Some things in math we choose to believe even though we can’t know for sure.
Michaelangelo, Bernini, Lloyd Wright (or, the students wondered, was it simply Wright?), Zarah Hussein, Native American geometry designs, mapmakers art, the feng shui compass, and a photo of a piece of jewelry covered the table as the students tried to figure out what the pictures all had in common. One picture had many “swirls” (spirals) from nature, sparking a brief conversation about fingerprints. After close guesses of octagons, circles, and symmetry, R said, “You can make them all with a compass.” So began the first session of our middle-school math circle.
“Who do you think invented the compass?” I asked. No one was sure, but Z suspected Galileo. After we talked about Galileo’s famous accomplishments and his pretty awesome full name (which A knew), I recounted an anecdote about his sour experience in medical school lectures. We discussed G’s comment that lecturing is the least efficient form of instruction. Then I showed them a painting of Euclid holding a compass 1800 years before Galileo was born, and told a story from Greek mythology about Daedalus’ assistant Perdix and the invention of the compass. I told them that it says on Wikipedia that Galileo invented it, and we all agreed that you can’t trust everything you read there. I told them that it seems certain that Galileo made some significant improvements in the compass, but that I am still trying to find an authoritative source on its true provenance.
“Would you like to use these compasses?” “Yes!” So we spent some time with paper, pencils, and assorted compasses: trying them out, comparing and contrasting. People drew concentric circles, overlapping circles, big and small circles, and things that looked like eyeballs or venn diagrams. Then we looked more closely at the picture of the piece of jewelry. It was a gold “flower of life” - a geometric design with history in many cultures and religions. The students debated whether it was constructed with any straight lines, and tried to make their own flowers of life. J came very close, with a similar center, but a more rectangular outward flow, almost like a compass rose. We traded compasses so that everyone could try each type. (Some were much harder to hold in position than others.) The students studied that photo of the necklace again and again, trying to come up with strategies on how to make the design.
As everyone worked, we looked more at the work of Islamic artist Zarah Hussein. I told of her mathematical studies, of the goals of her art (to understand her religion), and of the great influence of mathematics on Islamic culture and religion. When I mentioned that in this culture, the circle was once used as a unit of measure, J suggested that a rope was used. I said that his comment made me think of a dog tied on a rope in a yard, and the pattern pressed into the grass as it ran. We also talked about Euclidian constructions, also referred to as “compass and straightedge constructions.” We wondered whether it was true that you can make every geometric figure with a compass and a straightedge.
At this point, students wanted a hint about how to make the flower of life, so I told them to put the compass point where the circles crossed. As the designs blossomed into flowers, G still wasn’t convinced that the pattern had no straight lines. We had a hearty conversation about just what is the definition of a line anyway? It was impossible to define without using circular reasoning, a concept that everyone found fascinating. When Z said, “So tell us your definition of straight,” I told them that a line is considered undefined in math, but that we all agree to use the term with an understanding of what we mean. We talked about how some things in math we choose to believe even though we can’t know for sure. For instance, how can we really know that a line goes on forever, or even, suggested Z, that something can be infinite? We ended our circle with requests to do more drawing next time, and a promise to collaboratively define the term “circle” next week.