*A tad early for ice skating, perhaps, given the current weather in Chicago, but not for Shimer! Bev Thurber is a Shimer faculty person who holds a variety of degrees, from MIT to Cambridge University to Cornell, and from Nordic languages to math and other fields. Here's what she has to say on "Euclidean Ice Skating." As a non-skater -- who has never taken IS 2, this is the start of a wonderful education for me. How about you? *

Figures are what gave figure skating its name, though this seems to be peculiar to English --- in other languages, figure skating is called artistic skating. They're geometric designs that skaters draw on the ice with their blades. The official set of figures used in competitions, called compulsory figures, are all based on the figure eight. Figures have not been required in competition since 1991, and few people do them now.

Skating figures requires a lot of concentration. The goal is to get the line on the ice exactly right, then to go over it two more times. The lines left on the ice, called tracings, are the important part. In figures competitions, the judges stand on the ice and watch the figure being skated. Then the skater leaves and they examine the tracings carefully. Tiny mistakes, invisible to a spectator watching from the sidelines, can determine the outcome of a competition.

One example of a figure is the backward serpentine, which consists of three circles in a line, just touching one another. The skater starts at the intersection of two of the circles, skates (backward on one foot) halfway around the middle circle, then, without changing feet, glides all the way around the end circle. Once she or he has reached the other intersection point, she or he changes feet ("push...with vigor," advises Maribel Vinson Owen (*The Fun of Figure Skating*,98)) and completes the remaining 1.5 circles on the other foot.

This figure isn't very advanced --- it has no turns, and the backward change of edge, its main feature, is one of the basic building blocks used in more advanced figures. But it's quite difficult to learn, and many skaters have struggled with this figure. In fact, T. D. Richardson writes that the backward change of edge in figure skating "has been aptly described as the *pons asinorum* of skating" (*Modern Figure Skating*, 1st ed., 34).

Richardson's statement links this figure directly to Euclidean geometry. The *pons asinorum* (`asses' bridge') is the nickname of the fifth proposition of Book I of Euclid's elements, which is read in Integrative Studies 2, a course (also called Foundations of Math and Logic) that we offer at Shimer. It says: "In isosceles triangles the angles at the base are equal to one another, and, if the equal straight lines be produced further, the angles under the base will be equal to one another." (Euclid, *The Elements*, trans. Heath, I.251)

Euclid's proof of this proposition is, like skating a back serpentine, notoriously difficult for beginners. The Shimerian student assigned to do it on the board in IS 2 always has a hard time with it. The difficulty and the diagram of the proof, which looks something like a bridge, earned the proposition its nickname by 1780 (Euclid, I.415).

Euclid is hard for the same reason that figures are hard: you have to get it right. In geometry, you get a set of building blocks in the form of axioms and previous propositions. In figures, they're circles,

turns, and changes of edge. A new proposition, or a new figure, is a combination of the building blocks at hand, and no more. Studying Euclid teaches students (and all of his readers) how to think logically, just like studying figures taught many people how to skate well.

Euclid may not be very popular these days but his *Element*s remains an excellent introduction to logical reasoning. Students (and perhaps many otherds) aren't always enthusiastic about studying him, but they realize how important understanding his work is when they get to Lobachevsky's*Theory of Parallel Lines*, the next reading in IS 2. That's the one that makes them want to poke their eyes out.

And you?

For access to some of Euclind's *The Elements*, click here. For more on Bev Thurber, click here.

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