
A new section of The Tau Manifesto: Getting to the bottom of pi - mhartl
http://tauday.com/tau-manifesto?new
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tungstentim
"pi is a lie" and "pi is wrong" are incredibly disingenuous slogans. I would
be really excited about this site if it weren't for these sensationalist
remarks at the top. It's sad, because their main content is all about
mathematical reasoning, which overall they do a fine job presenting.

pi is not a lie. It's not part of some conspiracy theory among math elites to
mislead the masses. It's not about math bureaucrats refusing to admit they
were wrong. Everything they told you about pi is true (or it isn't, but in
that case it's not true of tau either). pi and tau are really just two
different notations for the same concept. Only perhaps one is more elegant and
convenient than the other, making it more fundamental in some mystical way
(and perhaps preferable if you ever need to convince visiting aliens that you
are intelligent).

~~~
mhartl
_"pi is a lie" and "pi is wrong" are incredibly disingenuous slogans._

For the former, you just need to know your meme:

[http://www.urbandictionary.com/define.php?term=the+cake+is+a...](http://www.urbandictionary.com/define.php?term=the+cake+is+a+lie)

For the latter, you need to know that it comes from Bob Palais' original
article.

For both, you might need to lighten up.

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Cushman
The last section nails it in a nutshell:

 _...imagine we lived in a world where we used the letter h to represent “one
half” and had no separate notation for 2h. We would then observe that h is
ubiquitous in mathematics. In fact, 2h is the multiplicative identity, so how
can one doubt the importance of h? But this is crazy: 2h is the fundamental
number, not h. Let us therefore introduce a separate symbol for 2h; call it 1.
We then see that h=1/2, and there is no longer any reason to use h at all._

Color me converted as well.

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WiseWeasel
It seems like these guys are fighting an uphill battle, not that I'd want to
discourage it. It's simply much easier to visualize the path traced when you
relate the radius to the circumference of a circle rather than its diameter,
since there's only one point to focus on at a time as you rotate around the
axis. Pi is insidious as it's only much later that this foundation begins to
appear problematic.

[My analogy was not very good. The circumference is the only dimension of a
circle stated in terms of diameter, not radius, and so substituting tau for pi
is simply relating the radius of a circle to its circumference instead of its
diameter. Circumference = tau * radius. This has the benefit of bringing the
circumference formula in line with all the other circle equations.

When we look at the area of a circle, tau is not very intuitive, however. The
formula for the area of a circle uses the square of the radius, so there's not
an intuitive way to factor two into the equation without just making extra
mental work for yourself.

For volume, the effect is somewhat neutral. Volume = 4/3 pi * radius^3 = 2/3
tau * r^3. 2/3 vs. 4/3 doesn't seem to add much insight into the equation.

Circumference = tau * r

Area = 1/2 tau * r^2

Volume = 2/3 tau * r^3

4th dimension circle = 3/4 tau * r^4? : P

Maybe there's something to this...]

~~~
blahedo
No, the tau manifesto agrees: radius is the more fundamental concept. That's
why tau is better! Pi is an expression relating the diameter to the
circumference, which as you point out is a little counterintuitive.

~~~
Retric
The problem with tau for introductory math is area = pi * r^2 = (tau/2) * r^2.
Given the choice between having equations with 2x or other with 1/2 y most
people feel more comfortable with 2x.

EX: e^(i * tau / 2) = -1 wait what?

~~~
bascule
If you read the Tau Manifesto you'll see why tau * r^2 / 2 is actually AWESOME
(he actually refers to it as pi's coup de grace)

Just think of it like an integral (that's what it is, after all!), raise the
power and divide by the power. Suddenly the origin of this formula is no
longer obscured.

Regarding Euler's identity: e^(i * tau) = 1

~~~
Retric
Telling a 3rd grader to think of it like an integral seems to be putting the
cart WAAAAAAAY before the horse.

Edit: Also that's is hardly a more fundamental equation because you could also
say e^(1024 pi * i) = e^(4096 * tau * i) = 1.

~~~
bascule
It won't amaze third graders, but it would make a lot more sense the first
time you learn integral calculus

~~~
Retric
Meh, so did 2 * pi * r > 2 * pi * (r ^ 2) / 2 = pi * r ^ 2.

The great thing about math is it's it works out either way. So, it really
comes down to which is easier to deal with and students seem to have an easier
time remembering and understanding 2 _pi_ r and pi*r^2, but in the end it's
tau day so that's what people want to argue about.

~~~
bascule
"students seem to have an easier time remembering and understanding 2 pi r and
pi r^2"

Are you speaking from empirical experience as a math teacher?

What makes 2 pi r easier to understand than tau r?

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cygx
I live with 60 minutes per hour, 24 hours per day, 7 days per week, 360° in a
full circle, a speed of light of 299,792,458 m/s, to name only a few cases of
historical 'accidents'.

2π is arguably more fundamental than π, but given the things above, I don't
care much.

If you want to tackle something worthwhile, update theoretical physics
lectures to 'modern' notation (where modern means 1960s).

~~~
mhartl
_360° in a full circle_

That's one of the things tau can help solve. I suspect the main reason that
people still use degrees so much is because using pi undermines the beauty and
intuitiveness of radians.

