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The Tau Manifesto

Posted: Wed Jun 30, 2010 5:55 pm UTC
by SSteve
This makes so much sense to me.

http://tauday.com (Link unparsed due to forum rules. - gmalivuk)

I always thought it was crazy that one cycle of sin or cos was 2π instead of π.

All we need is a "Let τ = 2π" t-shirt and the movement will be unstoppable.

Re: The Tau Manifesto

Posted: Wed Jun 30, 2010 6:46 pm UTC
by Velifer
Pi is the ratio of the circumference of a circle to its diameter. 6 would be wrong. Also, all this stuff.

Of course, if it's that important, you can always just start your paper with things like "let tau = 2pi, let current direction = electron drift(flow) direction, let pluto=planet, let black = white (because otherwise you're a RACIST!) and then have your party. Alone. And poorly understood.

Re: The Tau Manifesto

Posted: Wed Jun 30, 2010 7:20 pm UTC
by LLCoolDave
I think you're missing the point there, Velifer, sort of. If you consider the proposition of the tau manifesto, the fundamental circle constant shouldn't be the ratio of the circumference to the diameter but rather the ratio of the circumference to the radius (which I find to be more natural as well).

In a certain sense I agree that tau feels like the more sensible choice for the fundamental constant instead of pi, but I really am not too bothered with a factor of 2 and changing this would certainly require a lot of work.

Re: The Tau Manifesto

Posted: Wed Jun 30, 2010 7:24 pm UTC
by mike-l
Velifer wrote:Pi is the ratio of the circumference of a circle to its diameter. 6 would be wrong. Also, all this stuff.

Of course, if it's that important, you can always just start your paper with things like "let tau = 2pi, let current direction = electron drift(flow) direction, let pluto=planet, let black = white (because otherwise you're a RACIST!) and then have your party. Alone. And poorly understood.


Palais' article is linked on the page ssteve linked. And tau is the ratio of the circumference to the radius. Where else do we ever see diamater other than in the definition of pi?

One place I know of where pi is nicer than 2pi, is the volume of the n-sphere: [math]C_n=\frac{\pi^{n/2}}{\Gamma(\frac{n}{2} + 1)}[/math]

I do think that 2pi comes up much more often though.

Re: The Tau Manifesto

Posted: Wed Jun 30, 2010 7:57 pm UTC
by Vorticity
It's true, they have a point...

...but it's too late.

Re: The Tau Manifesto

Posted: Wed Jun 30, 2010 8:01 pm UTC
by ++$_
They seem about even to me. On the side of 2[imath]\pi[/imath] you have Stirling's formula and Euler's formula. On the side of [imath]\pi[/imath] you have the Gaussian integral and the area of the circle (which are really the same thing, but there you go). (What he says about quadratic functions is well taken, but let's face it -- the 1/2 is a pain in the ass.)

A more important consideration may be that it is easier to talk about [imath]\pi[/imath] and [imath]2\pi[/imath] than about [imath]\tau/2[/imath] and [imath]\tau[/imath]. In fact, if it were up to me, I'd want the circle constant to be [imath]\pi/2[/imath] so that both the sine and cosine functions have roots at multiples of the constant.

In order to make this guy happy, we should tell him that pi is defined as the area of a circle of radius 1, rather than the ratio of circumference to diameter. That should shut him up.

Re: The Tau Manifesto

Posted: Wed Jun 30, 2010 8:21 pm UTC
by mike-l
++$_ wrote:In order to make this guy happy, we should tell him that pi is defined as the area of a circle of radius 1, rather than the ratio of circumference to diameter. That should shut him up.


1 what?

Re: The Tau Manifesto

Posted: Wed Jun 30, 2010 8:25 pm UTC
by ++$_
mike-l wrote:
++$_ wrote:In order to make this guy happy, we should tell him that pi is defined as the area of a circle of radius 1, rather than the ratio of circumference to diameter. That should shut him up.


1 what?
1 greeble. And the area is measured in square greebles.

Maybe I'm missing the point of your objection.

Re: The Tau Manifesto

Posted: Wed Jun 30, 2010 11:41 pm UTC
by antonfire
++$_ wrote:In fact, if it were up to me, I'd want the circle constant to be [imath]\pi/2[/imath] so that both the sine and cosine functions have roots at multiples of the constant.
If you hate division so much, why not make it be, say, pi/180?

