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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

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.

scythe 15y ago
>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.
- ianterrell 15y ago
  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?
  - scythe 15y ago
    >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.
    - ianterrell 15y ago
      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.
      - scythe 15y ago
        >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_using_Fourier_series 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/EZeta.pdf http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Zeta/EZe...
        ianterrell 15y ago
        I can't convince you, but that doesn't make anything you've said above correct.
        scythe 15y ago

scythe 15y ago

>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.

ianterrell 15y ago
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?
- scythe 15y ago
  >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.
  - ianterrell 15y ago
    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.
    - scythe 15y ago
      >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_using_Fourier_series 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/EZeta.pdf http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Zeta/EZe...
      - ianterrell 15y ago
        I can't convince you, but that doesn't make anything you've said above correct.
        scythe 15y ago