Pi is wrong

[jemidiah]

Pi day was March 14th (3/14). Some people, including me, think pi is the wrong constant to use. There's a movement to replace pi with "tau", where tau = 2*pi. This would also replace pi day with tau day on June 28th (6/28). There's an interesting list of reasons here. Since it's long, I'll pick a few to summarize below.

1. A circle has 2*pi = tau radians. It's convenient to say a quarter turn of a circle is tau/4 rather than pi/2. This also makes converting from radians to degrees cleaner--multiply by 360/tau instead of 180/pi.
2. The radius is more fundamental than the diameter (for instance, the equation of a circle of radius r is x^2 + y^2 = r^2) so we should use tau = circumference / radius instead of pi = circumference / diameter.
3. The identity e^(pi*i) + 1 = 0 can be replaced by e^(tau*i) = 1 + 0.
4. The area of a circle is pi*r^2 = 1/2 tau * r^2. Many other laws include the 1/2: kinetic energy is 1/2 m v^2, spring potential is 1/2 k x^2, distance fallen is 1/2 g t^2, ....
5. Numerous formulas (the *vast* majority of the ones I've come across since learning of tau, actually) are cleaner using tau rather than pi:
   
   • 1/1^(2n) + 1/2^(2n) + 1/3^(2n) + ... = B_n (2 pi)^(2n) / 2*(2n)! = B_n tau^(2n) / 2*(2n)!
   • Fourier transforms, which always annoyingly have that extra 2 floating about.
   • The nth complex roots of 1 are e^(2*pi*i k/n) = e^(tau*i k/n).
   • The volume of an n-dimensional sphere of radius r for n even is: pi^(n/2) / (n/2)! * r^n = (tau/2)^(n/2) / (n/2)! * r^n, or for n odd is: 2^((n+1)/2) pi^((n-1)/2) / n!! * r^n = 2*tau^((n-1)/2) / n!! * r^n. [Random fact: the 7-dimensional sphere of radius 1 has the largest numerical value of surface "area" of any dimension.]
   • The normalization factor for the standard normal distribution includes 1/Sqrt(2*pi) = 1/Sqrt(tau).
   • The BBP formula for pi is easily modified for tau, and the resulting spigot algorithm for the hexademical digits is basically unchanged.
6. A gradual switch can start by simply writing tau = 2*pi whenever it comes up and using tau instead of pi from then on. Conversion between the notations is not at all difficult.
7. The symbol for tau looks very much like the symbol for pi.
8. The symbol pi has numerous standard uses (as the circle constant; as a generic symbol for a function; as a generic symbol for a permutation) as does tau, so people already have to deal with figuring out which usage is meant from context.

I don't have any reasons for continuing to use pi, apart from the fact that everyone knows that symbol. If the world were to forget its knowledge of pi, I think tau would emerge instead as the value everyone knows. Thoughts?

[techgnome]

Your tau day is incorrect... if it truly is 2*pi... then it's 6/14 ... not 6/28... 6/28 is just another form of the same fraction of 3/14 unreduced. so it's June 14th... not the 28th.

at any rate... when I initially read it, I was thinking it was a crack pot scheme... but the changes in the formulas (save one) makes sense...
where it doesn't make sense is here: The area of a circle is pi*r^2 = 1/2 tau * r^2 .. I suspect that this this is the reason that pi is notable while tau isn't .... the pi version is boiled down to the bare basics... the tau version still requires the additional step of halving tau. Maybe it's just me, but mathematicians strike me as a lazy lot... not unlike developers who would do the same thing.

Just my $0.02...

-tg

[jemidiah]

No, 6/28 is correct. For instance, this page says "(Of course, since τ=6.28…, June 28, or 6/28, is Tau Day itself.)". The circle area formula as written is actually kind of strange amongst square laws. Many other laws include the 1/2: kinetic energy is 1/2 m v^2, spring potential is 1/2 k x^2, distance fallen is 1/2 g t^2, .... With the 1/2, you also get the nice relation dA/dr = tau*r--that is, tau is the constant of proportionality of the rate of increase of a circle's area as its radius increases. Using pi, that constant is 2*pi. Proving that statement is more elegant using tau, rather than pi.

More generally, one can consider the volume of an n-dimensional sphere (n=2 gives the area of a circle). I gave formulas for that volume above, and they're both nicer using tau.