~~~
cygx
No, the main reason why degrees are still in use is because most people have
an easier time with 51° instead of 1/7 τ.

~~~
waterhouse
51° is not 1/7 τ. 1/7 τ is (51 3/7)°, or 51.428571...°. If you round that to
51°, then presumably you're doing something where inexact numbers are fine, in
which case you might then use "0.89 (radians)". My intuition is better for
degrees here, but I think that would change if I spent ten minutes doing
calculations involving real radian measures.

~~~
cygx
51° is the latitude of the city I'm living in, and I chose 1/7 τ because it
comes pretty close.

I could have used 0.14 τ to make my point as well: If you want to avoid
fractions in common cases, you'll need to multiply with an arbitrary number
like, say 100 (which has obvious benefits in a base-10 system), but arguments
could be made for something like 400 (in which case we end up with gon) or,
say 360, which has the benefit that it's the established standard (even though
it's not the most obvious choice).

Actual radians (ie not expressed as fractions of 2π) are pretty user-
unfriendly in a base-10 system.

~~~
blahedo
> Actual radians (ie not expressed as fractions of 2π)

I think this is the source of your problem: you view an expression like "τ/8"
or "π/4" as less an "actual" radian measure than the inexact ".785". That's a
function of how a lot of us were taught math, I think, and I'm quite sure that
it's only made worse by the fact that the use of π camouflages the fact that
this is related to fractions of a circle. But your premise is faulty. Of
_course_ radians expressed in decimal are user-unfriendly, but radians
expressed in decimal are no less "actual" than radians expressed as fractions
of τ _or_ π.

~~~
cygx
A physical quantity is given by numerical value and unit of measurement.
Giving angles as fractions of τ changes the unit of measurement from radians
to turn.

There are applications for which neither radians nor turn are a good fit.
Similarly, oftentimes the 'messy' SI units are a far better fit than any 'more
fundamental' system of natural units.

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sallowswine
"For example, consider integrals over all space in polar coordinates:

\\[ \int_0^{2\pi}\int_0^\infty f(r, \theta) r dr d\theta \\]

The upper limit of the $\theta$ integration is always 2$\pi$."

This statement is false. If $r = r(\theta)$, then it can be the case (e.g.,
for a "looped" limacon $r = \cos(\theta)$) that $\theta$ will not range from
$[0,2\pi]$. The upper limit is not necessarily "always" $2\pi$.

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simcop2387
This should probably link to the actual section of the page that was added:
[http://tauday.com/tau-
manifesto?new#sec:getting_to_the_botto...](http://tauday.com/tau-
manifesto?new#sec:getting_to_the_bottom_of_pi)

~~~
mhartl
The reason I didn't link to the exact section is because the math needs some
time to render. If you go right to the section, on many browsers you won't end
up where you wanted to be. If instead people hit the top of the page, the math
renders in the background while they're looking for the new section link.

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blahedo
Huh. Section 5.1 (one of the new ones) is presented as this new epiphany about
hyperspheres and tau, but I made a post back in 2010 about this (
<http://www.blahedo.org/blog/archives/001083.html> ), emailed Hartl at the
time, and he said he'd already thought of that. Still, cool to see that my
observation was worth (eventually) including in the manifesto itself. (He did
elaborate on the idea considerably, of course.) ;)

~~~
mhartl
When I said I'd already thought of it, I was referring to the general
observation that _n_ -sphere volumes support tau, not pi. You can find my
original comments, dating from July 2010, at
[http://forums.xkcd.com/viewtopic.php?f=17&t=61958](http://forums.xkcd.com/viewtopic.php?f=17&t=61958).

The "epiphany" refers to the specific realization that there are three
families of constants, and in particular that pi is not a member of the family
of volume constants but rather is part of a useless family of its own.

In any case, you certainly deserve mention, so I've added you to the
acknowledgments. Thanks!

~~~
blahedo
Nifty! I wasn't trolling for recognition so much as being puzzled why this was
billed as a new epiphany---that makes sense though. I _love_ the idea of
recasting the presentation from "list of complex formulae for area and volume"
to "sequence of constants expressing ratios".

~~~
mhartl
I know you weren't trolling, but I'm a strong believer in giving credit where
credit is due. Thanks again for your comments!

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leot
I know that as a kid I was all too sensitive to things in math "making sense"
in a deep way. Any semi-arbitrary conventions that made my understanding less
elegant, or which obscured the beauty of a concept, were fiercely protested.

So, up with tau!

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rsanchez1
It's just semantics. Pi and tau are the same thing. You don't see physicists
going around saying h is wrong and h-bar is what they need to use.

~~~
mhartl
_You don't see physicists going around saying h is wrong and h-bar is what
they need to use._

Sure they do—or at least, they did. Physicists realized that h is "wrong",
i.e., confusing and unnatural, because it's off by a factor of, um, 2 pi. They
introduced h-bar precisely to rectify the problem. And the substitution
worked: if you open up a standard book on quantum mechanics, the ratio of uses
of h to h-bar is just about epsilon. (You'll also see lots of (2 pi)s.)

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wolfpackk
Happy Tau Day

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mcnemesis
Am in and converted!

