Control-X: Polygonal numbers

Polygonal numbers belong to one of the somewhat arcane branches of mathematics that concern themselves with the physical representation of abstract concepts. Which is to say, while I imagine there is some aesthetic pleasure to be had in determining a series of polygonal numbers, it seems unlikely that there's any deeper mathematical truths to be unearthed in the process. It's math for beauty's sake, or order's -- a manifestation of the mathematician's desire to calculate all that can be calculated, for the purpose of itemizing existence to the fullest extent possible.

A polygonal number, according to Wikipedia, is any number that can be arranged as a regular polygon. Which is to say, n is a polygonal number if you can take n pebbles, or seeds, or -- in extreme mathematics -- hippos, and arrange those n physical objects into a regular polygon (i.e. a square, or a triangle, or a pentagon).
For any given polygon, there are a series of corresponding polygonal numbers that can make said regular polygon. These series always begin with one, and then proceed upwards: for example, you can make a triangle of hippos with one hippo, with three hippos, with six hippos, with ten hippos (pictured), with fifteen hippos, and so on. So the series of polygonal numbers for a triangle goes 1, 3, 6, 10, 15...

There is a formula, of course, for determining a polygonal number. If s is the number of sides in one's polygon, then the formula for the nth polygonal number in the series of polygonal numbers for an s-gon is:




Midway through the first semester of my freshman year of college, I was alarmed to discover that I could no longer learn math.

I had always been an excellent math student throughout elememtary school and high school. Though I fancied myself an artsy type -- a future writer, or maybe an editor, or possibly a jazz trumpeter, or perhaps a director, or maybe if all the cards played out right an actor -- I always scored higher on the math sections of standardized tests than the verbal sections. In seventh grade, I chose not to take advanced math, as the class met at a time I otherwise would have science with the foxy, low-cut-dress-wearing Miss Holton; this entirely rational decision would have long-lasting ramifications for my academic career. Not taking advanced math in seventh grade meant I went through middle school and high school on the standard mathematics track, with algebra in 9th grade, geometry in 10th, algebra 3/trig in 11th, and pre-calculus in 12th. I aced most of these classes, finding math a subject I could breeze through without breaking much of a sweat.

However, toward the end of senior year, I had some trouble with pre-calculus -- particularly with series theory. I remember loving the concept of factorials -- it was fun to envision eight factorial as 8!, pronounced simply by shouting the number eight -- but having difficulty with the formulas used to calculate them. I chalked these problems up to senior laziness and my antipathy toward Mr. Young, my pre-calculus teacher, who had scuttled my somewhat lame attempt, toward the end of senior year, to land on the varsity baseball team despite not having played organized baseball in four years.

But once I reached college, and calculus, I realized that I was now studying math at a level I was incapable of comprehending. At first I tried to blame the problem on my teaching assistant's impenetrable Chinese accent, but I soon realized that even when classmates explained the lessons to me in unaccented English I still didn't understand a single thing they were saying. It was as if the agreement I'd had with numbers all my life had been revoked, and suddenly those simple numbers were behaving in all sorts of odd ways. New rules and formulas sloughed from my memory, and math seemed like a difficult foreign language -- one in which I was taking some kind of advanced grammar seminar, despite not knowing the basic rules of speech.

I got an F on my first calculus test, the first F I'd ever gotten on a test in my life. In a panic, I took advantage of my friend Jamie, who had seized on our mutual safe unavailability -- she had a boyfriend in San Diego, I a girlfriend in Madison -- to nurture a freshman-year crush on me. She was a nice girl, but more critically to my fortunes, she was a math major, and we spent endless Platonic (Leibnizian?) nights with her patiently explaining calculus to me in terms more appropriate to a fourth-grader. When I needed a break, I'd play Nintendo baseball while she kept score in a scorebook she would utilize that spring as the scoreboard operator for our college's baseball team. I got an A on my second calculus test, and an F on my third. My grade -- in my mind, really, my entire academic future -- rested on my final exam.

Jamie slave-drove me through my studying, forcing me to put off the Nintendo World Series until such time as she was satisfied I could swing better than a C on the final. It worked: I got a B+, secured a C+ in the course, and never took math again. A subject that once was as easy as breathing disappeared forever, to be pulled out only occasionally when I needed to use basic cross-multiplication to convert fractions to percentages. Chief Toasohcah would never again be a part of my life.

Here's what I don't quite understand about polygonal numbers. The rules, as detailed in Wikipedia's definition, are such that the polygonal number series for a hexagon goes like this:



Why, in order for a number to qualify as a polygonal number, doesn't the number in question need to fill up the polygon? Why isn't the second hexagonal number seven, instead of six? Why isn't the third nineteen, instead of fifteen? Doesn't it seem something of a violation of the orderly nature of mathematics? Doesn't it all seem kind of arbitrary?

What happens to all those extra hippos?

Jamie moved back to San Diego at the end of freshman year, shortly after she slid a journal detailing the mortifyingly sexual fantasies she'd been having about me under my dorm room door. She remains the only girl who's ever been obsessed with me, even if she was obsessed not with me, really, but with the idea of a romantic escape from her doomed relationship. Soon my high school relationship started following the roller-coaster pattern of my relationship with calculus, but this time it all ended in disaster. Jamie got married, and then divorced. We fell out of touch.

I got married too, and seven years later had a daughter. She, my wife, and I make a nice little triangle together.