5 ms·

I have found the idea of tau useful even though I have never used it in writing. One argument in favor of tau is that in many formulas pi often has the multipl

I have found the idea of tau useful even though I have never used it in writing. One argument in favor of tau is that in many formulas pi often has the multiplier 2 in front of it. If these formulas are written in terms of tau, they may become slightly easier to memorize and manipulate. Perhaps so, but I don't really care about this. It’s not a big difference. Besides, there are also lots of formulas that are easier to memorize and manipulate using pi instead of tau. The probability density function of the standard Cauchy distribution f(x) = 1/pi * 1/(1 + x^2) is one example. However, to get a deep understanding of mathematics, I want to understand the connections between different parts of mathematics. If I see a mathematical formula with the constant pi in it, I ask myself: "How is this connected to circles?" The idea of tau taught me that that 2pi is the natural state of affairs. If I see pi by itself, I need to ask myself: "How is this connected to half-circles? Or has the multiplier 2 been cancelled away?” So, why does the pdf of the standard Cauchy distribution above contain pi instead of 2pi? What is the standard Cauchy distribution anyway? Take a gun that shoots particles in random directions, and place it one unit distance away from an infinitely long wall. Standard Cauchy distribution is where the particles will hit the wall. The particle will only hit the wall if it is shot in a direction towards it. This corresponds to 180 degrees - and there you have it: the connection to half-circles. Of course, you also have to work out the technical details. But on an intuitive level, when I see pi in the pdf of the standard Cauchy distribution I don’t think about how the missing multiplier makes it easier to remember a bunch of symbols; I think of particles hitting a wall.

jeffwass 10y ago
Your Cauchy argument is actually misleading and runs right into my joking example elsewhere in this thread about needing to define a new constant Sigma=4pi to work better with steradians. Pi doesn't always represent only a pure circle. Eg, in solid angles there are 4pi steradians over a sphere. Or 2tau. Or what I define as Sigma. This extra factor of two when using Tau should be just as disconcerting to tau enthusiasts for steradians as pi is for radians. Your example is talking about shooting particles in all directions over 3D space. This calls for a solid angle approach. Which means you should be integrating over steradians just as you'd use radians for an angular system. There are 4pi steradians over a sphere's full solid angle, the wall only covers 2pi steradians. Meanwhile the extra factor of two comes from the integration of decreasing infinitesimal wall cross sections over the azimuthal angle. The fact that it comes out to 1/2 tau is merely happy coincidence. Ie, the geometry introduced an extra factor of two because it's just as easily 1/4 sigma, and for a solid angle system (shooting particles in all directions) you should be using steradians. Hence sigma.
- tgb 10y ago
  Surely the Cauchy distribution you're replying to doesn't have any steradians going on, right? It's a 1D distribution and the gun described would point towards a random point on a circle, not on a sphere.
  - jeffwass 10y ago
    Argh, you may be right. I was actually thinking of a separate example from physics (years ago) calculating the classical flow of particles through an aperture. Where you assume particles have maxwell-Boltzmann distribution of speeds and consider them travelling in all directions over 3D and also the effective size of the aperture vs the azimuthal angle of travel. It's quite similar to this example actually but given the speeds you also must account for which velocity range [v,v+dv] as a function of angle will put a particle through the hole in time dt.
- MrManatee 10y ago
  As you already realized, my example was supposed to be in 2D, but I forgot to mention it. Sorry about that. But your point about steradians is still spot on. If I had been talking about the 3D case, there would have been a 1/(2pi) factor in the front of the pdf. Using tau would not have helped anyone realize that this particular 2pi is most naturally thought of as 1/2 of the full 4pi steradians. Using tau is not a substitute for careful thinking. And I am not even actually advocating switching to tau; I am not sure if it is worth the effort. But I still think that tau is a good way of conveying the idea that the current definition of pi as basically a historical accident. This can be enlightening even if one doesn't start using tau. As a more extreme example, I think it is useful to recognize that the decimal base (as opposed to binary, hex, etc.) is a historical accident, but I wouldn't advocate changing it.
Confusion 10y ago
One argument in favor of tau is that in many formulas pi often has the multiplier 2 in front of it. And an argument in favor of pi is that many formulas do not have that any multiplier in front it. So this cannot be resolved without quantitative evidence of the number of formulas using either one, including their frequency of use. Of course you can construct infinitely many formulas, but most of those are used a vanishingly small number of times. The question is what common formulas use pi or tau. How often would people actually have to write, read or say a factor?
- kazinator 10y ago
  In any formula where the 2 in 2π has been canceled by some irrelevant 2, the cancellation is not really an advantage; it obscures semantics. Also, when counting formulas don't forget to expand "angular frequency" ω to the 2πf it denotes.