To me, the area formula doesn't generalize well enough and doesn't fit with other square laws, so I prefer it with tau. Even if one preferred it with pi, I don't think that law alone is sufficient to cancel the similar radian<->degree conversion ugliness when using pi, since in practice I bet that conversion occurs much more often.

[dbasnett]

If you are serious then define pi in terms of tau

tau = circumference / radius and
pi= 1/2 tau


Having just finished College Algebra we were taught the equation of a circle of radius r is

(x-h)^2 + (y-k)^2 = r^2

for what it is worth.


This http://tauday.com/ was interesting. As a novice it makes sense.

[jemidiah]

Yes, I am serious. I usually write tau = 2*pi because that's the more common substitution in current practice (coming from a pi-centric world). Of course if tau catches on, people will learn it first and, if they ever learn pi (most likely as a historical idiosyncrasy) it'll be defined for them as you suggest, with pi = tau/2. Also, my circle was of course centered at (h,k) = (0,0), though your equation is more general.


I'm curious, does this whole issue seem like a practical joke on everyone else, perpetrated by mathematicians just for kicks? If so, that's hilarious.

[dbasnett]

"I'm curious, does this whole issue seem like a practical joke on everyone else, perpetrated by mathematicians just for kicks? If so, that's hilarious."

Not a practical joke, but maybe something only mathematicians care about. As a non-traditional college student the reasoning seems sound and if it(PI) comes up in a class I will certainly question why not tau. I am taking Physics (basic) this summer so who knows. It certainly seems like it would be easier to use. If you could get a calculator to have it as a constant you would have a winner I think.

IMHO your title was effective at getting my attention, but PI wasn't wrong, just not the best choice.

[jemidiah]

You'll probably get to a few square law formulas in a basic physics course. When you do, you should mention how strange it is for A=pi r^2 not to have the 1/2 out front.

I stole the title from my link. It's shameless attention grabbing on both our parts .

[zaza]

Speaking as a physicist, rather than a mathematician, I would point out that the area of a circle isn't a "square law", and you can't really go comparing it to the formulae for KE, springs etc. Compared to other formulae for area, such as the area of a square (!), it makes a lot more sense to keep pi r^2.

Personally, I would say that the primary reason for keeping a constant should be to ensure simplicity of understanding at the level at which it is taught. If learning about pi at age 10 or so, a similar age to doing advanced multiplication, division etc, I reckon keeping it straightforward and in keeping with laws of area makes a lot of sense.
If you can't cope with 2pi when you're doing Fourier analysis or complex numbers, you should probably think about pursuing a course in the arts.

[jemidiah]

I don't quite agree that the area of a circle isn't a square law. At least, the derivation of each of the ones I listed (KE, spring potential, distance fallen, and circular area) involves integrating from linear rates of change, where the linear constant of proportionality is the constant other than 1/2 in the resulting law (m, k, g, and tau = 2pi, respectively). I'm not sure how else one would define a square law.

You've reminded me of a famous quote from David Hilbert (stolen from Wikipedia) that always makes me chuckle:

"Good, he did not have enough imagination to become a mathematician".
—Hilbert's response upon hearing that one of his students had dropped out to study poetry.

Certainly people in higher level math can fend for themselves. Probably the only effects most people would feel from a switch are in the circular area formula and radian measurement changes (including conversions). I think the benefits from radian changes outweigh adding a 1/2 to the area formula, though I agree that it's convenient for that formula to be simple since it's usually taught early. It's cleaner to be able to directly convert tau * n/m into an n/m'th turn of a circle, rather than pi * n/m into a n/2m'th turn. I imagine the biggest gains for most people would be in high school trig. I always have trouble just remembering the special angles and have to rederive them (or, anymore, look them up online).

[dbasnett]

Speaking as a physicist, rather than a mathematician, I would point out that the area of a circle isn't a "square law", and you can't really go comparing it to the formulae for KE, springs etc. Compared to other formulae for area, such as the area of a square (!), it makes a lot more sense to keep pi r^2.

Personally, I would say that the primary reason for keeping a constant should be to ensure simplicity of understanding at the level at which it is taught. If learning about pi at age 10 or so, a similar age to doing advanced multiplication, division etc, I reckon keeping it straightforward and in keeping with laws of area makes a lot of sense.
If you can't cope with 2pi when you're doing Fourier analysis or complex numbers, you should probably think about pursuing a course in the arts.

