Tag: Scrolls

The Real Helicoid (Scrolls II)

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.

AssociateCatenoid2

A such, it possesses a deformation into the catenoid
AssociateCatenoid3

which every textbook on the geometry of surfaces mentions at least as an exercise. Half way the helicoid will look like this:

AssociateCatenoid1

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.

HeliocoidScroll2

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.

HeliocoidScroll

The Other Helicoid (Scrolls I)

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.

HyperboloidalScroll0

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.

HyperboloidalScroll2

We can bend further, making the generators truly horizontal. This gets us to the other helicoid:

HyperboloidalScroll

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.

HyperboloidalScroll top

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.

HyperboloidalScroll3

Here the idealized paper is slicing through itself, which looks interesting, but will, like most ideals, require some trimming in reality.