoid transforming into a catenoid. A helicoid was first described by Euler (you know, the guy who brought us e, the base of natural logarithm and so many other mathematical discoveries) 

A helicoid can be made by taking a two-dimensional plane, choosing an axis (line) on the plane and turning it around the axis. Because they are both minimal surfaces, a catenoid can be into the shape of a helicoid (and back) without stretching the surface.  

minimal surface locally minimizes its area, having a mean curvature of zero. 

Image source: here