I’ve written recently about several different crackpots who insist, for a variety of completely ridiculous reasons, that is wrong. But the other day, someone sent me a link to a completely serious site that makes a pretty compelling argument that really is wrong.
The catch is that they aren’t saying that the value of is wrong. They’re saying that is the wrong constant for talking about circles.
There’s a pretty good case for that. The fundamental measure that we use for a circle is the radius – and there are a lot of good reasons for that. But is based on the diameter: it’s the ratio of the circumference to the diameter. If you use the radius as the fundamental measure, and you go to define a circle constant, the natural choice isn’t . It’s . In the linked article, the author proposes naming this .
It really does make a lot of sense. Look at basic mathematical systems that use , and you find an awful lot of 2s – and the argument is that those 2s are all over the place because of the fact that we’re using the wrong damned constant.
For example, what’s the equation for a fourier transform?
Why are those 2s there? Because is wrong. We should use , which gives us the cleaner equations:
It’s no a big deal, but it does actually make those equations make more more sense. As we’ll see below, it actually helps to clarify the meaning of those equations!
So, suppose we decide to use as the fundamental circle constant. What effect does it really have? A whole lot of things actually make a lot of sense. What’s one full turn around a circle in radians? . And that’s quite beautiful and natural. A quarter circle is radians. That’s lovely.
What’s the width of a sin curve? .
Even Euler’s equation is more beautiful using :
And it even makes sense! What that says is: the complex exponential of turning around a full circle is 1. Which means: if you multiply a complex number by , that’s basically the same thing as treating it as a vector, and rotating it by an angle of : so what means is rotating it all the way around the circle: Euler’s equation becomes a very clear statement of the fact that the complex plane is symmetric with respect to that rotation.
It really does make sense. I actually think that he’s right. I think that human inertia is going to make it close to impossible to convince people to change – but I think he’s right, and the correct fundamental circle constant is .