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.
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.
>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.
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.
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.
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.
>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:
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:
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.
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.
>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.
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?
I'm going to give you the benefit of the doubt and assume that you're not trolling but are frustrated. In which case I must also assume that you haven't really spent the time to understand the argument at tauday.com. It is not that the ratio of the circumference to the diameter (called π) will change, but that the ratio of the circumference to the radius (now called τ) is more useful.
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.
Why does smallest in terms of absolute value matter? In any event, I can define the circle via the center and r/2 or center and r/4, etc... I don't see any value in caring about the absolute value.
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.
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!