Catenoid to helicoid

At θ = 0 it is a catenoid, the shape of a soap film between two rings. Cut it along the gold meridian and drag θ up: the surface twists as it unrolls, becomes a helicoid at θ = π/2 and closes back into a catenoid at π. Every small square of the grid only bends, never stretching or shrinking, and the change in length measured in the corner stays at zero; at every point the two principal curvatures still cancel exactly, so the largest |H| stays at zero too: every shape along the way is a minimal surface. Turn the view to watch the twist from another side.