# Of What Use is a Torus?

Knitted Tori

I have been knitting tori. A torus is a doughnut shape. It has particular geometrical and typological properties that make it quite interesting. For instance, in a video game like Pac Man, the Pac Man exits what looks like a flat plane and re-enters on the opposite side: the left and right or top and bottom portions of the screen are connected. This is the geometry of a torus. Pac Man isn’t in a flat plane but is cruising around on a torus. You just get to see a small portion of this. Pretty cool.

It was even more cool when I found that the torus might be a solution to a statistical problem I have. I am an ant ecologist currently studying the local distribution of the colonies of the Comanche harvester ant, Pogonomyrmex comanche. These ants nest in open prairie surrounded by oak forest and nowhere else. The real, hard boundary of the forest presents a difficulty for determining the spatial pattern of colonies since a colony at the boundary cannot have a near neighbor within the forest. By modeling the prairie area as a torus – so connecting the edges like the Pac Man game – may be an appropriate way to remedy the problem.

The torus is also a form that an attractor (from chaos theory – a set of physical properties to which a system moves) can have in a dynamical system, like ecological systems. So, I became even more enamored of the torus. This is a very useful shape besides being a doughnut, a bagel, or a lifesaver candy.

I have been using the book Making Mathematics with Needlework (by Sara-Marie Belcastro and Carolyn Yackel) to explore mathematics through some of my favorite forms of needlework. So far I have only done knitting projects. I have been learning a lot about knitting (knitted my first sock) and the mathematics. There is a project for knitting a torus. So, of course I took this up.

There are only two stitches that will create a torus. Sounds simple but I found the pattern a bit of a challenge to get into my head since it is so different to be knitting in the round with short rows when you are not making a sock, a hat or a sweater sleeve. After many, many hours of pulling things out and beginning again, I now have 2 finished tori and am about to finish a third. I have learned a lot about knitting and mathematics. Then a friend asked, “what exactly does one do with this item?”

Here is the progression of knitting a torus. Cast stitches on and distribute on 3 double pointed needles, then knit in a round using short rows to make the curved shape. When complete, the torus shell is stuffed and the ends of the tube knitted together.

So, of what use is a torus? My immediate answer is whatever you want. Your imagination and creativity are the limits. After all, a torus is an awesome shape. But I understand the need of many for some obvious utility (though this strikes me as a refusal to play the game, the intellectual game). So, I’ll humor you for a bit.

A knitted torus is great. It has at least these utilities:

1. The construction of the torus is an exciting way to learn knitting and mathematics. The pattern given does not actually tell you how to knit the torus together. So there is quite a bit of creativity and insight required to complete the project. This is a good project in problem solving.

2. The finished torus is the result of a creative process and rightfully is called fiber art and sculpture. It is beautiful.

3. It is also worth contemplating. What kind of shape is this torus? But also, what is the knitting that makes up the torus (It is an unknot! That this torus is made up of an unknot should trip you out. If it doesn’t you are very naïve.). What are the implications? How might this shape be used? Philosophical, craft, mathematical and other considerations are found in the torus.

4. The torus can be used as a discussion starter, as a paperweight, as bagels or doughnuts for dolls, as a coffee table decoration, etc.

5. The torus can be used to teach or demonstrate the math, to demonstrate how Pac Man moves.

6. It can be another example of the accomplishments of a child. My Dad has my first knitted torus on a table in his living room. I imagine he tells visitors that I knitted it and … whatever comments he might want to make about a torus (he is a theoretical physicist) – he probably likes that he can do this since it evens him up with Mom and her posting photos and drawings (like my molecular drawings of endo- and exo- brevicomin) on the refrigerator.

7. Like most items, you can collect and trade them; you can give one as a gift.

8. A torus can be used as a marker, a game piece, a token.

In the end, a torus is functional, abstract, and beautiful,  in some ways akin to poetry: if you ask what one does with it, you haven’t understood.