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The Pi Manifesto

thepimanifesto.com

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Re: The Pi Manifesto

#61

Post by ianterrell »

scythe wrote: post #60
ianterrell wrote: post #40

Earlier quoted context omitted.

> The area of the traditional unit circle is π 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 th…

>And it was chosen hastily. 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.

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.

Re: The Pi Manifesto

#62

Post by scythe »

ianterrell wrote: post #48
mikeash wrote: post #47

Earlier quoted context omitted.

I guess the pertinent question is whether this stimulates your intellectual curiosity because you find it to be an interesting question, or at least because it raises interesting questions, or whether it stimulates your curiosity because you think it's exceedingly important and making this change will have big consequences. I think the dismissive types see people obsessing over it as the latter, and this is, at least…

I won't belabor the point any further than this, but the argument really is that it would provide enormous benefit. It's not just a "what would happen?", and it's not that τ "extracts" meaning from equations. It makes the intrinsic meanings of equations drastically more clear . For instance, what does sin(x) mean? What does Euler's equation mean? They're both eminently simple concepts, but they remain obscured by π.…

>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 not enter the definition.

Re: The Pi Manifesto

#63

Post by scythe »

ianterrell wrote: post #61
scythe wrote: post #60

Earlier quoted context omitted.

>And it was chosen hastily. 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.

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.

>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.

Re: The Pi Manifesto

#64

Post by ianterrell »

scythe wrote: post #63
ianterrell wrote: post #61

Earlier quoted context omitted.

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.

>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.

Re: The Pi Manifesto

#65

Post by scythe »

ianterrell wrote: post #64
scythe wrote: post #63

Earlier quoted context omitted.

>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 peri…

>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.

Re: The Pi Manifesto

#66

Post by ianterrell »

scythe wrote: post #65
ianterrell wrote: post #64

Earlier quoted context omitted.

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 peri…

>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.

Hardly. It's acknowledging that both choices are definitions.

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?

Re: The Pi Manifesto

#67

Post by omaranto »

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…

  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.
Life is too short to argue over notation. I for one will show my "lack of intellectual curiosity" and go back to learning mathematics (with short breaks to argue with people on HN :P).

Re: The Pi Manifesto

#68

Post by scythe »

ianterrell wrote: post #66
scythe wrote: post #65

Earlier quoted context omitted.

>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.

Hardly. It's acknowledging that both choices are definitions. 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?

>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 [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.

Re: The Pi Manifesto

#69

Post by ianterrell »

scythe wrote: post #68
ianterrell wrote: post #66

Earlier quoted context omitted.

Hardly. It's acknowledging that both choices are definitions. 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?

>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 help correct them.

Re: The Pi Manifesto

#70

Post by ianterrell »

scythe wrote: post #62
ianterrell wrote: post #48

Earlier quoted context omitted.

I won't belabor the point any further than this, but the argument really is that it would provide enormous benefit. It's not just a "what would happen?", and it's not that τ "extracts" meaning from equations. It makes the intrinsic meanings of equations drastically more clear . For instance, what does sin(x) mean? What does Euler's equation mean? They're both eminently simple concepts, but they remain obscured by π.…

>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 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.

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