The Anti-Cone and its Doppelgänger (Scrolls IV)

The literature has not many interesting examples of ruled surfaces in Euclidean space besides cylinders, cones, hyperboloids, and the helicoid. Let’s fix that. A cone, or more precisely a frustum (latin for piece), can be described by following a horizontal circle (as a directrix) counterclockwise and rotating the generators also counterclockwise, horizontally pointing in the same direction as the point on the directrix, and with a fixed vertical component.

Cone

The result is a flat surface and thus not interesting for making a curved book. But we can also follow the circle and let the generators rotate the other way. I will call the resulting surface an anti-cone. It is certainly not flat anymore.

Anticone

Take for a ruled surface all of its generators (the straight lines) and shift them so that they pass through the origin.
Their intersections with a sphere centered at the origin is called the spherical indicatrix of the ruled surface.

In this case, the spherical indicatrix is a pair of horizontal circles, both for cone and anti-cone, but differently orientated.

An old theorem about ruled surfaces states that you can deform any ruled surface by changing its spherical indicatrix pretty much arbitrarily. It turns out that there are typically two different solutions to do this, even if we trace the indicatrix in the same direction.

In other words, for a given ruled surface, there is a second ruled surface along a different directrix but with pairwise parallel generators. Let’s call this the doppelgänger of the ruled surface. For the anti-cone, it looks like this:

Anticonedual

One can visualize the relationship between the two surfaces by putting them together and coloring corresponding (i.e. parallel) lines with the same color.

Anticone3

If we had paper ribbons shaped this way, we could bend one into the other.

Here is another image of the generators of the anti-cone doppelgänger. Giving up on clarity can increase the esthetical appeal when we add ambiguity.

Anticone1

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.