

The State of the Tau - mhartl
http://tauday.com/state-of-the-tau

======
gaze
After conferring with some fellow mathematicians, nobody had heard of the "tau
movement", and nobody cares. Tau is for people who want to feel like they're
taking stances on things that matter in math without actually putting in any
of the work to do math.

~~~
B-Con
I think tau is really just motivated by people who like things to be "clean".
You see them in all walks, wishing programming language syntax, math notation,
spelling, grammar, etc, had been chosen to be cleaner or make more sense from
the beginning.

I don't have a strong opinion, but from what I've seen I think tau makes the
most sense. These things are fun to think about, because it requires that you
think about how pi is used in mathematics and contemplate how we organize and
use the concepts related to pi. It's a good brain exercise.

But it's hardly worth getting into a fight over.

~~~
dajohnson89
But how much cleaner is using tau, beyond grade-school geometry? Euler's
equation is arguably the most beautiful relationship in mathematics: e^(i
__pi)= -1. I 'm afraid tau would dirty it up.

Edit: OK, e^(i __tau)=1, which is quite nice. :- /

~~~
epidemian
Not only that. If you think about multiplying any number z by e^(ia), where a
is a real number, as rotating z by a radians in the complex plane, then i
think the tau version of the Euler's equation conveys a more fundamental
meaning: rotating any complex number by a whole turn yields that same number,
it's the same as multiplying it by the multiplicative identity (i.e. doing
nothing).

    
    
        e^(iτ) = 1
    

And, as rotating something by one whole turn is the same as rotating it by any
number of whole turns, you can get a nice intuitive series of equivalences
that are not as pretty if you where to use pi:

    
    
        1 = e^0 = e^(iτ) = e^(i2τ) = ...

~~~
gaze
Can people really not separate out a factor of 2? There's weirdness in every
language. If I started SUDDENLY speaking English with a more uhh... orthogonal
grammar, people would maybe get what I was thinking, but also deem me a self-
righteous asshole. This is EXACTLY how mathematicians see tau. Math is no more
flexible than any other widely spoken language.

~~~
epidemian
Yes, once you have learnt the basics, factoring out a 2 is trivial. But that
doesn't make it right to make children that still haven't learnt that wonder
"what's the angle of 3/4 turns?". One and half pi? Why? It's 3/4 tau. It
should be trivial. There shouldn't need to be any conversion by a factor of 2,
especially for people that are starting with that stuff (angles and
trigonometry is where lots of kids get lost at maths).

------
thomasahle
Clearly the idea of pi and tau are both wrong, since they make the 2d circle
the reference instead of the 1d circle. Notice the formula for the volume of
an n-ball is proportional to pi^(n/2)r^n! Define tau properly, and could
become (tau r)^n!!

Also notice how many other cases outside of 2d geometry uses sqrt(pi) or
sqrt(2pi). The gamma function, normal distribution, stirling approximation
etc. are all full of it, and when you rarely need pi or tau, adding a ^2 is
much nicer than always having to draw a big ugly sqrt.

I don't mind tau especially, but it is seems the wrong battle to fight.
Sqrt(pi) also has mythical powers:
[http://www.squarerootofpi.com/](http://www.squarerootofpi.com/)

~~~
merlincorey
I'm not against being open to things like ancient aliens, but I think the fact
that squarerootofpi references that and astrology, etc puts it on completely
different footing from tau, which the post we are discussing right now
mentions being included in programming languages and scientific papers. [edit:
obviously this says nothing about square root of half tau and its uses!]

I put tau into my math-related code last year, and I have enjoyed the results
(which are much the same as before, of course).

------
codeflo
No love for pi/4? It has (IMHO) the most beautiful infinite series:

pi/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - ...

~~~
gizmo686
If you re-arrange the terms, you can find that that series equals tau/4.

~~~
Retric
No, no you can't it's tau/8.

~~~
gizmo686
Its a conditionally convergent series; you can re-arrange the terms to add up
to any real number you like (or diverge). [1].

[1][http://en.wikipedia.org/wiki/Conditional_convergence](http://en.wikipedia.org/wiki/Conditional_convergence)

~~~
Retric
You can construct a new series such that every term in the old one is there.
However, if you try and rearrange the terms you end up with a backlog that
coverages to the difference between the series. Which is why the actual series
converges in the first place.

------
schiffern
Best argument for 𝛕 – when we meet aliens, they'll laugh at us if we're still
using C/d.

