I did the project below with the boys on Sunday before they went off to camp for a week. The idea wasn’t to get into heavy math, but rather just a relaxed walk through some fun shapes. We got one detail wrong in the 4th video which I was sort of kicking myself for, but then I saw a tweet from Nassim Taleb showing some of the geometry in a different problem that Alexander Bogonolny had posted and it made me realize the connection between the algebra and geometry in our problem was still fun to show:

— Alexander Bogomolny (@CutTheKnotMath) July 11, 2017

Below are the videos showing our walk through the geometry. First, though, here’s the quick introduction to the problem:

After that intro we looked at the region described by the constraint in the problem. We have to thicken up the region a little bit using the absolute value function in order to see it, so the Mathematica code looks a bit more complicated than in the problem, but that extra complexity is just to make the picture easier to see.

One cool thing about our discussion here is that my younger son thought there should be 3 fold symmetry in the shape because there was 3 fold symmetry in the equation 🙂

Now we looked at the situation in which the surface achieves the maximum value subject to the constraint in the problem. My younger son made the nice observation that the two surfaces appeared to be “blending together” at certain points. That “blending” is an important idea in Lagrange Multipliers – though, don’t worry, we aren’t going down that path today.

Next we looked at the minimum value of the surface subject to the constraint in the problem. The error I made here was accidentally reversing the two surfaces. The fixed surface – the one describing the constraint – is now on the outside rather than the inside.

Finally, I asked the kids to pick a value smaller than 45/4 for the curve so that we could see what happened. Unfortunately they picked 7 which is too small – there’s no surface! – so they chose 10 and that allowed us to see that the shrinking surface inside of the original shape. Also we can see fairly clearly (after some rotation) that the two shapes do not intersect.

Definitely a fun project showing the boys a beautiful side of a really challenging problem.