
The Pi Manifesto - zerovox
http://www.thepimanifesto.com/
======
ianterrell
1\. This is a good discussion to have. Those who are dismissive show, in my
opinion, a lack of intellectual curiosity. Elegance for the sake of elegance
is a worthwhile goal.

2\. From a pragmatist point of view, you're right, it doesn't matter. You
continue reading and writing PHP and using π. They both get the job done. You
don't have to participate any further.

3\. There will be 2s floating around some equations forever, whether using π
or τ. That's not the point. The point is not "cleanliness" or even teaching
efficacy. The point is elegance and that comes from _meaning_. What does the
equation _say?_ Equation cleanliness and ease of understanding are both
worthwhile side effects, but it's _meaning_ that's important.

4\. Going from π to τ would be nontrivial, and would involve confusion of its
own. That makes it not worth it to some people, and that's a valid opinion.

5\. This article suffers from more selection bias than the Tau Manifesto. The
radius is the undisputed king of the circle; it defines it. The area of a
circle is not, after all, π * (D/2)^2. But it's not about prettiness, it's
about _meaning_! Area is a property _defined by the integral_ , which has a
natural meaning and result with τ. The result may be a little equation that's
pretty or not depending on your point of view, but it's just a shortcut.

6\. The other examples in the article similarly fall apart when _meaning_ is
considered.

~~~
MostAwesomeDude
The area of the traditional unit circle is π, which has strong ties to the
definition of every trigonometric function, and the reason that radians of
common fractions of the unit circle are expressed in terms of π is related to
the integrals used to derive arc length.

Just as an exercise, try setting the area of the unit circle to 2π, and then
see how meaningful your radian measurements are. How many radians are in a
quarter arc of the circle with area 2π? :3

~~~
ianterrell
> _The area of the traditional unit circle is π_

The area of the unit circle is 3.14(etc) units squared. The result is a
number; it's how you get there that's important. You get there by integrating.
The "right" equation for area is not πr^2 or τr^2/2; it's the integral that
leads to either of those equations. The 1/2 in the τ version is meaningful
because it is an artifact of the integration. The lack of the 1/2 in the π
version shows why it's "wrong"—it's not as meaningful.

> _try setting the area of the unit circle to 2π_

This is a nonsensical statement. As is "traditional unit circle," but I let
that one slide already.

> _the reason that radians of common fractions of the unit circle are
> expressed in terms of π_

The reason they're expressed in terms of π is because a circle constant is
needed, and π was chosen. And it was chosen hastily.

Your arguments are unsound!

~~~
scythe
>And it was chosen hastily.

It was chosen here:

<http://arxiv.org/abs/math/0506415>

Elegance is found in arguments and proofs, not in results, and so any attempt
to look at equations is really missing the point. If you believe Euler's
methods might be simplified by using 2pi instead of pi, first consider a look
at the methods themselves.

~~~
ianterrell
Your reasoning is circular (get it!). If the circle constant were τ, Euler
would have found zeta(2)=τ^2/24 instead. The proof is the same.

The definition of the circle constant comes first.

~~~
scythe
>Your reasoning is circular (get it!). If the circle constant were τ, Euler
would have found zeta(2)=τ^2/24 instead. The proof is the same.

>The definition of the circle constant comes first.

You didn't read the paper, did you? The "circle constant" wasn't even defined
when it was written. He picked it out of thin air in that very paper in order
to make his arguments more clear.

~~~
ianterrell
I'll rephrase using his words. He wrote: "Namely, I have found for six times
the sum of this series to be equal to the square of the perimeter of a circle
whose diameter is 1."

The reason he uses π is due to his choice of diameter. Had he looked at the
unit circle instead with a radius of 1, he would have written: "Namely, I have
found for twenty-four times the sum of this series to be equal to the square
of the perimeter of a circle whose radius is 1."

Again, the proof is the same, but he chose to use a unit diameter rather than
radius. This is exactly equivalent to saying π=C/D instead of τ=C/r.

~~~
scythe
>Again, the proof is the same, but he chose to use a unit diameter rather than
radius. This is exactly equivalent to saying π=C/D instead of τ=C/r.

So it does not make it more clear? This contradicts your original assertion.