Re: The Tau Manifesto

Posted: Thu Jul 01, 2010 3:18 am UTC
by ++$_
antonfire wrote:
++$_ wrote:In fact, if it were up to me, I'd want the circle constant to be [imath]\pi/2[/imath] so that both the sine and cosine functions have roots at multiples of the constant.
If you hate division so much, why not make it be, say, pi/180?
Because you rarely use subdivisions that small, so you would end up needlessly multiplying by large numbers. If you made the constant pi/2, then the multiplier would typically range between 4 and 1/6 for most applications. That seems about optimal in terms of the simplicity of the representation of the multiplier.

Re: The Tau Manifesto

Posted: Thu Jul 01, 2010 3:27 am UTC
by letterX
++$_ wrote:
antonfire wrote:
++$_ wrote:In fact, if it were up to me, I'd want the circle constant to be [imath]\pi/2[/imath] so that both the sine and cosine functions have roots at multiples of the constant.
If you hate division so much, why not make it be, say, pi/180?
Because you rarely use subdivisions that small, so you would end up needlessly multiplying by large numbers. If you made the constant pi/2, then the multiplier would typically range between 4 and 1/6 for most applications. That seems about optimal in terms of the simplicity of the representation of the multiplier.


One wonders if (s)he was perhaps referring to a little-used unit called 'the degree', and if perhaps you have been had.

Re: The Tau Manifesto

Posted: Thu Jul 01, 2010 3:45 am UTC
by ++$_
letterX wrote:One wonders if (s)he was perhaps referring to a little-used unit called 'the degree', and if perhaps you have been had.
What's a "degree"?

EDIT: I looked it up on Wikipedia. Apparently it's some ancient Babylonian unit for angles :/

Re: The Tau Manifesto

Posted: Thu Jul 01, 2010 5:27 am UTC
by antonfire
++$_ wrote:Because you rarely use subdivisions that small, so you would end up needlessly multiplying by large numbers. If you made the constant pi/2, then the multiplier would typically range between 4 and 1/6 for most applications. That seems about optimal in terms of the simplicity of the representation of the multiplier.
This depends on how you define simplicity. To the Babylonians, it apparently seemed preferable to be able to represent angles pretty closely with whole numbers. You seem to be more comfortable than they with allowing division when you need it. I, it seems, am yet more comfortable with it than you.

Even if the whole point was to make written expressions as short and simple on average as possible, what convention is to be adopted depends on what it is you're doing. For an engineer, perhaps naming pi/2 would be convenient. For an astronomer it is perhaps nicer to use a smaller measure, by say, naming pi/180, pi/10800, and pi/648000. When you do mathematics (and I am not referring here to mathematics homework), I'd say giving a name to 2pi really does save you the most work.


But anyway, the point isn't to just pick a convention that saves you ink. It's desirable to pick one which makes things conceptually clear.

The standard primitive 6th root of unity is exp(pi i / 3). Why 3? Well, because a sixth is a third of half a turn.
The standard primitive 6th root of unity is exp(2pi i / 6). Better.

The area of a circle with radius r is pi r2. Okay.
The area of a circle with radius r is 2pi/2 r2. Why divided by 2? Because 1/2 x2, not just x2, is an antiderivative of x.

If you adopt the convention of naming and using pi, or pi/2, questions about unexpected terms tend to lead you back to this convention. If you adopt the convention of naming and using 2pi, questions about unexpected terms tend to point you in other more interesting directions.

Re: The Tau Manifesto

Posted: Thu Jul 01, 2010 12:57 pm UTC
by Velifer
It's a question of standards and legacy. If we're opening up the definition of pi, why not open up the definition of 6? It's not the inverse of 9, and that's problematic. You're getting way ahead of yourselves with the whole pi discussion.

It has little to do with sensibility. It has quite a bit to do with cultural inertia. Some Babylonians decide how we count time, and everyone shares a common understanding of that, and sticks to it. Not because it's best, but because it's commonly adopted. Some people pipe up with antisexagesimal rants now and then, but standards like these are entrenched.

You want to use tau, go for it. You'll have to explain yourself over and over and over and over, but if enough people find it useful, it will slowly creep in as a standard, perhaps even the standard. Until then, get off my lawn.

