Let’s drop pi

Has pi had its day?

Physicist Michael Hartl thinks so. He argues in The Tau Manifesto that there are many more formulas that need "2π", rather than "π", so we should consider dropping pi.

Actually, credit should go to Bob Palais of the University of Utah, who appears to have thought of it first. [See Pi is Wrong (PDF)]. He has some follow-up thoughts here.

Even Good Math Bad Math's Mark CC (who loves to debunk bad math) agrees and thinks Pi really is wrong.

Summary of the Proposal

The simplest and strongest argument for a new circle constant involving "2π" rather than "π" is that in radian measure, a quarter of a circle (a "quarter turn") is written:

But we would expect to see a 4 in the denominator of that fraction.

The proposal is to define (the Greek letter) tau in terms of circumference and radius of a circle (rather than circumference and diameter, as in the case of pi). So tau becomes:

If our circle constant was really based on "τ = 2π", then a quarter turn would be written:

This is certainly more intuitive.

Check out The Tau Manifesto for the rest of his arguments.

My take?

(1) I'm not convinced there are more formulas involving "2π" than "π".

How about the area of a circle? Using pi, we write

If we use this new constant tau, we need to write the area as:

A random search of several text books and other sources certainly showed "2π" was commonly in use in formulas, but I'm not sure we can conclude it is dominant.

(2) Tau (τ) is a terrible symbol to use.

Tau looks too much like "t" (or upper-case "T") and would suffer the same idiotic fate as "x" and "×" have suffered over the years - too similar and too often used together.

(The great math notation warrior Leibniz didn't like "×" for multiply either. And have you seen how most Chinese write "x"? Looks an awful lot like "×". Enough said!)

I've had far too many students who've struggled with formulas involving "t" and τ, since they are too alike. Why add to a student's burden?

As well, tau also looks too much like the letter "r" (see the above Area image). Since the 2 symbols will be used together a lot, this is not a good idea.

Actually, for the area formula above, I had to put a space between the tau and the r to make it readable. Without the space, this is how it looks (almost like "pi squared". Not good.

Hartl has pretty good reasons for choosing tau (towards the end of his article), but if this idea of dropping pi has any legs, let's have a better symbol.

Palais originally proposed the following symbol, derived from pi, but I agree with Hartl that it's not very good:

I propose (somewhat tongue in cheek) the following symbol for the circle constant:

Why?

1. It embodies the 2 key concepts - circumference of a circle and its radius
2. It is a common symbol in use already (it's the registered trademark sign), so it will exist in most fonts and can be typed on a computer
3. It is easily distinguished from the letters "t", "T", and "r"

It works in LaTeX, too:

Conclusion

We should constantly debate about the math symbols we use and be willing to throw out the ones that are no longer relevant. We should introduce new ones that are more meaningful and easier to read (and hopefully easier to write and type on a computer).

In this case, I'm not totally convinced about throwing out pi, but if we do, let's not use tau as the symbol!

What are your thoughts?

[Happy Pi Day, everyone!]

See the 9 Comments below.