4 ms·
>And it was chosen hastily. It was chosen here: http://arxiv.org/abs/math/0506415 http://arxiv.org/abs/math/0506415 Elegance is found in arguments and proofs
by scythe 15y ago
>And it was chosen hastily.
It was chosen here:
http://arxiv.org/abs/math/0506415 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.
- ianterrell 15y agoYour 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.
- scythe 15y ago>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.
- ianterrell 15y agoI'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 agoHardly. 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 agoIt'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.