A long time ago, when I was a psychiatric resident, one of the patients on our ward entertained all the residents by giving a lecture on certain advances in mathematics that he had invented. We sat attentively while he drew on a blackboard with chalk and emphasized the salient points by hitting it with a long pointer. First he drew a large circle and then an infinity sign on the very top of the circle, where he said it had “potential energy.” Then, with his fingers, he made the infinity sign go flippety-floppety along the outside of the circle until it struck bottom, dissipating its energy. I immediately shot up my hand.
“What if you nested a smaller infinity inside the larger infinity?” I asked. “Would it have still more potential energy?”
But the patient had never heard that some infinities were larger than others. After I explained, he looked at me crookedly and said, “That’s crazy talk.” I think it was, but, still, his creative approach to mathematics made me think more carefully about geometry, in particular about the special qualities of a circle. I was used to thinking that mathematics had been handed down to us by an arbitrary deity of some sort, and there was no talking back; but maybe that wasn’t so.
If you look at the circle without any preconceived notions, you will discover that the circumference is almost exactly three times as long as the diameter. When the ancients first considered the circle, they thought the circumference might be just a teeny bit larger, but within the margin of error. They experimented a lot with long pieces of string, and each time—no matter how big the circle—the circumference was almost exactly three times as long as the diameter. Having even in those days a sense for the beauty and simplicity of mathematics, they concluded that any deviation from three was a mistake and possibly immoral.
There are various henges strewn about Europe, the most well-known being Stonehenge in England. These are Neolithic structures of wood or stone often built in vaguely circular shapes. No doubt the builders appreciated the purity of Pi being exactly three (although, of course, it wasn’t called Pi in those primitive days.) Because the henges were so big, they were no longer plotted with pieces of string but with very long pieces of rope, the one measuring circumference being three times the length of a shorter (but still very long) rope which measured the diameter. Sometimes the henges were oval just a little, or squashed in a little, in order to make the mathematics work out just right. Because these were such creative works of art and worship, I consider them examples of what I call “creationist math.”
A much more recent example of creationist math occurred in 1897 when the Indiana state legislature came within one vote of making Pi equal to 3.2, a half-way measure at best. And in the wrong direction. There is no record of their deliberations, but it is thought they were responding to the obvious disadvantages of Pi being defined the way it is. I mention some of these below:
- The special appeal of mathematics is its exactitude. Pi, on the other hand, is just an approximation. No matter how much time you spend calculating Pi—to trillions of digits—it is never exactly right, making calculations clumsy and dubious.
- Pi is unpredictable. No matter what digit you are looking at, you can’t guess what the next digit will be. Sometimes you can come across a run of seven or eight sixes in a row—and you begin to think the calculation has settled down. All sixes from then on. But then the next digit turns out to be a three or a seven, or whatever. By the way, it is often asked why there are billions of sixes and sevens, but there is never a zero. The reason is that when Pi was first invented, no one had ever seen a zero. Even Archimedes, who was a great geometer, never used a zero. Because of that tradition, modern calculations never include zeros. (The zero was invented by some Arabs in the third century B.C.)
- Pi is too long. Even people who are good at memorizing Pi can never get past the first few hundred digits. So you need a table—a very long table that goes on for thousands of pages.
- Finally, perhaps most telling, mathematics is supposed to be aesthetically pleasing; and a long, bumpy mathematical structure like Pi is not pleasing.
The Indiana legislature may have thought also that lengthening Pi somewhat would mean some savings in building materials when they built domed structures; but, if so, they were misinformed.
I think there can be no question but that Pi would be improved if it were shortened exactly to three. All those digits would disappear, but, speaking strictly numerically, not much would be lost—less than one per cent. Pi shows up in a lot of other mathematical places so the advantages would be compounded.
There is, then, the critical question—can it be done? Some mathematicians say “no”; and theirs is currently the majority opinion. And there are those among the public, as there always are, who accept slavishly everything that mathematicians and scientists say about, for instance, evolution or global warming; but there others who are more open-minded. These individuals have the true scientific attitude: skepticism of authority. Of course, it can be done! Just as sure as our ancestors rode about on dinosaurs. Here is just one such method:
Take two rods of equal length and lay them one upon the other at right angles, making a cross. Then draw a circle around them with a rubbery, but firm, but very thin, bit of hose. Really thin. Attach the circle to the ends of the cross. Then put an object right under the center of the cross and enlarge it until the rim of the circle is pulled inwards slightly. Voila! Now the circumference of the circle has shrunk to a proper size! So much for the circle. There is a similar proof for the sphere, but there is too little space in the margin here to write it down.
There is not written down here, perhaps, sufficient evidence to convince the committed formal mathematician. But surely there is enough reason to allow creationist mathematics (sometimes called revisionist mathematics) to be taught in the classroom alongside formal mathematics so students can get a rounded picture of the circle. So to speak. (c) Fredric Neuman