3 ms·

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 w

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.

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.