Playing With Blocks

Start with a square. To keep things simple, I’ll say its area is exactly 1 square unit. Since it’s a square, that means each side is 1 unit long.

I’ll describe a procedure for growing this square into a very large rectangle. The first step is to place a second square (also with area 1) adjacent to the first, so that together they form a rectangle with width 2, height 1, and area 2.

That was easy enough, but what comes next is more complicated, because we’re going to start adding rectangles instead of squares. However, at each step the area of the new addition will always be equal to 1, but one of its sides will be long enough to match the existing rectangle. This time, we place a rectangle on top of the two squares, so it must have width 2 and height $\frac{1}{2}$ to meet the side length and area requirements.

Now we make an addition to the side, this time matching the existing height of $1 \frac{1}{2}$, so the addition must have a width of $\frac{2}{3}$.

Repeat this process of adding rectangles, alternating between top and side, ad infinitum. The total area grows and grows, but each added rectangle is longer and skinnier. If the viewing window keeps zooming out so that the big rectangle always has the same width, we see something interesting.

After 1,000 steps, it looks like this:

After 10,000 steps, it looks like this:

After 1,000,000 steps, it looks like this:

The fact is that these all look pretty much the same–a mathematician might say that the ratio of width to height approaches a limit.

What is that limit? Well, it’s approximately 1.57001 after 1,000 steps, but grows to 1.57072 at 10,000 steps and 1.570795 at 1,000,000 steps, which probably pretty close to its true value. That number may look unremarkable until we multiply it by 2 to get 3.14159.

Happy Pi Day!

(The explanation of this phenomenon requires Calculus. Is anyone interested?)

Update: Here’s an explanation.

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10 responses to “Playing With Blocks”

1. Dimitri Tishchenko

Very interesting. Yes, please post the explanation.

• rothlmar

I’m working on the explanation, but I won’t have time to finish it until tomorrow. Thanks for your patience! In the meantime, several of the comments on the reddit thread provide good pointers.

2. AW

3. Foo

Good god. Your whole blog entry demonstrates the reason why Tau is the proper constant, not Pi. And yet you conclude by hacking in a “multiply by 2” to get to Pi. Tsk, tsk tsk.

• rothlmar

I am a non-partisan–as far as I’m concerned, any rational multiple of $\pi$ or $\tau$ is just as good as any other. I multiplied by 2 for pragmatic reasons–so the result would be most recognizable.

• DM

If the blog was written in terms of Tau, he’d have multiplied by 4 at the end to get a recognizable number. I see no reason Tau/4 would be better than Pi/2.

If the blog were to demonstrate why Tau is the “proper” constant, the result would need to have been divided by 2, not multiplied by 2.

• fh

Eh, I’ll assume this is humorous but still:

tau = 2 pi.
2 * “his constant” = pi
4 * “his constant” = tau

So you’d need a hack twice the size to get to tau.

4. Damokles

I worked it out. You don’t necessarily need a lot of calculus to get the result. (Depending on how you prove Wallis’s formula.)