Beauty and joy found in a math proof


There may be a perception among high school age learners and some adults that math classrooms are not creative spaces. I have heard with my own ears from both groups of people. How large the groups of people are who believe this is up for debate, but not the goal of this post.

Instead, I want to draw your attention to this image. You can click to embiggen it.

visual proof that e^pi > pi^eSource: https://arxiv.org/pdf/1806.03163.pdf 

The beauty of the proof is immediate and powerful. No mathematics beyond the image is necessary to understand the idea, but it helps. The 1 page paper also has the following math to support the proof.

math for visual proof. 2 lines. not much there! Source: same as above

That is it. The author gives only those two lines. Which really, are three lines, because the author abuses the equal sign notation in line 1. I spent some time deconstructing this, and came up with the following, more easily followed, version of the proof.

detailed proof of the factclick to embiggen

Brilliant. It is just beautiful in its presentation. Such an amazingly approachable proof. But notice that red square. There is some magic going on in that square.

Could this proof be done with a function f(x)=-x? Or how about f(x)=(x=pi)^2? The visual element appears to yield the same information.

The area of the rectangle is greater than the area under the curve. But does this yield the same proof? I worked it out, and I ended up with some cool math, but nothing which approached e^pi or pi^e. Visually good, mathematically on the wrong track.

So what is going on in the red box in the hand drawn proof above? What is so different?

The difference is that in the original paper, the author recognizes that out of infinite number of functions that are decreasing from the interval { e, pi } the one function that will yield the area under the curve as log(x) + constant is f(x)=1/x. Recognizing that and realizing that one must work in base e, and from there we can create the necessary exponential.

That is very creative.

Amazingly creative.

I have been enamored with this problem ever since Taylor Belcher posted it. It is obvious at one level, and so not obvious at other levels. The graph and areas are obvious, but it is not clear how that translates to the idea that pi^e < e^pi. That leap, that jump from any decreasing function, to a specific decreasing function which creates the necessary area is the creative leap that can be difficult.

I see that with the college learners as the move from the calc series to the proofs based courses like groups, fields, and rings. The approach to the proofs classes must be different because instead of procedures, the learners must push creativity into their work. They must think of the options they have for an approach and pick and choose with intentionality.

The balancing of  the goal with the multiple paths which can approach the goal, and choosing made me think of Bob Ross and his painting show on PBS. “I need a happy tree here, so I will use this brush and …” He talked about what outcome you wanted, and decisions that had to be made along the way. He spoke of the joy of selecting color, making mistakes, fixing mistakes, and not becoming trapped in decision, but pushing forward to obtain the outcome you wanted.

That is the beauty and joy of math. The art that can be found in math.

I really need to stop procrastinating, but I found so much joy in this proof that I wanted to share it.

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