Sudo make me a pseudosphere
You may have seen it before: a magician spreads out a deck of cards on a table into a long row. He tilts the endmost card upward, and the neighboring cards rise accordingly, resting on the adjacent card. The next thing you know there is a beautiful shape, an array of cards coming to a sharp point which can be moved back and forth with the magician’s finger.
This is a tractrix. It’s actually the same shape as the curve that you get when you drag a pole along a line in the ground. You can imagine this by imagining each card in the spread as a pole at a different instant in time. The pole is being dragged by the lower side of the card, and the upper side is the side being dragged, as shown in this animation.
To make things more interesting, what would the cross-section of these cards look like when rotated about a horizontal axis? Each card would rotate to become a cone, so we would be left with a collection of nested cones that together form the surface of revolution of the tractrix.
This surface is called the pseudosphere, and it is a classic example of constant negative curvature (saddle-shaped curvature). What is cool is that we have built a surface with negative curvature out of many cones, which are intrinsically flat in the sense that they could be built from flat waffles to make waffle cones. This presents itself as a wonderful way to build a pseudosphere from paper: cut out a bunch of identical circles from paper, and roll them up into variously sloped cones, and then stack them from most shallow to steepest.
I calculated the optimum sizes of paper circles and cut them up and created a lovely model of this. It looks like a dunce cap with a very sharp point at the top. In reality, it goes infinitely far up, but I didn’t have enough time to build it that far. I did, however, have enough paper to build the infinite version, because it turns out that the surface area of the pseudosphere is finite despite its infinite extent, which is kind of cool.
The pseudosphere is a series of stacked cones, each with slightly differing slopes. The thing that I found particularly interesting about this model is that the cones have the interesting property that they can all be cut from equally sized disks of paper.
The following video is what this Pseudosphere model looks like in all its glory.