Re: The Tau Manifesto

Posted: Thu Jul 01, 2010 3:48 pm UTC
by phr34k
There already is a symbol for 2π. Ready? It's 2π.

If we use τ= 2π, then we'd also have to keep talking about τ/2 whenever π would pop up (zeros of sin, sum of angles in a triangle, half-plane polar parametrizations, area of a unit circle,etc.) And fractions are always more cumbersome (typographically) then products.

So I remain unconvinced by any of the Manifesto's points: none of the simplifications offered are of any significance for 'professional' mathematics.

Re: The Tau Manifesto

Posted: Thu Jul 01, 2010 6:00 pm UTC
by BlackSails
Also, e^i*T/2=-1 is much less nice looking.

Re: The Tau Manifesto

Posted: Thu Jul 01, 2010 6:20 pm UTC
by Ended
phr34k wrote:So I remain unconvinced by any of the Manifesto's points: none of the simplifications offered are of any significance for 'professional' mathematics.

Sure, it won't make any difference to professional mathematicians (since it's just a trivial substitution). But what about the pedagogical benefits, particularly when it comes to teaching students about trigonometry and radians? It makes complete intuitive sense to define a full turn as T radians. For me this is really the strongest case made by the Manifesto.

BlackSails wrote:Also, e^i*T/2=-1 is much less nice looking.

eiT = 1 looks quite nice.

Re: The Tau Manifesto

Posted: Thu Jul 01, 2010 9:04 pm UTC
by antonfire
Velifer wrote:It's a question of standards and legacy. If we're opening up the definition of pi, why not open up the definition of 6? It's not the inverse of 9, and that's problematic.
Oh I agree that changing it now is probably impractical. I just also happen to agree with the fact that it is a better notation. Right there along with temperature being backwards and Lp spaces being backwards. It's one of those things that's too minor to change but still kind of annoying.

phr34k wrote:If we use τ= 2π, then we'd also have to keep talking about τ/2 whenever π would pop up (zeros of sin, sum of angles in a triangle, half-plane polar parametrizations, area of a unit circle,etc.) And fractions are always more cumbersome (typographically) then products.
And there's a good reason to expect a factor of 1/2 in each of those cases.


At any rate, if you don't change the notation, it pays to get into the habit of thinking of 2pi as a unit, and pi as half that.

Re: The Tau Manifesto

Posted: Thu Jul 01, 2010 9:29 pm UTC
by BlackSails
antonfire wrote:
Velifer wrote:It's a question of standards and legacy. If we're opening up the definition of pi, why not open up the definition of 6? It's not the inverse of 9, and that's problematic.
Oh I agree that changing it now is probably impractical. I just also happen to agree with the fact that it is a better notation. Right there along with temperature being backwards and Lp spaces being backwards. It's one of those things that's too minor to change but still kind of annoying.


What is backwards about hilbert space?

Re: The Tau Manifesto

Posted: Fri Jul 02, 2010 12:48 am UTC
by Suffusion of Yellow
The best part of the this paper is when he points out that pi is both transcendental and irrational. Spooky

Re: The Tau Manifesto

Posted: Fri Jul 02, 2010 3:30 am UTC
by B.Good
BlackSails wrote:Also, e^i*T/2=-1 is much less nice looking.

Is it wrong that this was my first thought when I was reading the website?

Re: The Tau Manifesto

Posted: Fri Jul 02, 2010 7:10 am UTC
by antonfire
BlackSails wrote:What is backwards about hilbert space?
Lp spaces aren't all Hilbert spaces.

Anyway, I would have what's normally denoted Lp be denoted L1/p instead, L2 would be L1/2, L1 would still be L1, and so on. This particular discussion really belongs in here, though. Then again, maybe so does this whole thread.

Re: The Tau Manifesto

Posted: Fri Jul 02, 2010 3:12 pm UTC
by phr34k
antonfire wrote:Oh I agree that changing it now is probably impractical. I just also happen to agree with the fact that it is a better notation. Right there along with temperature being backwards and Lp spaces being backwards. It's one of those things that's too minor to change but still kind of annoying.
...
At any rate, if you don't change the notation, it pays to get into the habit of thinking of 2pi as a unit, and pi as half that.


