So much that's mathematical

And how fax machines remind me of calculus

Have you ever thought, so much around us is mathematical? It comes to me while crossing the road, while hitting a tennis forehand, while looking at patterns in the sand, even while noting that the planet Saturn's North Pole is surrounded by a gigantic hexagon.

Try this: Throw a ball into the air, and that's no ordinary or random trajectory it traces. It's a parabola, a pretty curve you can describe with a precise mathematical formula. Drop the ball into a placid pool and watch the ripples spread in perfect circles. Unless they meet other ripples from another dropped object - in which case, watch both sets of ripples "interfere" to produce elegant patterns. Patterns: fundamentally mathematical.

There's much more in that vein in nature ... but what about that forehand, you want to know? Well, it's like this. As I hit the ball, I aim to brush up on it, giving it topspin. If I get that right, the spin affects the path of the ball through the air, bringing it to earth sooner than would have happened with a flat shot. This is why most top players play with topspin. If I am skilled enough to brush the ball differently - I am not - I can get the ball to swerve left or right in the air, again because of the spin the ball takes. All phenomena that those who play the game know well; all described well by mathematical equations.

Then there's the time a friend asked how a fax machine works. But wait, who even remembers these devices? To me, they were actually a huge step backward in technology, but I'll leave that complaint for later. For now, think of what happens as you feed a sheet of paper into a fax machine. It's not that I know exactly what goes on inside, but it must go something like an analogy I like suggests.

Imagine drawing a grid of lines on the sheet, horizontal and vertical, closely spaced. That is, you have divided the sheet into a large number of tiny squares. According to whatever's on the sheet - whatever you're aiming to fax - any given square will be either empty or filled in. So now you call your friend Annelise and ask her to draw an identical grid on her sheet of paper. When she's ready, you "read" out your rows to her, starting at the top, one row at a time.

What do I mean by "read" the rows? Simply, you go from left to right, one square at a time, calling out either "empty" or "filled in". With each "empty", Annelise does nothing except shift focus one square to the right. But with each "filled in", she first fills in the corresponding square and then shifts focus.

Think of it: when you finish with the last square in the last row, Annelise should have a pretty good facsimile - thus the name "fax" - of whatever's on your sheet. Admittedly, this is a rather tedious way to get the information across to her, which is why I complained above. But hey, it works. And this is essentially how a fax machine must do its job. One thin strip of tiny squares at a time.

Yet again, essentially mathematical: the idea of a grid and its implicit coordinates to mark each square; each square an individual "pixel" that's either empty or filled in, off or on, 0 or 1. And consider that the smaller you can make the squares, the more accurate the image that's transmitted, the better the fax machine.

In fact - and maybe this is where I was going with this - in fact, in that last sentence is a hint of that particularly mathematical pursuit that some among us quail at: calculus. In essence, calculus is about breaking things down into tiny bits, and putting those tiny bits back together to make a whole.

What's on the page you want to fax becomes, when you draw that grid, a series of individual squares, each one a small part of the whole. Mathematicians wıll typically speak of these parts as "elements" of the whole - the whole page in this case, of course. In calculus, they also like to think of the parts as small without limit. For the smaller they get, the closer the grid hews to the original, the more accurate the reproduction at the other end. That's when the fax machine there takes the stream of incoming squares and puts them all back together.

Differential calculus has to do with that breakdown into infinitesimal parts. Integral calculus is about summing up those parts. Because if you add up all the parts of a whole, what you get, no surprise, is the whole. A fax transmission is composed of that breakdown and a subsequent summing up.

This is not to say that a fax machine performs some intricate calculus operations as it hums and whistles along, as your page rolls through. But it is a reminder, an oblique insight, into calculus. In the other direction, the essence of calculus can help us understand how the transmission works.

A little too far-fetched for you? All right then, take all this for what's at the beginning: so much is mathematical. Start with that premise, and your thoughts might also move seamlessly from hexagons on Saturn to topspin tennis, from fax machines to calculus.

And you too might start looking around, discovering patterns. Speaking of hexagons, check out France.