~~~
ianterrell
Hardly. It's acknowledging that both choices are definitions.

One provides clarity and is related directly to the unit circle, the other is
related to the circle with radius 1/2. Which is more intuitive?

~~~
scythe
>One provides clarity

How so? Neither is more intuitive. The unit circle is _itself_ a definition
you have grabbed. The notion of defining a circle by its radius comes to us
from Euclid:

>"Let the following be postulated":

>1\. "To draw a straight line from any point to any point."

>2\. "To produce [extend] a finite straight line continuously in a straight
line."

>3\. "To describe a circle with any centre and distance [radius]."

>4\. "That all right angles are equal to one another."

>5\. The parallel postulate: "That, if a straight line falling on two straight
lines make the interior angles on the same side less than two right angles,
the two straight lines, if produced indefinitely, meet on that side on which
are the angles less than the two right angles."

In fact, pi/2 itself is sitting right there in the fourth axiom, and pi is in
the fifth. 2pi is nowhere to be found.

~~~
ianterrell
It's tau/4 that's sitting in the fourth postulate, and tau is right there in
the fifth (the sum of all four angles formed by the "straight line falling" on
the side of the two straight lines' intersection).

Besides, Euclid would have been a tau advocate, as he defined circles with
their radius, which is clearly superior to the diameter.

I can't help people chose the poorer constant for so long; I can only hope to
help correct them.

~~~
scythe
>tau is right there in the fifth (the sum of all four angles formed by the
"straight line falling" on the side of the two straight lines' intersection).

...yes, but that makes the postulate meaningless! You have to look at one side
of the line in order for the postulate to have any relevance.

>Euclid would have been a tau advocate

Oh yeah? Well.. well... Ramanujan would have been a pi advocate! Ha!

>which is clearly superior to the diameter.

It is expedient in the process of mathematical argumentation. Looking at
expedience, though, we see that using a constant 2pi introduces an untoward
amount of fractions into just about every mathematical calculation -- see for
example here:

[http://en.wikipedia.org/wiki/Basel_problem#A_rigorous_proof_...](http://en.wikipedia.org/wiki/Basel_problem#A_rigorous_proof_using_Fourier_series)

Irrespective of the definition of constants, which is long since forgotten at
this point (how much of a pain is it to define a circle, starting from ZFC?),
it is kind of disappointing to see you refusing to read the proofs which you
claim to be clarifying -- most of them get uglier moving to tau, on a cursory
examination of the seminal work _Proofs from THE BOOK_. Go on, mentally
replace every instance of "2pi" with "tau" and "pi" with "tau/2" in, say, this
paper:

[http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Zeta/EZe...](http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Zeta/EZeta.pdf)

~~~
ianterrell
I can't convince you, but that doesn't make anything you've said above
correct.

~~~
scythe
Oh come on. You appealed to elegance and failed to show any. I could just as
easily define a circle as the shape which encloses the most area for a given
perimeter. If you think this is confusing, consider that it is the same as
defining it as the shape which a small water droplet forms on a piece of
glass.

The thing is that most of us knew what a circle was before we knew what a
radius was. You've probably been encountering circles since before you could
speak, and the term was certainly in your vocabulary long before you ever took
a course in geometry. Appealing to the definition of a circle as elegant is
weird when you consider the intrinsic _inelegance_ of trying to formally
define an intuitive concept. It makes _more_ sense to measure it, which could
be why Archimedes, Liu Hui, and Brahmagupta all ended up studying the same
number.

Euclid's formalization of geometry was a landmark achievement in mathematics
and possibly the most important single technique of antiquity. However, it was
superseded multiple times before set theory became the foundation of
essentially all of modern mathematics. Today, a circle is not an axiom but a
construct itself derived from the distance formula and the definition of R^2
(a collection of points all the same _distance_...).

What got me involved in this argument is the assertion that it would make life
easier for students learning mathematics. I, like most HN'ers, regard with
serious concern the deterioration of mathematics education in the United
States, but, also like most HN'ers, am not apt to solve my problems with snake
oil. As a student myself, I regularly got pi/3 confused with pi/6, as the
latter was a third of a right angle. Since angles and their respective sines
and cosines were always diagrammed in class as portions of a right angle, I
slipped up a few times between pi and pi/2.

This doesn't mean anything, though, other than a vagary of the way I used to
think at the ripe old age of ten. Students of mathematics quite often have
their own individual approaches and understandings of the concepts as
presented, and this switch of constants is not really likely to make things
any easier. This is why I kept pressuring you (unreasonably I do admit) to
demonstrate that some essential proof or argument is simplified by using tau.

It is _more_ annoying, though, when good, practical, tested, and effective
solutions to educational problems go ignored in favor of something that geeks
find interesting.

<http://jumpmath.org/>

This is _an example of_ a far more effective use of our collective time than
the definition of any fundamental constant, be it pi (perhaps tau/2), e
(perhaps 1/d, where d is the decay constant), i (perhaps -i), gamma (perhaps
log(gamma-prime), since e^gamma appears as often as gamma), etc...