What is backwards about temperature??? :shock:

And I already do think of 2pi as a unit :-)

Re: The Tau Manifesto

Posted: Fri Jul 02, 2010 4:19 pm UTC
by BlackSails
antonfire wrote:
BlackSails wrote:What is backwards about hilbert space?
Lp spaces aren't all Hilbert spaces.


I know, but L^2 is the only space I have ever worked with, so thats the only one I know enough to ask about.

Re: The Tau Manifesto

Posted: Fri Jul 02, 2010 9:07 pm UTC
by antonfire
phr34k wrote:What is backwards about temperature??? :shock:
The way it is now, negative temperatures are hotter than positive temperatures. That is, from hot to cold, we have -0.1 K, -10 K, "infinity K", 10 K, 0.1 K.

If we used inverse kelvins instead, from hot to cold would be -10 K-1, -0.1 K-1, 0 K-1, 0.1 K-1, 10 K-1.

You also see a lot of 1/T in statistical mechanics computations and so on.

Re: The Tau Manifesto

Posted: Sat Jul 03, 2010 3:26 am UTC
by BlackSails
Yeah, temperature should really be measured as the thermodynamic beta (1/kT) instead of T.

Re: The Tau Manifesto

Posted: Tue Jul 06, 2010 3:37 am UTC
by mhartl
B.Good wrote:
BlackSails wrote:Also, e^i*T/2=-1 is much less nice looking.

Is it wrong that this was my first thought when I was reading the website?


There's nothing wrong with your reaction; it's to be expected after being exposed to a lifetime of pro-[imath]\pi[/imath] propaganda. ;-)

Geometrically, the equation [math]e^{i\tau/2}= -1[/math] says that a rotation by half a turn is the same as multiplying by [imath]-1[/imath]. And indeed this is the case: under a rotation of [imath]\tau/2[/imath] radians, the complex number [imath]z = a + ib[/imath] gets mapped to [imath]-a - ib[/imath], which is in fact just [imath]-1 \times z[/imath]. Written in terms of [imath]\tau[/imath], the "original" form of Euler's identity has a transparent geometric meaning that it lacks when written in terms of [imath]\pi[/imath].

Re: The Tau Manifesto

Posted: Tue Jul 06, 2010 4:00 am UTC
by mhartl
Suffusion of Yellow wrote:The best part of the this paper is when he points out that pi is both transcendental and irrational. Spooky


Of course, transcendental implies irrational, and you're right that the paper's phrasing implied that the two categories are distinct. I've updated the manifesto with clearer language. Thanks. :-)

BlackSails wrote:Also, e^i*T/2=-1 is much less nice looking.


It looks nicer if you think about it geometrically: [imath]e^{i\tau/2} = -1[/imath] says that a rotation by half a turn (i.e., by [imath]\tau/2[/imath] radians) is the same as multiplying by [imath]-1[/imath]. This is in fact the case: rotating the complex number [imath]z = a + ib[/imath] by half a turn yields [imath]-a-ib = -(a+ib)= -z[/imath]. Q.E.D.

In any case, as noted in the manifesto, the equations [imath]e^{i\tau}=1[/imath] and [imath]e^{i\tau}=1+0[/imath] look nicest of all. The former has a transparent geometric significance ("A rotation by one turn is [imath]1[/imath]"), while the latter combines the additive identity, the multiplicative identity, the imaginary unit, the exponential number, and the circle constant in a single equation.

Note: User PM'd about not double posting (but posts were queued so they couldn't edit anyway). -Lanicita

Re: The Tau Manifesto

Posted: Tue Jul 06, 2010 10:06 am UTC
by Patashu
Wow, this is actually mind-opening. I've always found it really annoying how 2pi is a full circle, and tau = one turn!

My only complaint is that the facetious insertion of e^(i*tau) = 1 + 0, while hilarious, dulls the blow the manifesto is meant to be. In fact, you could defy the pi-establishment further, by establishing a set of euler identities to not just present an interesting mathematical relation, but educate the layperson about what exactly is being done here:

e^(i*0) = 1
e^(i*tau/4) = i
e^(i*tau/2) = -1
e^(i*tau*3/4) = -i
e^(i*tau) = 1

Re: The Tau Manifesto

Posted: Wed Jul 07, 2010 6:20 am UTC
by mhartl
mike-l wrote:
Velifer wrote:Pi is the ratio of the circumference of a circle to its diameter. 6 would be wrong. Also, all this stuff.

