A Math Circle is, by definition, a conversation about math.

“How many points should our polygon have?” R asked G. They were playing with the dynamic geometry software Geogebra* before today’s Math Circle began.

As I was setting up, I had mentioned to them that I hadn’t yet found an obvious way to insert a diameter into a circle on this geometry program that I was trying out. Having never seen the program before, they got right to work constructing a Baravelle Spiral in GeoGebra.

“I can move my point! This is so awesome!” said R.

“This could go on forever,” observed G as they repeatedly enlarged the spiral to continue the pattern inwards. “This is better than anything on paper!”

“Way better,” agreed R.

“We have to decide when to stop, though,” continued G. We discussed whether patterns could get infinitely smaller. Then Z walked in and looked at their work. “We’re doing it freehand, without a compass!” announced both G and R. It took me a few minutes (and some explanation on their part) to realize that they were using the term “freehand” ironically, using a bit of Euclidian humor to say that this program delivered the goods in an immediate and precise manner.

“I think we ought to get started with our Math Circle,” I told them, even though not everyone had arrived.

“But this is Math Circle right now,” protested Z. I agreed, but explained that since today was this group’s last Math Circle for a few months, I did want to spend some time showing them something else too. With a stricken look on her face, Z replied dramatically, “It’s our last one? My poor, poor aching heart!”

J and M arrived as I was showing pictures of various compass designs and spirals: Baravelle Spiral quilts; nautilus shells; and mandalas from 7 different cultural traditions.** As we were looking and discussing, most of the kids were already creating their own compass designs on paper. I told them the story of the artist Giotto’s perfect circle and the Pope’s reaction to it, and we also spent some time defining the term “polygon.” As I finished showing the mandala pictures, Z asked “Do we get to make our own mandalas?”

“Yes,” I replied, and I showed them an unfinished mandala that I had created. They oohed and aahed until I told them that I couldn’t finish coloring it symmetrically because the coloring step had revealed a major flaw in the design. I asked them to find the flaw and suggest corrective actions.

G wanted to create a mandala similar to mine, so she studied it and brainstormed how to create a dodecagram inscribed in a circle using the Euclidian convention. M and J were each creating BB gun targets. J’s target was a Baravelle Spiral compass construction using a ruler (versus straightedge) to quickly find midpoints, so he was coloring his intricate design before most of the kids had gotten a handle on their plans. Z was struggling with where to begin and requested, “I need a challenge!”

“Want to copy mine?” offered M generously. She didn’t, but accepted the same challenge as G, the 12-pointed-star mandala. I told of how that pattern had come to me in the middle of the night, after I had been brainstorming mandala designs for 2 weeks. We debated whether “Eureka moments” were fact or fallacy. I told of the circumstances of Archimedes supposedly coining the word Eureka. I played an audio file of John Philip Sousa’s telling of how his composition “Stars and Stripes Forever” had come to him in a dream. I also relayed related anecdotes about Einstein and Darwin. I concluded with an excerpt from an article about science writer Steve Johnson’s book Where Good Ideas Come From. The kids agreed with this new thinking about the Eureka moment: it isn’t “a flash of insight, but what Johnson calls a ‘slow hunch.’”

A Math Circle is, by definition, a conversation about math. And what a great conversation the past six weeks have been. I look forward to resuming our conversation later in the winter. Dates and topics for our next Math Circles are posted on the Talking Stick website, but, as these children know, we will also talk about whatever math we want to. Thank you, parents, for sharing your kids with me here.

— Rodi

*You can download Geogebra for free online.

**I can email you the file with these pictures if you’d like to see them.