Lauren. 20, Colorado, Homestuck, LGBTQWTFBBQ. Married to science. Ceaselessly affectionate, compulsively creative, borderline incompetent.
I am a citizen of the Internet. I do things sometimes. Some of those things will go here. Yeah.
So, I wrote up a long-ass post about triangular numbers and how I started out thinking about them and part of why they’re so awesome. I put it behind a read more because 01. It’s long, and 02. I know some of you hate maths (philistines!).
I also decided to write it in this sort of amusing but slightly-condescending way like a Vi Hart video or a chapter from The Number Devil, because I’m not even sure why. That’s just how it came out in my head. Hopefully it’s not too annoying, and you’ll come out of it with at least some of my appreciation for both triangular numbers and the shapes for which they’re named. So, if you feel like reading it, click below to give it a go.
Alright, so we’ve all had those times now and then when we zone out and decide to do some sort of repetitive mathematical or linguistic operation in our head for boredom’s sake. Let’s say one night you’re lying awake in bed and you decide to do just that to help you drift off to slumberland because you certainly don’t want to get up or start a book or something that will just end up capturing your attention and keeping you awake even longer.
You could just do doubling, but no that’s too boring for you; you already know all the tricks! That will obviously just give you the powers of two, and that’s not very interesting at all. You’ve been hearing a lot about factorial numbers lately (that’s when you multiply every number up to and including a number), so maybe you’ll try that. 1, 2, 6, 24, 120, 720 no, that’s far too much multiplication for you. But…what if you tried the same thing, except with addition? One’s far too unique and weird, and it seems that it always ends up back at itself seemingly no matter what you do to it, so you’ll just skip that one. 1+2=3, 1+2+3=6 (hey, there’s six again!), 1+2+3+4=10, 1+2+3+4+5=15, 1+2+3+4+5+6=21.
Wait a second. There has to be a faster way to do this; there’s got to be some sort of pattern. Spotting it is easier than you’d thought: the difference between each number is increasing, but obviously only by one. So the second difference is always one, and a function with a first difference that increases and a second difference that doesn’t, well, that’s just a parabola. Could all this repetitive addition really just be a quadratic equation? Shit, you haven’t used one of those since 9th grade. You decide you’ll figure it out in the morning.
You look it up. Sure enough, the sum of every number up to and including a number is called The Triangular Number, and Triangular Numbers always equal 0.5(n2+n). But why are they called triangular numbers?
Well, this has got to be a visual thing, and you’ve always been a visual learner, so you decided to figure this out by graphing them. Fiddling about with the quadratic graph doesn’t really bring anything triangle-y to mind, but then you figure you can look at each number individually with a simple bar graph.
This is literally the easiest graph you have ever made, and it shows the triangular numbers! Since each one is represented by the adding of consecutive numbers, you can just represent each number with its own area, and then you can just add up all the areas to get the triangular number. In fact, wait…
You don’t have to do any adding at all if you just draw one line! Then you have a nice, neat triangle. And since the area of a triangle is just half its side lengths multiplied, and the side lengths will always be the same, it’s just 0.5n2.
But oh no, you’ve forgotten all the bits that got cut off by the line.
Well, maybe you can just add those back. They’re only half a block each, and there’s always going to be just as many of them as there are blocks, so that’s another 0.5n, which means the whole thing is just 0.5(n2+n). That’s the same formula! Wait…did you just prove that? And you did it with triangles! You draw a happy triangle to celebrate.
(Happy triangle courtesy of the infinitely (but countably) wonderful Vi Hart)
Now that you’re looking at that formula again, though, it doesn’t look quite right. Your triangle back there was 0.5n2. Surely, if you double-add every number up to a number it doesn’t make the number’s square. That doesn’t make any sense; why would that work? You try it, though, and it totally works! That’s crazy. You write down the first several like this.
Holy shit. It’s another triangle…and wait! While the rows are all the squares, the columns are all the triangular numbers! It’s all connected, like some crazy number conspiracy! Your mind is spinning so hard, you decide to take a break and go do something that doesn’t involve triangles. You hope there’s still something left.