Of course, if it's that important, you can always just start your paper with things like "let tau = 2pi, let current direction = electron drift(flow) direction, let pluto=planet, let black = white (because otherwise you're a RACIST!) and then have your party. Alone. And poorly understood.


Palais' article is linked on the page ssteve linked. And tau is the ratio of the circumference to the radius. Where else do we ever see diamater other than in the definition of pi?

One place I know of where pi is nicer than 2pi, is the volume of the n-sphere: [math]C_n=\frac{\pi^{n/2}}{\Gamma(\frac{n}{2} + 1)}[/math]

I do think that 2pi comes up much more often though.


Actually, [imath]\tau[/imath] shows up even in the example you cite. It's true that the usual way of writing the general formula for the volume of an [imath]n[/imath]-sphere uses [imath]\pi[/imath]:
[math]V_n(r) = \frac{\pi^{n/2} r^n}{\Gamma(\frac{n}{2} + 1)}.[/math]
This expression uses the [imath]\Gamma[/imath] function, which (apart from nonpositive integers) is defined for any real number. But in this case [imath]n[/imath] is a nonnegative integer, and we should use this fact when expressing the volume of the [imath]n[/imath]-sphere.

The exact form of the volume depends on whether [imath]n[/imath] is even or odd, and it's prettiest when written in terms of the double factorial function [imath]n!![/imath]:
[math]n!! = \begin{cases}
n(n-2)(n-4)\ldots5\cdot3\cdot1 & \text{if } n \text{ is even}; \\ \\
n(n-2)(n-4)\ldots6\cdot4\cdot2 & \text{if } n \text{ is odd}.
\end{cases}[/math]
The volume of an [imath]n[/imath]-sphere is then
[math]V_n(r) = \begin{cases}
\displaystyle \frac{(2\pi)^{n/2}\,r^n}{n!!} & \text{if } n \text{ is even}; \\ \\
\displaystyle \frac{2(2\pi)^{(n-1)/2}\,r^n}{n!!} & \text{if } n \text{ is odd}.
\end{cases}[/math]
Here we see the ever-present factor of [imath]2[/imath] in front of [imath]\pi[/imath]. Making the substitution [imath]\tau=2\pi[/imath] then yields
[math]V_n(r) = \begin{cases}
\displaystyle \frac{\tau^{n/2}\,r^n}{n!!} & \text{if } n \text{ is even}; \\ \\
\displaystyle \frac{2\tau^{(n-1)/2}\,r^n}{n!!} & \text{if } n \text{ is odd}.
\end{cases}[/math]
Q.E.D.

Re: The Tau Manifesto

Posted: Wed Jul 07, 2010 8:31 pm UTC
by ++$_
Your formulas are WAY more complicated than the one with the gamma function. They're defined piecewise and they have the product of the odd numbers in them rather than the much nicer gamma function. In fact, in order to get the tau in there you had to squeeze a power of 2 into the denominator, which is what resulted in the presence of the double factorial.

Any formula with tau can be converted into one with pi, and vice versa. This is obvious. The question is whether doing so gives us some useful information. In this case you are not gaining any.

Re: The Tau Manifesto

Posted: Fri Jul 09, 2010 12:51 am UTC
by mhartl
++$_ wrote:Your formulas are WAY more complicated than the one with the gamma function. They're defined piecewise and they have the product of the odd numbers in them rather than the much nicer gamma function. In fact, in order to get the tau in there you had to squeeze a power of 2 into the denominator, which is what resulted in the presence of the double factorial.

Any formula with tau can be converted into one with pi, and vice versa. This is obvious. The question is whether doing so gives us some useful information. In this case you are not gaining any.


First of all, I should have mentioned in my original comment that at least in the case [imath]n=2[/imath], i.e., for a circular disk, we know there's a hidden factor of [imath]2[/imath]. As argued in the Tau Manifesto, the area formula [imath]A = \pi r^2[/imath] is more naturally written in terms of [imath]\tau[/imath]: [math]A = \pi r^2 = \textstyle{\frac{1}{2}}(2\pi) r^2 = \textstyle{\frac{1}{2}}\tau r^2.[/math] This gives us reason to suspect that the volume of an [imath]n[/imath]-sphere might have natural factors of [imath]2\pi[/imath] for [imath]n > 2[/imath] as well.

