Sunday, September 4, 2011

PI is wrong

At the core of any Lispers understanding is the fact that mathematical notation sucks. We recognise just how painful it is to have to parse expressions using a horrid order of operations table like PEMDAS. We reject the use of such syntax in program source code, which is written the luxury of a keyboard, but we are tolerant of such vices when writing with pencil and paper.

As we have a skeptical outlook of mathematical notation from the get-go, you will be hard-pressed to find one of us that isn't also receptive of new mathematical symbols like Tau:

$$\tau = 2\pi = 8 \sum _{n=0}^{\infty} \frac{(-1)^n}{2n+1}$$

I am one of those people who has constantly failed to make sense of the mind boggling concept of pi-radians, so I when I read Micheal Hartl's "Tau Manifesto" I was a quick convert.

The reason that Tau makes more sense then Pi in most cases is that Tau is based upon the distance from the center to any point in the circle, or the radius, rather then the diameter which is relatively rarely used.

$$\tau = {C \over r}$$
$$\pi = {C \over D}$$

Tau makes trigonometry much easier, because tau is the periodicity of the sin and cosine functions, and Euler's identity becomes $e^{\tau i} = 1$ which is far more elegant. Even the circular area makes more sense with tau, as it becomes tau times the antiderivative of r:

$$A = \frac{\tau r^2}{2}$$

Regardless of these considerations mathematicians still overuse pi, which is another example of a flaw in our notation systems. However, I would still say that most mathematical theorems are factually correct, despite our flawed attempts at expressing them.

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