The Pi Manifesto • Hacker News Board

Re: The Pi Manifesto

#71

Post by scythe » Tue, Jul 05, 2011, 9:00 PM UTC

ianterrell wrote: post #69

scythe wrote: post #68

Earlier quoted context omitted.

>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 [ra…

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 hel…

>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_...

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...

Re: The Pi Manifesto

#72

Post by MostAwesomeDude » Tue, Jul 05, 2011, 9:02 PM UTC

ianterrell wrote: post #56

MostAwesomeDude wrote: post #53

Earlier quoted context omitted.

> hastily 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 incre…

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…

Your assumptions are invalid.

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.

Re: The Pi Manifesto

#73

Post by ianterrell » Tue, Jul 05, 2011, 9:08 PM UTC

scythe wrote: post #71

ianterrell wrote: post #69

Earlier quoted context omitted.

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 hel…

>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…

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

Re: The Pi Manifesto

#74

Post by simcop2387 » Tue, Jul 05, 2011, 9:22 PM UTC

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...

Re: The Pi Manifesto

#75

Post by scythe » Tue, Jul 05, 2011, 9:31 PM UTC

ianterrell wrote: post #73

scythe wrote: post #71

Earlier quoted context omitted.

>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…

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

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...

Re: The Pi Manifesto

#76

Post by scythe » Tue, Jul 05, 2011, 9:42 PM UTC

ianterrell wrote: post #70

scythe wrote: post #62

Earlier quoted context omitted.

>They're both eminently simple concepts, but they remain obscured by π. 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 n…

> The meaning does not depend on the definition of a circle but on the properties of the exponential function 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…

>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!

Re: The Pi Manifesto

#77

Post by ianterrell » Tue, Jul 05, 2011, 10:02 PM UTC

MostAwesomeDude wrote: post #72

ianterrell wrote: post #56

Earlier quoted context omitted.

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…

Your assumptions are invalid. 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.

I said smallest amount of information. The diameter, circumference, and area are all functions of the radius.

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.

Re: The Pi Manifesto

#78

Post by ianterrell » Tue, Jul 05, 2011, 10:08 PM UTC

scythe wrote: post #75

ianterrell wrote: post #73

Earlier quoted context omitted.

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

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 enco…

> 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.

Re: The Pi Manifesto

#79

Post by colanderman » Tue, Jul 05, 2011, 10:17 PM UTC

MostAwesomeDude wrote: post #36

ianterrell wrote: post #22

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 forev…

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…

The circumference of the traditional unit circle is τ, 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 τ is related to the periods of sine and cosine.

…it's a poor argument that supports your opponent after switching key nouns.

Re: The Pi Manifesto

#80

Post by scythe » Tue, Jul 05, 2011, 10:20 PM UTC

ianterrell wrote: post #78

scythe wrote: post #75

Earlier quoted context omitted.

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 enco…

> 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 (pro…

>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.