~~~
ianterrell
> _Oh come on._

Really I've just tired of it today, and also had to cook dinner. :)

> _You appealed to elegance and failed to show any._

You trot out Euler and then don't find it inelegant that his proof uses not a
unit circle but a circle with a diameter of 1? Even as you also trot out
Euclid who defines circles with radii? Fixing even just that _is_ elegant.

And if you do find a few cases where π is super convenient (probably because
you only care about half the rotation of something), feel free to substitute
half tau. :)

The forest really is there, in addition to the trees.

But I really am done. Feel free to leave your last (I'm sure to be
exceptionally) clever rebuttal for posterity.

~~~
scythe
>then don't find it inelegant that his proof uses not a unit circle but a
circle with a diameter of 1?

I don't know, do you find it elegant? It's like a goddamn footnote, that's the
whole point!

For reference, you trotted out Euclid when you defined the circle. I only
pointed out whom you referenced.

>And if you do find a few cases where π is super convenient (probably because
you only care about half the rotation of something), feel free to substitute
half tau. :)

 _yawn_

It does go without saying that you won't read this post, doesn't it? You
didn't read the previous one.

------
natural219
I just want to ask all of the Pi/apathetic people-- how long did it take you
to understand radians? For me, it was a week before I was comfortable naming
any angle in radians in a reasonable amount of time (this is after a week of
drilling).

This is just my point of view, but calculating radians was a _significant_
roadblock into making quick trigonometric calculations. In fact, I'd have to
say it was the biggest roadblock. This has nothing to do with how "clean" it
looks or how I "feel" about how it's presented.

That said, I don't think it's worth it to make the switch because of all the
hassle. I'm just curious about all of the hostility towards Tauists.

tldr; it has nothing to do with any mathematical formula looking "cleaner,"
but everything to do with teaching math effectively.

~~~
amalcon
It only took me about a day, but I suspect that's because I was already used
to cycles at the time (1 cycle=2pi radians). Getting used to cycles took me
about a week.

To my mind, the problem is not with pi; the problem is with degrees. Everyone
learns about degrees first, and then must "un-learn" these artificial numbers
and begin thinking in fractions of a circle (with an extra constant thrown in
there one way or the other, in the case of radians). If we began labeling
globes, protractors, and the like in radians (or fractional cycles), this
problem would go away.

~~~
roundsquare
"the problem is with degrees"

I partly disagree. For kids, defining something using irrational numbers would
probably be very confusing. Using integers is much easier.

So, why 360? Because we talk about right angles a lot and we want to have a
third, a half, etc... and (I'm guessing) they wanted it to be a multiple of
10.

I think the point is to be able to teach geometry to kids without worrying
about them getting confused by fractions and/or irrational numbers.

~~~
amalcon
So just use cycles instead of degrees. Simple fractions or a circle. Fractions
are already taught to young children as "how many pieces a circle[1] can be
divided into." It would seem to kill two birds with one stone.

I think most kids who even study pre-Calc could handle applying a conversion
of "2pi" from there.

