From Problem to Question to Proof to Problem
I just had an absolutely wonderful meeting with a student I’m working with, a second grader by the name of Millan. The kid is a natural mathematician, and a joy to work with. Allow me to describe what happened today.
Every time we meet, he brings me a question; this is his central duty in between our meetings. In the past he’s asked me questions about cutting slices of spheres (inspired by an exhibit at the Pacific Science Center) and how multiplication by 9 works, and today, thinking of our previous work on projections, he asked:
Is there a shape you can shine a light on such that no matter how you turn it, you always get a square?
He was inspired by a particular property of spheres that we discussed: they always cast circular shadows. (This isn’t immediately obvious, and the implications are actually serious: Aristotle used this fact, along with the observations of the Earth’s shadow on the moon during lunar eclipses, to argue that the Earth was a sphere!) It’s a beautiful question. Probably too hard for a second grader to answer rigorously, but worth thinking about. If Millan wanted to think about it.
In fact, he did, and together, we managed to solve it. First, he noticed that the shape must have all square faces (and he clarified that square shadow should not change as the object rotates, so all the square faces must be identical). Why? Because you could always have your light source close enough to the face that the shadow is just a projection of the face. So now we had a new question:
What 3D shapes have all identical square faces?
This is still tricky, but more tractable than the other question. The cube is the obvious one (it unfolds in the drawing below) I observed that there must be a band of squares all in a row, like the four down the middle in the picture.
So you get some kind of shape on the sides after you fold that band together, and then you need to fill in the empty parts of it with squares. After Millan did some careful analysis of the possible ways to do it, he realized that if the shape had any angle besides a right angle (or a straight angle), it wouldn’t work. We were able to dispose of all the options. The cube was the only choice (implicit here was a desire for convexity. Something to return to later).
The whole thing was elaborate enough that I thought we should write it up. Millan opened his notebook and wrote:
The only 3D shape with equal square faces is a cube.
To which I added, at the beginning:
And then, underneath:
We spent the next twenty minutes recollecting the argument and writing it down. It was an amazing accomplishment for a second grader, in my opinion. After it was done, I told him about the phrase “quod erat demonstratum,” and how mathematicians write QED at the end of their proofs. Since he’s now a working mathematician, I told him he deserved to write it at the bottom of his proof. And so he did.
But we weren’t done yet! He returned to his original question and polished it off. The only option for the shape that projects squares must be the cube, by his theorem, and the cube doesn’t work (shine a light from the corner and you get a hexagon). So after the followup question was answered, the original question was answered!
Does a mathematician stop there? Never! A mathematician sees in every solved problem an opportunity for a more general question or extension. I thought of asking Millan about what shapes could be built from triangles, but before I could, he suggested hexagons. And can you build a shape from hexagons?
Almost… there’s just a few pesky pentagons in there. Why don’t they just use hexagons?
He’ll think about that for next time. What a pleasure, the young mathematician.