#visual proof

What can be learned from the fact that √2+√3 ~ π ?

The aesthetic

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I’m sure the thing I’m about to show y’all has a different (possibly cooler) name, but I’m calling it “distillation”. Because it makes sense to me on some level. You’ll see why in a minute.

Let’s begin with a picture:

This is a representation of the relatively well-known fact that the difference between two consecutive squares is always odd. More specifically, the difference between any two square numbers n^2 and (n+1)^2 is 2n+1.

If you’re like me, you’re not at all satisfied with this. You’re thinking something like: why the hell should I trust you and your pretty pictures? Sure, 4^2 - 3^2 is 7 and 3*2+1=7. And sure, that’s true of all the other numbers you tried. But why should you believe that this will go on forever? (If you know the proof of this already, feel free to skip to the next section) This is actually kind of easy to prove visually. If you look at the picture above, you can see that creating another square involves adding the side length of your original square to both sides, and then putting in a corner piece:

As you can see, there are two pink dots, for the two black dots that make the side length of the square. And there is one new dot, the green corner dot. If you extend the diagram to the region of the ellipses, you can see that, as long as the shape is square, a side length n will yield n pink dots on either side--plus, no matter how long the sides are, exactly one green dot. And because 2n (the total number of dots in 2 rows of n pink dots) is even, 2n + 1 must be odd.

A couple of weeks ago, I was going through some much cooler math (you’ll see it in the next section) with a friend of mine, and we ran into the thing where the difference of consecutive squares is 2n + 1. For some reason I’d either never thought of or totally forgotten the explanation for the visual proof I just gave, and so when my friend tried to show me the visual proof I asked if there was anything more rigorous. He then showed me an algebraic proof that’s really pretty much the same thing as the visual one:

(Pretty obvious, right? I can’t believe it never occurred to me... (as an interesting side note, “occurred” comes directly from the latin occurrerre, means “to run to”--so the idea never ran or came to me--but now my tangents are growing tangents, and I must be getting on to the point...))

This next bit is the fun part. If you read through the last section, you know there’s going to be some “much cooler” stuff here. We’ll jump right in:

So, this is interesting. In the second row we’ve got our old friend the difference-of-squares (speaking of which, you can also prove the 2n + 1 thing using what your algebra teacher calls the “difference of squares,” x^2 + y^2 = (x+y)(x-y). Give it a shot.)

Then in the next row we have the cubes: 1, 8, 27, and so on. If you haven’t figured it out already, what we’re doing here is taking the difference between adjacent numbers and marking it down in the next sub-row, until we get to a point where all the numbers are the same and the difference in zero. (These last rows are colored in turqoise-y blue.) Anyway, you’ll see it takes one more iteration--”distillation”, as I’ll call it--to get to blue, with the cubes.

The fourth row is the powers of four, unsurprisingly. This time it takes us four distillations to get to blue/zero.

Notice any other patterns? I’ll let you think about it for a minute...

Go on...

It may or may not have something to do with factorials...

Ok, here you are. You’ve gotten it, or given you’ve up. You’ll notice that the number in the first blue row, 1, is equal to 1! (that’s pronounced 1-factorial). The blue number in the second row, 2, is equal to 2!. The blue in the third row is 6 = 3! (and now I can explain what a factorial is--n! = n*(n-1)*(n-2)*...*1). The blue number in the fourth row is 24 = 4!.

A very neat visual proof for the formula of the first n square numbers.