The Talking Stick Blog

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The Perfect Circle, the Perfect Rock, the Perfect Perpendicular Bisector

Interesting. Surprising. True.


“Well, we don’t want to waste erasers here in Alexandria, so this is the best method,” argued R (playing the role of Euclid), after she demonstrated how to perpendicularly bisect a line with only a ruler, a straightedge, and a piece of chalk. She faced off in debate against characters who might have different perspectives on this task: an Engineer (M, who spent the time before class theoretically improving upon the compasses we use); Artist (our visitor P); Computer Programmer (A); and Origami Maker (G). Since we didn’t have enough assigned roles for every child, and, moreover, since the various perspectives weren’t obvious, the kids immediately helped each other and played every role. They enjoyed imagining how the different characters would tackle the task.

“Code it!” said Computer Programmer A. “Measure it with a tape measure!” said Engineer M. “Just look!” And so on. Origami Maker was the trickiest perspective to fathom. Then I asked if they could think of someone who might have a method different from all of these. “Fashion Designer,” was the reply. The kids explained that such a person would undertake a hybrid method involving eyeballing, measuring, and folding. (Interesting. Surprising. True.) Then we returned to the Euclidian method.

Last week several kids had vehemently urged me to allow constructions done using the eyeballing method, and did not grasp how that method was different from the Euclidian method. Now they understood why some people might prefer Euclid’s method over just looking. “It’s not perfect,” said J about eyeballing. Then they contrasted the Euclidian method with measuring: which method would be more accurate in which situations? A then explained to P that the Ancient Greeks “didn’t have a number system” that could effectively divide lengths in two, so it’s a moot point historically.

We returned to our weekly challenge of attempting to define geometric terms. This week’s term was “spiral.” Several children gave definitions which I wrote on the board. Then I read to them a formal, authoritative definition, which contained elements of each concept that was on the board. Smiles appeared on faces as M proclaimed, “We really got it!”

Since P was visiting, we took a few minutes to show her some of the compass art we had been making at Math Circle and at home. As I was paging through my sketch book, a few students noticed the Baravelle spirals I had made and said “I want to make that!”

“Show us how to make that!” demanded G. I asked the group if they all agreed, and heard a resounding “yes!”

As the rest of the group studied the Baravelle spirals I had drawn and colored, and some that I had printed out, J got right to work. Before the rest of the group had collectively remembered how to construct a hexagon with a compass and straightedge, he was halfway done with his spiral. I told the group how I had been attempting these spirals at home without a ruler, which is tricky since every line segment must be bisected. I told them that it doesn’t matter to me how they did theirs. It was interesting to see the purists in the group engaging in the Euclidian method, while others used rulers, and yet others eyeballed it. Our discussion returned to the pros and cons of each method. I mentioned how artists for ages have attempted to draw a perfect circle freehand. I also told of how I had filled years of boredom in school attempting this, unsuccessfully, on my own. And of how I spent years poring through stones on beaches to find nature’s perfect circle in a rock. I did find one once, I thought, but was always afraid to check it with a compass in case it wasn’t perfect. The kids begged me to bring the rock to Math Circle next week. I said I would look for it in my basement, as long as they promised to lie to me and say it’s perfect even if it isn’t.

As they worked, I told of the life of Hypatia. Then time was up. Everyone seemed excited to continue working on their spirals at home. Everyone, that is, except for me. I’m off to my basement to look for that rock.

-- Rodi

PS:  Thanks to Maria Droujkova for the role-playing idea and to Rudy Penczer for suggesting that our Circle investigate Baravelle Spirals.