If, at an early age, I had been taught that the area of a circle was

1/2 tau * r^2

instead of

pi r^2

that is what I would be comfortable with. If using tau will make things simpler in later math courses, then why not. Of all the math things you are required to remember this, as a novice, would not be hard.

[zaza]

Nah, Pi is probably one of the earliest introductions in one's life to a fundamental constant. I reckon adding a factor of 2 in there is just bodging and fiddling.

Besides, there's actually nothing to stop anybody from using tau if they really want to, although personally I have enough trouble finding sufficient Greek letters for my other variables........

[jemidiah]

To be fair, pi itself is used in physics/chemistry for a few things already (eg. pi bonds, pions, osmotic pressure), though tau is almost certainly used for more (eg. torque, proper time, tauons). Aside from usage as the circle constant, my guess is that in math they're both used about as often as each other. They're often used together to denote two different instances of the same idea, like a monomorphism from A to B.

I think people are usually introduced to pi as the ratio of the circumference to the diameter. Introducing tau instead as the ratio of c to r would probably be just fine pedagogically. After the initial definition, the area formula comes up, at which point I think the 1/2 doesn't do much harm since it's already magical and (to my knowledge) beyond proof at such an early stage.

[NickThissen]

I like many of the reasons you state for using tau = 2pi, except for the one that many formulas are cleaner using tau. For example, take a look at the wiki page for List of formulas involving pi. There's a lot of formulas there, but I don't see very many involving exactly 2pi (in which case they would look cleaner using tau). For example the physics formulas often involve 4pi or 8pi (or just pi) in which case you'd still have to write 2tau or 4tau (or 1/2 tau) and you wouldn't benefit from that at all.

While many formulas would benefit from using tau, at the same time many others would become less clear.

The only good reason to use tau instead of pi in my opinion is reason 1 (I've always found it stupid that a half turn is pi instead of pi/2).


Edit:
Also, I think the original Euler identity (e^i pi + 1 = 0) is better than e^i tau = 1 + 0, simply because "1 + 0" is redundant; the "+ 0" is just an excuse to add the 0 in there. But that does't really matter I guess

[jemidiah]

With regards to Euler's identity, I had the same initial reaction as you did--the +0 is kind of a trick. But actually, the identity is

e^ix = cos(x) + isin(x)

and with x=tau this is e^i*tau = 1+0 [= 1+0i]. It's not so cheap a trick after all--the 0 [or 0i] was there all along. Some might even prefer "e^i*tau = 1+0i"; the formula then involves not only the 5 most fundamental constants, but also the complex version of the three most fundamental arithmetic operations (addition, multiplication, and exponentiation), whereas the other formula doesn't have complex addition.

Conversely, using pi, you end up with e^i*pi = -1. Subtracting off the -1 is actually a bit artificial, and is really just done to make the result prettier. Derivation-wise, Euler's identity with tau instead of pi is more elegant. It's difficult to judge, though, since we're so used to the pi version. I actually really like the 1+0i version now that I noticed it. It's a bit strange now that the exponentiation and multiplication used complex numbers while the addition didn't.


I went through the list of formulas you linked, since I was curious how pi and tau stacked up. I'll write my impressions on each one in the following post. Feel free to skip them; the summary statistics are as follows:

strictly better with tau: 13
strictly better with pi: 8
no clear winner: 25
thrown out: 4

Of course this is very much a matter of taste. It's also not exactly fair to weight the integral of sech(x) the same as Cauchy's integral formula, for instance, but ah well. In any case, by my count tau wins. Selection bias presumably works against it as well. To be honest, a great many of those formulas, to me, are rather unimportant. The only one I really wish tau had "won" was the error function integral (equivalently, gamma(1/2)).