The formula with the [imath]\Gamma[/imath] function isn't simpler than the one using double factorials; it just hides its complexity behind simple-looking notation. The double factorial function, though relatively obscure, is elementary: any smart ten-year-old could understand it. In contrast, the [imath]\Gamma[/imath] function, though notationally simple, is actually an integral over a semi-infinite domain—try explaining that to a ten-year-old. Moreover, the formula I quoted is more specific because it uses the fact that [imath]n[/imath] is a nonnegative integer. Indeed, if you look at a standard reference, such as Wolfram MathWorld's entry on the hypersphere (http://mathworld.wolfram.com/Hypersphere.html), you'll see formulas along the same lines as the ones I used:
[math]V_n(r) = \begin{cases}
\displaystyle \frac{\pi^{n/2}\,r^n}{(\frac{n}{2})!} & \text{if } n \text{ is even}; \\ \\
\displaystyle \frac{2^{(n+1)/2}\pi^{(n-1)/2}\,r^n}{n!!} & \text{if } n \text{ is odd}.
\end{cases}[/math]
The first thing we notice is that MathWorld writes the volume piecewise, as I did; the integral nature of [imath]n[/imath] results in a piecewise expression for the volume whether we want it to or not. We next notice that they use the double factorial function, also as I did—but, somewhat mysteriously, they use it only in the odd case. (This is a hint of things to come.)

Let's look at the odd case first. I don't know about you, but to me
[math]2^{(n+1)/2}\pi^{(n-1)/2}[/math]
looks an awful lot like
[math]2(2\pi)^{(n-1)/2}.[/math]
And at this point we immediately recover the formula I cited in my original comment.

Now let's look at the even case. Perhaps [imath]\tau[/imath] is more natural in the odd case, but [imath]\pi[/imath] is more natural in the even case, right? It's certainly possible, but one might reasonably ask why MathWorld chose to use a double factorial in one case and only a single factorial in the other. My suspicion is that (like most of the world) their pattern recognition is overly biased toward [imath]\pi[/imath], so they simply didn't notice that they could unify their formulas using the following identity:
[math]\left(\frac{n}{2}\right)! = \frac{n!!}{2^{n/2}}.[/math] Substituting this into the formula for even [imath]n[/imath] then yields
[math]\frac{2^{n/2}\pi^{n/2}\,r^n}{n!!},[/math] which bears a striking resemblance to
[math]\frac{(2\pi)^{n/2}\,r^n}{n!!},[/math] and again we recover the expression in the original comment.

We see from this discussion that using [imath]\tau[/imath] in place of [imath]\pi[/imath] does give additional insight into the relationship between the volume of an [imath]n[/imath]-sphere and the circle constant. I'm guessing from your tone that there's no convincing you on this point, but I hope you'll reserve the right to change your mind about [imath]\tau[/imath]. (I certainly reserve the right to change my mind about [imath]\pi[/imath]. After all, I've changed my mind once already.)

Re: The Tau Manifesto

Posted: Fri Jul 09, 2010 4:28 am UTC
by mhartl
Patashu wrote:Wow, this is actually mind-opening. I've always found it really annoying how 2pi is a full circle, and tau = one turn!

My only complaint is that the facetious insertion of e^(i*tau) = 1 + 0, while hilarious, dulls the blow the manifesto is meant to be. In fact, you could defy the pi-establishment further, by establishing a set of euler identities to not just present an interesting mathematical relation, but educate the layperson about what exactly is being done here:

e^(i*0) = 1
e^(i*tau/4) = i
e^(i*tau/2) = -1
e^(i*tau*3/4) = -i
e^(i*tau) = 1


You'd be amazed at the number of people whose biggest complaint about [imath]e^{i\tau} = 1[/imath] is that "it doesn't include [imath]0[/imath]", so the addition of [imath]0[/imath] to form [imath]e^{i\tau} = 1 + 0[/imath] wasn't facetious—though it was meant to have just a little undercurrent of snark. :wink: In any case, I love the idea of adding in the other Eulerian identities, and it's likely that a future revision of the manifesto will include a table containing them. Please email me off-list (I am easy to find) with your full name (and an optional personal URL) if you would like to be credited in the acknowledgments.

