In 1760, Leonhard Euler studied the curvature of intersections of a surface with planes perpendicular to the surface, and showed that the maximal and minimal values of their curvature are attained along orthogonal curves. In 1776, Jean Baptiste Marie Charles Meusnier de la Place showed that for minimal surfaces these principal curvatures are equal with opposite sign. He went on to show that both the catenoid and the helicoid satisfy this condition, thus exhibiting the first two non-trivial examples of minimal surfaces. Euler had discussed the catenoid as a minimal surface before, but only in the context of surfaces of revolution.
In its standard representation as a ruled surface, the parameter lines are the asymptotic lines of the helicoid. For a change, here is the helicoid parametrized by its curvature lines:
The purpose of this note is a little craft, similar to what I explained earlier using Enneper’s surface: A ruled surface that has as directrix a curvature line of a given surface, and as generators the surface normals, will be flat and can thus be constructed by bending a strip of paper. Doing this for an entire rectangular grid of curvature lines results (for the helicoid) in an attractive object like this one:
To make a paper model, one first needs to find planar isometric copies of the ribbons. This is done by computing the geodesic curvature of the curvature lines of the helicoid, and, using the fundamental theorem of plane curves, then finding a planar curve with the same curvature. The (planar) ribbon is then bounded by parallel curves of this plane curve:
Using four (due to the inevitable symmetry of things) copies of the template above, carefully cut out & slit, allows you to easily build the model below, which also makes a nice pendant. Print out the template so that the smallest distance between two slits is not much wider then your fingers, otherwise assembling the pieces will be tricky.
Begin with the largest J-piece and use the four copies to build a frame, by sliding the hook into hook and non-hook into non-hook. Then continue inwards, adding four copies of the second largest J, by placing the hook of a new J next to a hook of the old J.
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.
I have been thinking for a while to make a book out of curved paper, and my new year resolution for 2016 is to make this happen.
Usually, a book consists of a few rectangular pieces of paper that are attached to each other along one side of the rectangles to form the spine of the book. The fact that we can turn a page nicely uses the fact that flat sheets of paper can be bent into cylindrical or conical shapes without the need to bend the spine as well. A good choice of a shape for curved paper that behaves similarly is that of a ruled surface or scroll. The latter name is not in common use anymore, but I like it better.
For instance, we could take paper in the shape of a hyperboloid of revolution. This consists of a family of generators (the orange straight lines) that are attached to a directrix (the waist circle, for instance). We will now cut open this hyperboloid along one of the generators and bend it a little along all generators simultaneously, thus making them more horizontal.
We can bend further, making the generators truly horizontal. This gets us to the other helicoid:
That it is not the standard helicoid that you get by lifting and rotating a horizontal straight line along a vertical axis becomes evident in the top view.
Cross sections of this helicoid with vertical planes are graphs of the reciprocal of the sine function, in case you have wondered.
We can deform further, arriving at more scroll like images.
Here the idealized paper is slicing through itself, which looks interesting, but will, like most ideals, require some trimming in reality.