[jemidiah]

My impressions on whether tau or pi is better in the formulas listed on Wikipedia's list of formulae involving pi: (wrapped in code tags so it scrolls)

C = 2*pi*r = pi*d
more elegant with tau using radius; radius is preferred in general as "more fundamental"

A = pi*r^2
consistent with square laws if tau is used; derivation cleaner with tau; generalizes better with tau (surface "area" of a hypersphere formula is nicer with tau)

V = 4/3 pi*r^3
generalizes better with tau (volume of a hypersphere is nicer with tau)

A = 4*pi*r^2
generalizes better with tau, as above

integrals of sech, sqrt(1-x^2), 1/sqrt(1-x^2), sin(x)/x, and x^4(1-x)^2/(1+x^2) -- all cleaner with pi; random integrals (as opposed to general formulas) have never seemed particularly fundamental, so I don't really count these. Perhaps that's a little unfair, since the first three are quite elegant. I'll add them to pi.

integral of e^(-x^2) -- better with pi, and somewhat fundamental. However, the normal distribution formula is better with tau and is even more fundamental

Cauchy integral formula -- better with tau

infinite series -- many of these can be converted to use tau with no loss [eg. divide Chudnovsky algorithm by 2, or multiply BBP formula by 2; Ramanujan's is strictly better with tau]. In order, they are too close to call, too close, better with tau, too close, too close, and too close.

zeta(2n) -- strictly better with tau; I'm ignoring zeta(2) and zeta(4) as special cases

sum of ((-1)^n / (2n+1))^k for n from 0 to infinity for k even is just (2^k - 1)*zeta(k), and so is strictly better with tau in general. I haven't found a formula for the odd case, so that's a tossup right now

Leibniz formula for pi -- either pi/4 or tau/8 is used; neither seems better than the other

Madhava formula is strictly better with pi

Euler's product for pi/4 starting with 3/4*5/4*... also gives tau/8; no clear winner

Euler's sum starting with 1 + 1/2 + 1+3... is strictly better with pi (though that doesn't rule out a version with tau that yields a cleaner positive/negative rule)

various Machin-like formulae -- all use pi/4; neither wins

Wallis product -- gives pi/2; neither wins

Vieta's formula -- gives 2/pi; neither clearly wins; with tau, it gives 2^2 / tau, preserving 2's everywhere, and adding a nice symmetry between square roots on the left and squares on the right, division on the left and multiplication on the right

continued fractions -- the first one uses all 6's with tau, but then the 1^2, 3^2, ... pattern is borken; with pi, there's a 3 hanging around. Neither clearly wins. The second and third really give pi/4 or tau/8, so again neither clearly wins

Stirling's approximation -- tau clearly wins

Euler's identity -- too close to call, neither clearly wins

Totient function approximation -- tough to call; 3n^2 / pi^2 = 3*(n/pi)^2 vs. 12n^2 / tau^2 = 3*(2n/tau)^2; I suppose pi wins

sum of phi(k)/k approximation -- also tough to call; 6n/pi^2 vs. 24n/tau^2; I suppose pi wins

gamma function -- the same as the error function, which favored pi

strange arithmetic-geometric mean and zeta function identity -- strictly better with tau

strange modular arithmetic/limit formula -- strictly better with tau (1-pi^2/12 vs. 1-tau^2/3)

strange limit formula with 555...5 -- strictly better with pi, but presumably a very similar formula for tau also exists, and this one is listed only because of selection bias

cosmological constant -- perhaps better with tau; arguable

Heisenberg's uncertainty principle -- better with tau, particularly since it's usually written with hbar's, which is defined to be h/(2pi)

Einstein field equations -- perhaps better with tau; arguable, as with the cosmological constant

Coulomb's law -- perhaps better with tau; arguable

Magnetic permeability of free space -- uses 4pi or 2tau; neither wins

Simple pendulum period -- strictly better with tau

[krtxmrtz]

Replacing a well established constant is liable to bring about a lot of trouble, like going from a foot/mile/pound/Fahrenheit system to meter/kilometer/kilogram/Celsius. Takes time and pain to get accustomed to. Errors of -50% and +100% would be likely to creep in.

[jemidiah]

Yes, switching people over would certainly be a big thing. One very nice thing about pi and tau is that both notations can very peacefully coexist. It's simple to switch between the two--add or divide by a factor of 2. This contrasts to miles/kilometers, where the conversion is much more involved. It also contrasts with the charge of the electron, for instance. It would have been nice if the electron were called "positive" (since they're the most-used charged particle). Making that switch now would require adding or removing lots of negative signs in existing formulas, and it's very unclear which convention a given formula uses if both are common.

As an example, if I made a math library for some programming language, I would include both the constants pi and tau. I imagine almost nobody would care, and a few people would gradually use tau instead of defining their own 2*pi constant. Maybe after long enough tau would be used more than pi.