jeffwass 10y ago

Your Cauchy argument is actually misleading and runs right into my joking example elsewhere in this thread about needing to define a new constant Sigma=4pi to work better with steradians. Pi doesn't always represent only a pure circle. Eg, in solid angles there are 4pi steradians over a sphere. Or 2tau. Or what I define as Sigma. This extra factor of two when using Tau should be just as disconcerting to tau enthusiasts for steradians as pi is for radians. Your example is talking about shooting particles in all directions over 3D space. This calls for a solid angle approach. Which means you should be integrating over steradians just as you'd use radians for an angular system. There are 4pi steradians over a sphere's full solid angle, the wall only covers 2pi steradians. Meanwhile the extra factor of two comes from the integration of decreasing infinitesimal wall cross sections over the azimuthal angle. The fact that it comes out to 1/2 tau is merely happy coincidence. Ie, the geometry introduced an extra factor of two because it's just as easily 1/4 sigma, and for a solid angle system (shooting particles in all directions) you should be using steradians. Hence sigma.

tgb 10y ago
Surely the Cauchy distribution you're replying to doesn't have any steradians going on, right? It's a 1D distribution and the gun described would point towards a random point on a circle, not on a sphere.
- jeffwass 10y ago
  Argh, you may be right. I was actually thinking of a separate example from physics (years ago) calculating the classical flow of particles through an aperture. Where you assume particles have maxwell-Boltzmann distribution of speeds and consider them travelling in all directions over 3D and also the effective size of the aperture vs the azimuthal angle of travel. It's quite similar to this example actually but given the speeds you also must account for which velocity range [v,v+dv] as a function of angle will put a particle through the hole in time dt.
MrManatee 10y ago
As you already realized, my example was supposed to be in 2D, but I forgot to mention it. Sorry about that. But your point about steradians is still spot on. If I had been talking about the 3D case, there would have been a 1/(2pi) factor in the front of the pdf. Using tau would not have helped anyone realize that this particular 2pi is most naturally thought of as 1/2 of the full 4pi steradians. Using tau is not a substitute for careful thinking. And I am not even actually advocating switching to tau; I am not sure if it is worth the effort. But I still think that tau is a good way of conveying the idea that the current definition of pi as basically a historical accident. This can be enlightening even if one doesn't start using tau. As a more extreme example, I think it is useful to recognize that the decimal base (as opposed to binary, hex, etc.) is a historical accident, but I wouldn't advocate changing it.

Confusion 10y ago

One argument in favor of tau is that in many formulas pi often has the multiplier 2 in front of it. And an argument in favor of pi is that many formulas do not have that any multiplier in front it. So this cannot be resolved without quantitative evidence of the number of formulas using either one, including their frequency of use. Of course you can construct infinitely many formulas, but most of those are used a vanishingly small number of times. The question is what common formulas use pi or tau. How often would people actually have to write, read or say a factor?

kazinator 10y ago
In any formula where the 2 in 2π has been canceled by some irrelevant 2, the cancellation is not really an advantage; it obscures semantics. Also, when counting formulas don't forget to expand "angular frequency" ω to the 2πf it denotes.