It’s fun to be whimsical, and one way to achieve that is to take a topic – however silly – and write about it, simply as an exercise to see where it leads and what angles can be illuminated on the journey. Now, various people have expressed the notion that mathematics would be a lot simpler if only pi were exactly three, and there are even stories in which the idea of pi being equal to three forms part of the plot (one modern example being Going Postal by Terry Pratchett). So it occurs to me that it might be fun to imagine a world in which pi really is equal to three, and to mentally explore some of the consequences.
A trivial way to render true the statement “pi equals three” is to invent a language in which the word for “three” is pronounced “pie“, but that is not only cheating, it also lacks subtlety. One way or another, cheating is a necessary part of the exercise, but we can at least be more subtle about it. Pi is the ratio of a circle’s diameter to its circumference, and if we want to imagine a world in which pi equals three we should at least preserve that definition.
A shape for which the perimeter really is exactly three times the “diameter” (defined here as the distance from corner to diametrically opposite corner) is of course the regular hexagon. A good way to show this is to divide the hexagon into six equilateral triangles. If we agree not to mess with the geometry of straight lines, then a universe in which pi equals three must be a universe in which the circumference of a circle is equal to the perimeter of the largest hexagon that can be inscribed within it. How can we make this be?
Straight away another trivial solution presents itself, which is to redefine “circumference” to mean the perimeter of the largest inscribable hexagon. That makes them equal all right, but is only slightly more subtle than our first attempt, and not very satisfying. It fails to respect the idea that the circumference is the length along a curve, and that the length along half a circle should be half the length around a whole one. Whatever result you get by trying to inscribe a hexagon inside half a circle (not a well-defined problem to begin with), it certainly isn’t that.
Let’s take the image of the hexagon inside the circle, and magnify the bit between one corner and the next, as shown below. If pi equals three, and we’re to insist on a more subtle means of achieving this than we’ve tried so far, then the length of the upper curve must be the equal to the length of the line segment underneath it. Our third and final attempt will involve redefining “length” in such a way that this can be true.
Here’s the idea. We’ll redefine the “length” of a path as something along the lines of the energy it takes to draw it, and we’ll put this in the context of an imaginary physics where you get bonus energy for moving in curves. This is now getting much more subtle, but you may already be able to see where it’s going to get unstuck.
We need a conversion factor for translating between “length” as we know it and “length” as redefined in our imaginary world. Given that our bonus energy is acquired by moving in curves, this conversion factor ought to be a function of curvature. Curvature, as defined by mathematicians, is the reciprocal of the radius of the circle that fits most snugly into the curve.
For a circle of radius one (or in other words a curvature of one), the conversion factor has to be 3/pi. This is because in the real world the circle has circumference two pi, whereas in the imaginary world it has circumference six, so 3/pi is the factor we need to make one equal to the other. But for a circle of radius two (curvature one half), the conversion factor is once again 3/pi. We’re forced to the conclusion that the conversion factor is 3/pi for any curve, with the exception of a perfectly straight line (curvature zero), in which case the conversion factor is plain old one.
To convince yourself of this result, study the diagram below and think about both the green circle and the blue one. Incidentally, if the appearance of pi in the conversion factor bothers you, feel free to call it pi subscript mundane. It’s the value pi has in the real world, not in the imaginary one.
A problem with our third redefinition is that it fails to respect the principle of continuity, that a very small change to a shape should make only a very small change to the length of a path along it. Consider, on one hand, a straight line segment, and on the other hand, a segment of the circumference of a really, really large circle, so large as to be indistinguishable from a straight line (e.g. a line drawn in the sand on the spherical Earth). We are forced to conclude that the curved line is 3/pi times the length of the straight line, no matter how impossibly close to being straight the curved line happens to be. This will not do.
Worse still, in such a universe we could define something I’ll call the pseudiameter, which is the length of an impossibly close to straight but not actually straight curve stretching from one side of a circle to the other. We could then ask about the ratio between the pseudiameter and the circumference of the circle, and … Oh! Look who’s back!
At this point I’m inclined to give up. Whatever you do to redefine pi so that it can be equal to three, one way or another the definition is going to fall to pieces. It isn’t possible to keep a good irrational number down.
Of course, we knew that all along.