"No no, we were transmitting on hydrogen * 𝛕!"

~~~
gizmo686
I don't think so. The stupid thing about pi is that it is the only place in
math that you ever talk about the diameter of the circle, everything else is
expressed in terms of the radius.

Admittedly, the radius does feel like the natural way to define a circle (all
points r away from the center), however that is not the only way to define a
circle, and we have no other intelligent species to use a reference for how
objectively natural that way is.

~~~
schiffern
We're saying the same thing. I'm using that as an example of why it _seems_
like radius is the right one. Of course we have no examples, but we do know
that an alien intelligence would be free from the historical accidents of
humanity. "Mathematics is the only truly universal language", as Sagan would
say.

Here's a similar argument: In mathematics, do we measure angles in radians or
diameterans? Why is that?

> _that is not the only way to define a circle_

Ok, let's hear 'em! What rigorous definition of a circle is more simply
expressed in terms of diameter?

The only one I can think of is, "the smallest volume shape that can contain a
length D line segment at any angle", but that seems contrived to me. I shutter
at using the solution to optimization problem as a definition, except when
there is no other choice. What's worst, it seems like a trivial derivative of
the radial definition, just with extraneous concepts tacked on.

~~~
gizmo686
The most natural, diameter based, definition I can think of is, for any given
point on the circle, the distance to the farthest point is a constant, D. You
probably need to add additional restrictions to make this rigorous (such as
requiring the circle to be form a continuous, closed loop).

You could also imagine rotating a line segment about its center. This has a
nice sense of symmetry, as we are rotating about the center, not an arbitrary
end. This also implies a generalization to off center rotations.

------
thetwiceler
Wow this is hilarious. Truly a "manifesto" to set a constant for 2pi. It
definitely has a great point, and I totally agree that it's a better constant
than pi, but it's not the worst historical accident of mathematics.

The thing is, tau is really my go-to variable when I need a 2nd constant to
compare with t. It is already used as a time constant, a dummy integration
variable that substitutes for t... The manifesto has a pre-made counter-
argument to this... But I'm not sure it's convincing.

The thing is that we like to say some variation of e^(-tau) a lot, and we also
like to say e^(2 pi i) a lot. And often we combine these (rotation and
exponential decay), and we get e^(-tau+2pi i). And this would ruin tau
notation! It would be e^(-tau'+tau i)...

~~~
jessriedel
Yea, the thing about a circle constant (whether pi or tau) is that it's so
ubiquitous you can't use the symbol for anything else. It's pretty rare to use
lower case pi for anything in physics and math, and most of the exceptions
including something like boldface or a vector hat to distinguish it.

There are no available symbols. The only option is to introduce a new one,
like ת suggested by samatman, or possibly by using a weird typeface (like
\mathfrak in LaTeX).

Incidentally, my friend has pointed out to me that sigma is a much better
choice than tau if we're going with existing symbols. Who cares about tau
sounding like a "t" for "turn", when sigma _actually looks_ like someone
trying to measure the circumference of a circle with the radius of the circle:
σ.

~~~
mhartl
σ has potential, but there's a pretty bad conflict with standard deviation.
The circle constant shows up an awful lot in statistics.

 _Who cares about tau sounding like a "t" for "turn"_

Pi comes from perimeter, phi comes from Phidias, etc. There's a long tradition
of naming constants using a linguistic root (typically Greek). To my
knowledge, there's no precedent for using a symbol based on its visual
appearance to a geometric object. (Of course, you could always break with
tradition.)

~~~
sesqu
To be completely fair, π turns up occasionally in statistics too, as a symbol
for probability.

------
sixbrx
Very interesting that Euler even sometimes used π as C/r!

------
mkl
Pi is just fine. It makes just as much sense as tau, has history on its side,
is simpler for many formulas, etc. The Pi Manifesto has good counterarguments
and heaps of examples:
[http://www.thepimanifesto.com/](http://www.thepimanifesto.com/)

~~~
mhartl
Many of the examples in _The Pi Manifesto_ are wrong. See the rebuttal I
posted last year:

[http://tauday.com/tau-manifesto#sec-
the_pi_manifesto_a_rebut...](http://tauday.com/tau-manifesto#sec-
the_pi_manifesto_a_rebuttal)

Indeed, the author of _The Pi Manifesto_ essentially rebutted his own work in
March of last year:

[http://spikedmath.com/fact-005.html](http://spikedmath.com/fact-005.html)

