I find it surprising that this article, and many mathematicians asked to comment in the media, pull out the area formula as their prime counter-example, seemingly forgetting that using pi is only obscuring the mechanics behind it.
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.
So no, you don't have to consider every question that's raised "no matter what" to be considered intellectually curious, but this is hardly such a trivial, general case as that implies.
I think the dismissive types see people obsessing over it as the latter, and this is, at least in my opinion, way more than is called for. There's nothing wrong with considering the consequences of such a change just for some mathematical fun. (After all, who here hasn't drawn out a complete system of units based on furlongs, fortnights, etc. as the fundamental units, rather than mks or cgs? What's that? None of you? Oh....) But the original manifesto seems to be overstating its case severely in terms of the consequences of such a change, and I think that's what others are reacting to.
Ignoring the boring costs of switching the installed base of mathematicians, software, etc. from one constant to the other, the benefit of a switch seems to be that certain things become easier and other certain things become harder. At best, there's a small net gain. Hardly seems worth the discussion when considered in that light.
If you're considering it as an interesting exercise to extract meaning from equations, well, go for it! But that seems to fit more under the banner of "What would happen if we switched this?" rather than "It would provide enormous benefit if everybody switched this!" as the original manifesto seemed to be saying.
It's exactly like refactoring code: sure, it does the same thing, but now it's more compact, more concise, more clear, more elegant; the parts of the system all fit together better, and people coming onto the project will be able to learn it faster. If you don't care about those things, then you won't refactor your code, or see the point of τ.
They are equally obscured by tau, or whatever other arc-length you might choose! The meaning does not depend on the definition of a circle but on the properties of the exponential function:
http://en.wikipedia.org/wiki/Euler%27s_formula#Proofs
As for sin(x), it is a function based on circles and arc lengths, but specific values of arc length do not enter the definition.
Actually, you can go backwards and say that its circular properties define the exponential function. Euler's formula describes the rotation of the unit vector through the imaginary plane.
> As for sin(x)...specific values of arc length do not enter the definition
It's not about the definition, it's about the meaning. Sin(x) is the height of the circle at x radians. And it's super awesome with tau: one tau is full circle, and one period.
Well, if you've already made the jump to understanding negative heights as going below the real line. This is a topic for an introductory course in geometry? When I learned about sine and cosine, it was first defined in terms of SOHCAHTOA!
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
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!
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.
The definition of the circle constant comes first.
>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.
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.
So it does not make it more clear? This contradicts your original assertion.
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?
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.
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.
...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_...
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...
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.
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...
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.
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.
Was two millennia not long enough? We could take another couple centuries, I guess. I don't think we're ever going to get a new answer for the ratio of the circumference of a circle to the diameter of that circle, though.
> The lack of the 1/2 in the π version shows why it's "wrong"—it's not as meaningful.
Frankly, I don't know what this means. I've read it several dozen times, and each time, it seems increasingly more inane. I can't help but wonder what you would say about the derivative (and antiderivative) of e^x; would you complain that it's not meaningful? Does it need more coefficients? More exponents?
Which shouldn't come as too much of a surprise because the radius is the smallest amount of information that determines what a circle is, as well as the basis for how we define radians.
The radius is not the smallest amount of information which determines a circle. The radius and origin can determine a circle. So can the diameter and origin, or the circumference and origin, or the area and origin. Don't forget your geometry.
Sure, you can write any in terms of any of the others, but the radius is the smallest: both in terms of absolute value as well as dimensionality.
As for dimensionality, how are you using the word? No matter what, we need 3 numbers to define a circle in R^2. Two numbers define its position and one number defines it size. I don't see what you're getting at. If you mean that because radius is a measure of length (one dimension) and area is a measure of area (two dimensions) then I have two questions: 1) Why does that matter? Its still just a single number. 2) Even if it does matter, why is radius more fundamental than diameter.
…it's a poor argument that supports your opponent after switching key nouns.
A quarter arc of a circle is π/2 radians, regardless of area.
A quarter arc of the unit circle is π/2. A quarter arc of the circle with area 2π, though... Here, let's do it in τ for fun. :3
So, the circle with area τ has radius sqrt(2) (from A = πr^2), diameter 2sqrt(2), and circumference τsqrt(2) (from C = Dπ), thus a quarter arc of that circle is τsqrt(2)/4. That's not a pretty number at all to work with.
Maybe keeping the unit circle at its current size is a better idea.
How much simpler could it get?
Defining the unit circle as a circle with area τ is a contradictory statement and is semantically invalid.
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.