Re: The Tau Manifesto

Posted: Fri Jul 09, 2010 4:14 pm UTC
by the tree
mhartl wrote:You'd be amazed at the number of people whose biggest complaint about [imath]e^{i\tau} = 1[/imath] is that "it doesn't include [imath]0[/imath]", so the addition of [imath]0[/imath] to form [imath]e^{i\tau} = 1 + 0[/imath] wasn't facetious—though it was meant to have just a little undercurrent of snark.
[imath]e^{i\tau} - 1= 0[/imath]
hoshitI'magenius.

Re: The Tau Manifesto

Posted: Fri Jul 09, 2010 6:20 pm UTC
by Dason
the tree wrote:
mhartl wrote:You'd be amazed at the number of people whose biggest complaint about [imath]e^{i\tau} = 1[/imath] is that "it doesn't include [imath]0[/imath]", so the addition of [imath]0[/imath] to form [imath]e^{i\tau} = 1 + 0[/imath] wasn't facetious—though it was meant to have just a little undercurrent of snark.
[imath]e^{i\tau} - 1= 0[/imath]
hoshitI'magenius.

but that's subtraction. Ewwwwww. None of the cool kids use subtraction.

Re: The Tau Manifesto

Posted: Fri Jul 09, 2010 9:18 pm UTC
by Patashu
Let's define co-peanu arithmetic by starting at omega and defining a 'predecessor' function!

Re: The Tau Manifesto

Posted: Fri Jul 09, 2010 10:57 pm UTC
by gmalivuk
Wouldn't that just give you something isomorphic to the nonpositive integers?

Re: The Tau Manifesto

Posted: Sat Jul 10, 2010 6:23 am UTC
by antonfire
mhartl wrote:The exact form of the volume depends on whether n is even or odd, and it's prettiest when written in terms of the double factorial function.
On the other hand, if you're the sort of person who prefers tau over pi, you ought to dislike the double factorial function just about as much. And you especially ought to dislike having to split anything into even and odd cases.

If you're a big fan of being all natural and whatnot, you should probably be comparing the volume of a unit n-sphere to an n-cube of side length 2 (not 1), and hence looking at the expression (pi/4)^(n/2) / (n/2)!.

Re: The Tau Manifesto

Posted: Sat Jul 17, 2010 1:51 am UTC
by jemidiah
The n-sphere formula can also be written as

[math]C_n = \frac{\frac{\tau}{2}^{n/2}}{\frac{n}{2}!}[/math]

It seems simpler to me to just remember "everything is divided by 2". [Yes, the factorial doesn't work for odd n; I'd argue the Gamma function is the source of that bit of ugliness.] The n-sphere Wikipedia page goes on to give a number of 2pi-laden formulas as well. I'm in favor of tau, especially since a lot of the simpler applications of pi/tau are cleaner using tau: Radian<->decimal conversions; special trig angles; period of complex exponential function; complex roots of unity.

Like the manifesto said, mathematicians can fend for themselves. It's the uninitiated who need simplicity and clarity. I haven't thought of a single "basic" example of pi being cleaner than tau.

Re: The Tau Manifesto

Posted: Mon Feb 07, 2011 6:49 pm UTC
by nwest
I am a recent convert to tau. The manifesto clicked with me immediately. Pi was derived in terms of diameter, yet diameter NEVER shows up in our equations. It's always the radius! If we used the diameter in our formulas, then Pi would be perfect! But instead we use the radius, so we have to have an ugly 2 next to Pi all the time. I think it's fairly obvious that if Pi was originally defined as 6.28 instead of 3.14, then we would not be having this conversation about whether Pi=3.14 would have been a better choice. Thinking back to my junior high math classes, everything would have made so much more sense if we talked about turns instead of 2 Pi. We deliberately confuse kids these days because we're too stubborn to recognize we've been doing it wrong all these years.

I'm team-Hartl all the way. He knows Matt Groening! You know, I would LOVE to see "Tau=6.283185..." show up in a Futurama episode someday. Maybe that will help spread the adoption of Tau!

Let Tau = 2 Pi!!!