After talking about the other helicoid first, it would be impolite to ignore the real helicoid, which is of course much more famous,
mainly because it also is a minimal surface.
A such, it possesses a deformation into the catenoid
which every textbook on the geometry of surfaces mentions at least as an exercise. Half way the helicoid will look like this:
I have chosen the size of the helicoidal paper so that the two spiral edges almost touch. This deformation is, however, problematic for book making, because no curve could serve as a spine.
But there are other ways to bend the helicoid, that are not anymore discussed in text books. We can keep the horizontal generators of the helicoid as straight lines, but let them slope upwards a little, like above,
or like below, with steeper lines.
The Inner Frame 