Making a Pseudosphere

When I learned about models of the hyperbolic “plane” I was intrigued mostly by the Pseudosphere, and how it represents a surface of constant negative curvature, this grabbed my attention so much that I knew I had to have one. And thus began my epic quest to make a rotation of a tractrix about its asymptote.
One thing to note about the tractrix is that it is a curve that you can create through rather simple methods. All you need is to have a string, paper, a pencil, and an object of small but noticeable mass. The tractrix is created by tying one end of your string to the mass. Place the weighted object on the paper as far away as it can go so that your taut string makes a right angle with the edge of your paper. Holding your end of the string in your hand at the edge of your paper, move your string along the edge of the paper so that your hand follows a straight path. As you move the string, the weight will begin to move. Now mark the location of your mass every centimeter or so. This curve is called the tractrix.
There are many methods to create this shape, one of the more popular modern methods for making these sorts of shapes is on a 3D printer. Since I don’t have a 3D printer, I decided that I would have to make this Pseudosphere by hand which is obviously much cooler. The method I chose was a lathe, which is a tool that turns a chunk of wood at high speeds, allowing the woodworker to remove parts of the wood, yielding radial symmetry about the axis of rotation.
The first thing I had to do was determine how large I wanted my Pseudosphere to be. To do that, I looked up the equation for a Pseudosphere, which is defined parametrically.

Parametric equations for a pseudosphere.

Parametric equations for a pseudosphere.

The constant value “a” can be changed at will, so I decided to make “a” be 3 which would give me a main diameter of 6. At a length of 10 inches, five on each side, I would have sufficient width at the ends to maintain structural integrity. The units I decided to use were inches, due to the fact that I am already familiar with them, and my measuring tools are all in inches.

One-inch blocks of wood glued together before lathing.

One-inch blocks of wood glued together before lathing.

For my choice of wood, I had two options. The first was take a big log and try to cut it down to the desired size. Due to the fact that it would take a while to cut a six inch diameter log that is ten inches long down to the proper size, and since large logs often have splits in them, I chose another option. That was taking a series of one inch thick squares of wood and gluing them together in a pattern roughly resembling the shape of the Pseudosphere.
Using a trace of the tractrix, my father and I determined the placement of the blocks and began the process of gluing pieces together in the right configuration. The gluing process took two days to complete, because I didn’t want my project flying to pieces for a lack of patience on my part.  Essentially, the glue had to set up to full strength to withstand the turning.

An intermediate pseudosphere.

An intermediate pseudosphere.

Once the pieces were all glued together, I had the momentous task of taking squares, and rounding them out to a closer approximation of a Pseudosphere. The process wasn’t easy, throughout the whole project I had a sore hand from having to hold onto the tool through all the jerking motions made by the heavy wood slamming into the metal cutting tool.
After having removed all of the corners, I took calipers, and used them to determine the diameter that the Pseudosphere should have at every inch. Cutting down to those points, I began slowly, and carefully working inwards, trying to get the right curvature. Because the natural tendency of all of these tools is to cut a straight line across a surface, I had to pay special attention to the tool’s path, so that I could get a constantly changing curvature throughout the whole length of the Pseudosphere. There were some tense moments when the knife didn’t behave in the way I was intending, and I cut too deeply, but the wonders of foresight helped, because I had, at the beginning given myself a 1/8 inch cushion in my measurements, so that I could sand, and perform the finishing touches properly.

The finished pseudosphere.

The finished pseudosphere.

The sanding was a fun process, the high friction caused a heat buildup in the sandpaper, making it necessary to use a glove. Once we had it all sanded we applied a finishing oil called “Tung Oil” to give the wood a darker, richer color. It also smells pretty good.
In the end I was left with a wonderful mathematical object. There were some questions from people, such as, why I didn’t make the Pseudosphere infinitely long. My response is twofold. One I would have needed infinite wood. Despite the fact that a Pseudosphere has finite volume, the cylinder I would have needed to start with would have needed to be of infinite volume. Also structural stability had a role to play there, I can only make something so thin and rotate it at high speeds without it suffering a physical breakdown.
I do however hope you like my project, and remember, our lives are only as interesting as we decide to make them.

-Travis

Sources
http://mathworld.wolfram.com/Tractricoid.html

Image credits
(I made them myself, don’t tell anyone)

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