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Can You Make an Elephant out of Polyhedra?

Our Math Circle is about playing with, conversing about, and collaborating on math.

“We got those for Christmas,” announced G as she arrived and saw the Polydrons scattered across the table. “They were the only educational thing we got.”

 A brief flash of worry entered my mind as our new Math Circle (with mostly new students) seemed to be labeled “educational” right from the start. Our Math Circle is about playing with, conversing about, and collaborating on math, without a formal stated agenda. I needn’t have worried, though, since the students snapped up the pieces and immediately began building.

G knew that you can make nets and build solids with Polydrons, so we played with the possibilities. D, J, and G quickly constructed “triangular pyramids,” which we also called tetrahedrons. X built a net that I knew could be folded into a cube. As she folded, I said, “Looks like you’re making a cube.” The open-mindedness of children prevailed, though, as she corrected, “It doesn’t have to be a cube.” While N was constructing “either a hat or a house,” M created a pattern with pentagons and triangles that reminded us of a soccer ball. We spent a few minutes considering which polygons make up a soccer ball. Then I posited that perhaps every object in nature can be thought of as a construction of polyhedra.

“No they can’t!” answered D assuredly.

“What can’t be?” I asked.

“An elephant,” he replied. The conversation immediately turned to whether an elephant’s parts could be represented geometrically, with shapes all formed from straight lines. The kids agreed that using a sphere for the head was not allowed, so we’d need to build something ball-shaped from pentagons, hexagons, etc. Octagons were suggested. G’s face lit up as she realized “the more sides you add, the rounder it looks!” Everyone then agreed that we could make the head.

Other body parts were constructed in our minds. This was all done in our minds, in a great example of the mathematical goal of moving from the concrete (plastic manipulative polygons) to the abstract (a mental sphere consisting of octagons). The sticking point was the tail. No consensus was reached, and attention shifted when X help up a new creation and asked, “What’s the name of this shape?” M and X both constructed interesting irregular solids and asked the rest of the group to name them. A “kkkharrr?” A “shoe-gon?” An “eight-billion-a-gon?”

The students then compared and contrasted everyone’s constructions. “What’s the same, what’s different?” I asked occasionally. We practiced discrimination, since mathematics can be described as the art of looking for similarities and differences.

I introduced Schläfli symbols, a way to represent polygons, tessellations, and solids numerically. They symbolized the shapes we had created, and then I challenged them to figure out what shape a given symbol represented. We really just touched on this, but will do more next time.

All marveled at J’s creation, on which she had been working for nearly an hour: a large ball constructed almost entirely of pentagons. As she was close to finishing it, her end of the table ran out of pentagons, so she kept going with other shapes. We agreed to collaborate next week on continuing J’s work, but with pentagons alone.

As everyone worked, I told the life story of the mathematician Euler. N remembered something he had heard months ago about Gauss and connected it to Euler. G found a commonality between Euler and what she knew of Euclid. The whole class, especially N, enjoyed trying to anticipate the tragedies that befell poor Euler. We also had a lot of fun trying to pronounce funny names and words (not just Euler and Schläfli, but “tsar” and others). The kids were surprised to hear how many leaders of Euler’s time had the phrase “The Great” at the ends of their names. They were also surprised to hear that until Euler came along, math symbols were not international. Time flew quickly, and before we knew it, it was time to go. As we cleaned up, we talked about “Hershel’s planet,” as it related to Euler and to a mathematician we will discuss next week. Looking forward to resuming then!

-- Rodi

PS: Thanks to Sue VanHattum for introducing me to Polydrons, and to Maria Droujkova for the great working definition of mathematics for children.

(Photo Credit: Rachel Steinig)