The mailing list for my local LUG recently had a small flurry of posts on The Tau Manifesto, a proposal to replace of the constant π with τ, equal to 2π. Pro- and anti- camps quickly emerged, and much beer will likely be spilt over the issue at our next meeting.

Disregarding for the moment any conceptual elegance or expanatory power that Tau might have, I was interested to know if the move would really reduce redundancy in common mathematical expressions. Lets say (rather arbitrarily) that Tau simplifies a mathematical expression whenever π is preceded by an even constant - that means that 2π becomes τ, and 4π becomes 2τ, and so forth. I had a vague intuition that the majority of occurances of π in the wild fell into this category, which might indicate that τ is a more natural (or at least parsimonious) constant to use. Was my hunch right? This, I felt, was something I could quantify.

# Methodology

I wrote a small script to crawl all the articles linked to from the Wikipedia List of Equations page. For each page, I extracted all mathematical expressions, and checked the LaTeX source of each for occurances of the symbol π. A little bit of light parsing was then done to check if the symbol was directly preceded by an integer constant. Finally, I rendered the LaTeX source back to images to produce the equation tables below.

Of course, anyone of sound judgement will disregard what follows entirely, due to the many obvious shortcomings of this procedure and its underlying assumptions. Readers of my blog, on the other hand, may find the results interesting.

# Results

I found a total of 3173 equations, of which 133 contained the symbol π. Of these 133 equations, the distribution of constant factors preceding π looked like this:

I call this a straight win for Tau - the vast majority of expressions using π (119 of 133) are preceded by even integer constants.

# Equations

Below are all the expressions that included π, plus the detected constant factor. The headings point to the Wikipedia pages from which the equations were taken.

If nothing else, this list is a nice reminder of the mysterious ubiquity of a constant involving the diameter and circumference of a circle in all aspects of physics and higher math.

## Relativistic wave equations

constant expression
8

## Sine-Gordon equation

constant expression
2
2

## Fokker–Planck equation

constant expression
2
2
2
2

## Euler's equation

constant expression
None

## Friedmann equations

constant expression
8
4
8
8
8
4
8
8
8
8

## Vlasov equation

constant expression
4
4
4

## Screened Poisson equation

constant expression
4
2
2
4
4

constant expression
4
2

## Stokes-Einstein relation

constant expression
6
6
6

## Fisher equation

constant expression
None
None
None
None

## Einstein's field equation

constant expression
8
8
8
8
8
8
8
8
8
8
8
8
4
4
4
8
8

## Sackur-Tetrode equation

constant expression
4

## Laplace's equation

constant expression
None
4
4
4
2
4
4

## Cauchy-Riemann equations

constant expression
2

## Cubic equation

constant expression
2
2
2
4
2

## Partial differential equation

constant expression
2
2
2
2
None
2
None

## Lane-Emden equation

constant expression
4

## Heat equation

constant expression
None
None
None
None
2
4
4
4
4
4
4
4
4
4
4
2
2

## Wave equation

constant expression
4
4
4
2

## Primitive equations

constant expression
None
None

## Quintic equation

constant expression
2
2
2
2
2

## Black–Scholes equation

constant expression
2
2
2

## Fredholm integral equation

constant expression
2

## Poisson's equation

constant expression
2
4
4

## Helmholtz Equation

constant expression
4

## Van der Waals equation

constant expression
4
4
2
4
2

## Lorentz equation

constant expression
4
4
4

## Maxwell's equations

constant expression
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4