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

Quadratic equation

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