[1] Where circle="cake","pie","pizza",etc.

~~~
roundsquare
Good point. I forgot we learned about "slices of a pizza" that early.

By the way, the history of the degree is somewhat interesting:

<http://en.wikipedia.org/wiki/Degree_(angle)>

------
scythe
Damnit, this is just as bad as the tau manifesto. The point is that it doesn't
matter what the bloody constant is, we don't need any -more- goddamn
manifestos. Call it an arc constant or an angle constant or whatever you want.

However, there is one massive abuse of terminology that is driving me insane,
which is the use of the phrase "Quadratic forms". E = 1/2 k x^2 is not a
quadratic form. A quadratic form is a homogeneous polynomial of degree two,
and it's a topic discussed in number theory:

<http://en.wikipedia.org/wiki/Quadratic_form>

The vast majority of people will never encounter a legitimate _quadratic
form_. Call it a quadratic equation or whatever -- it's not a quadratic form.
Both the tau manifesto and the pi manifesto got this bit manifestly _wrong_.

~~~
cpa
I don't see why E = 1/2 k x^2 isn't a quadratic form. It's a quadratic form in
a vector space of dimension 1 (here, the real numbers), which matrix is just
(1/2 k). In fact here, we're dealing with a (degenerated) conic.

What's your point? Should the author have written E(x) = ...?

Besides, tau is pointless because a tau pie isn't nearly as fun as a pi pie.

~~~
scythe
At best, it's a cubic form, since the spring constant is also a variable.
Similarly 1/2 a t^2 is a cubic form. pi r^2 however is a quadratic form.

~~~
cpa
Nope, check the introduction section of your wikipedia link.

It's like saying that f(x) = ax is a function of two variables. Indeed it is,
but it's commonly accepted in mathematics to use it as a function of one
variable (x), with a parameter a.

~~~
scythe
Unless you take every spring in the Universe to have the same spring constant
(nonsense), k must also be treated as a variable. The energy relation for an
individual spring is a quadratic form, but the energy relation for all springs
is either a cubic form or a set of quadratic forms (or a bijection from the
real line to the set of unary quadratic forms, if you want to be really
technical and boring).

f(x) = a * x is an example of a function of x which only becomes a function
when a value is assigned to the variable 'a'. In other words, it is a _type_
of function.

>Indeed it is, but it's commonly accepted in mathematics to use it as a
function of one variable (x), with a parameter a.

In introductory calculus, sure; in those areas of mathematics where the term
'quadratic form' is actually _used_ , things have to be defined more
precisely.

~~~
pixcavator
Does this help: f_a (x) = a * x? Is this a function of x in your opinion?

~~~
fexl
There's my cue to drag in the lambda calculus:

    
    
      \g = (\a\x ...)
      \f = (g a)
    

Now f is is a function of x.

------
wccrawford
I disagree with change for change's sake. This whole tau thing is born of some
idealist that thinks things only make sense his/her own way.

The only thing I didn't see in the article is that the symbol visually looks
like a T, so when you see it in a formula, you have to really look at it to
know what's going on.

~~~
edw
I would add that this stupid tau thing speaks to the conspiratorialist
instinct common to many HN readers. You see, the self-evident truth of tau's
superiority has been masterfully obscured by powerful dark forces in an
attempt to protect their crude economic self interest, and if you don't agree,
you're obviously either part of the conspiracy or one of the sheeple that's
been snowed by it.

~~~
sesqu
I have had that thought, but find it incredibly hard to entertain. Do you
genuinely believe it, or is it just something that explains why others'
opinions might differ?

~~~
edw
I should have put quotes around everything following the first sentence and
attritubted it to a hypothetical (and straw-man) tinfoil hat-wearing tau
advocate.

------
kwantam
The practical problem with tau is, as was pointed out in the article, tau is
already used for other things. Shear stress, torque, time constants, you name
it.

pi is a notational freak in that it represents something so fundamental that
few dare tread upon the usage---pi truly is a globally reserved name. To a
lesser extent, the same is true of e, but even a number as important as i
doesn't enjoy this property: electrical engineers use j for sqrt(-1) because i
is current.

