I was walking to my bank after work the other day, in a zig-zag one-block-over-one-block-down sort of way, when it got me thinking about this weird mathematical problem it took me several years to tackle back in high school. I thought I'd write about it, for fun.

The issue I'd had trouble wrapping my mind around involved walking from Point A to Point B on a square grid, like a system of blocks, for instance. Of course, like any good human being with a pair of ears and a passable memory, I knew the old (true) adage that the shortest distance between two points is a~~shortline~~ straight line.

*Warning: Any of you who have even the most basic, instinctual understanding of how math works are probably going to find this hilarious or depressing or both.*The issue I'd had trouble wrapping my mind around involved walking from Point A to Point B on a square grid, like a system of blocks, for instance. Of course, like any good human being with a pair of ears and a passable memory, I knew the old (true) adage that the shortest distance between two points is a

From there the logic followed that if one was walking on a grid, one should zig-zag. If Point A is at coordinates (0,0) and B is at (5,5), the shortest distance would be to walk up one, over one, up one, over one, etc. A diagonal line. The jagged hypotenuse of a strange little triangle.

As it turns out, this is not true at all. If you count the blocks, both ways are exactly the same distance. Moreover, they are ALWAYS the same distance, whether you zig-zag from (0,0) to (5,5) or to (∞,∞). Likewise, it doesn't matter if those square units are blocks, miles, or microns. Well shit, right?

That was as far as I'd gotten for a while. I knew enough about calculus to realize that if the distance of the zig-zag route didn't change as the size of the units approached zero or the distance between points reached infinity, something was wrong with my reasoning, but I spent years not knowing what it was. I'd just ponder the matter while walking to a destination--taking the zig-zag route for admittedly superstitious reasons--and then get distracted by something else. Sometimes I'd decide to "work it out" on paper when I got home (whatever the fuck that would have involved) but never would.

Years later, randomly, it occurred to me that a diagonal route is only the shortest distance if you are actually moving on both axes

There's probably a moral in here somewhere, a point or something. Probably about being knowing certain facts without having any understanding of the logic beneath them, and about how that applies to life in the world at large, but mostly I just spent a few years not understanding an extremely basic mathematical truth and that's amusing, right?

As it turns out, this is not true at all. If you count the blocks, both ways are exactly the same distance. Moreover, they are ALWAYS the same distance, whether you zig-zag from (0,0) to (5,5) or to (∞,∞). Likewise, it doesn't matter if those square units are blocks, miles, or microns. Well shit, right?

That was as far as I'd gotten for a while. I knew enough about calculus to realize that if the distance of the zig-zag route didn't change as the size of the units approached zero or the distance between points reached infinity, something was wrong with my reasoning, but I spent years not knowing what it was. I'd just ponder the matter while walking to a destination--taking the zig-zag route for admittedly superstitious reasons--and then get distracted by something else. Sometimes I'd decide to "work it out" on paper when I got home (whatever the fuck that would have involved) but never would.

Years later, randomly, it occurred to me that a diagonal route is only the shortest distance if you are actually moving on both axes

__SIMULTANEOUSLY__, a detail one of you may have been screaming at the screen for a few minutes now. Fucking*duh*. If you are exclusively making either horizontal or vertical (from a grid perspective) process, you aren't actually going diagonally at all. It makes so much sense that I have trouble believing I didn't grasp it immediately.There's probably a moral in here somewhere, a point or something. Probably about being knowing certain facts without having any understanding of the logic beneath them, and about how that applies to life in the world at large, but mostly I just spent a few years not understanding an extremely basic mathematical truth and that's amusing, right?