So let's say we all start using tau. Then I decide I'm going to do some basic
rotational mechanics, and now I have two taus, one for torque and one for 2pi.
OK, that's a no-go. How about we just redefine pi=2pi? Well... how do we know
whether someone means pi=~6.28, or pi=~3.14?

It's just no good. Tau is not a viable candidate name for the constant equal
to 2pi. Find another character in another language. How about Pei (Hebrew)?

~~~
sesqu
Pi is not globally reserved. It is commonly used in, for example, statistics
to refer to multinomial probabilities, and wikipedia tells me it's also the
name of the prime-counting function and parallax.

I should note that this was annoying to me when dealing with IRT, in which
some models have 2pi in the normalizing constants. Not to say tau isn't used
in statistics already as well.

There just aren't enough greek letters. I have wondered about what the symbol
for tau _should_ be, and I keep thinking of a circle including a radius line
that extends slightly outside the circle, but tau is much easier to write,
especially when using ascii.

------
sophacles
Here is what is wrong with the "anti-tauist" rants:

They take a bunch of people who already learned the subject and presume that
those people are experts a teaching said subject. These people always assume
that the way they learned is best, because "dammit, it was good enough for
me". They just can't see any other way.

Sadly, this ignores all of the other people, who may be capable of
understanding and properly using the subject if presented in a different way.

For pedagogical purposes, Tau is worth a shot, if it helps some people get to
the point that they realize "for the math the constant doesn't matter".

Just like anything else: try to teach broadly, and let the experts do the
adjusting, not make the novice bend to the expert's will or be damned.

------
jblow
I stopped reading early, when he's claiming pi is better for the area of a
circle, because that revealed that the author hasn't really thought about this
very much.

If you look at the equations for the volumes of spheres in n dimensions (with
2D being just one of them), tau shows a clean pattern. pi leaves you with a
mess.

------
fexl
When you ask hard-core Tauists what the area of a unit circle is, do they
actually answer "tau over two"? Or do they just say "pi"?

By the way, this whole discussion reminds me of what W.V.O. Quine called
"mathematosis".

~~~
bascule
It's τ/2. The area of a circle is τr²/2. You may be familiar with the idea of
x²/2 from calculus: it's an integral, which can be used to compute areas.

~~~
fexl
Yes, that certainly is the integral of x * dx. So voila, there's your 1/2. And
tau relates the triangle to the circle. Nice trick.

Mind you, I don't have a dog in this hunt so I'm not all up to speed on it.
All I know for sure is that tau = 2 * pi, so I won't be terribly upset if I
see either usage. I generally favor the use of notations which better reveal
an underlying concept, but I don't like it when people get all high and mighty
about things.

------
willvarfar
I've considered refactoring my pathfinding code - 2.0 is this magic constant
that is floating around all over my code.

Inertia has prevented me.

~~~
aidenn0
Off topic, but if you're using floating-point for pathfinding code, you're
going to have different results depending on your distance from the origin.

------
simcop2387
All this will ever lead to is Indiana passing a new law that Tau is exactly 6.

[https://secure.wikimedia.org/wikipedia/en/wiki/Indiana_Pi_Bi...](https://secure.wikimedia.org/wikipedia/en/wiki/Indiana_Pi_Bill)

------
stayjin
For me this debate is the monumental evidence that when people get obsessed
over something their intellectual openness shrinks to a very small dividend of
pi, sorry I meant tau :)

------
jannes
TLDR:

\- The area of a unit circle is Pi.

\- The Tau Manifesto is full of selective bias. They pinpoint formulas that
contain 2π while ignoring other formulas that do not.

------
SonicSoul
eagerly awaiting π vs τ rap wars..

~~~
adavies42
i wonder if we can get monzy and mc++ to take opposite sides....

------
afhof
And I open my eggs on the little end...

------
ThaddeusQuay
Given the nature of the debate, I think it's funny that the HN URL here ends
in "404".

------
brudgers
Is the authors replacement of "Tauists" with "Taoists" and "Tau" with "Tao" an
indication that he is repressing his subconscious belief that Tau is the way?

