I’m a mathematician. Tau is bikeshedding. It’s far removed from being either interesting or useful. But it is a useful litmus test for whether a person’s options on math are worth listening to or not.
Disclaimer: I'm not a mathematician, and I only have a bachelor degree in science.
With respect: It's not bikeshedding.
Pi is wrong, and you know it, it's just that you (likely) feel that it is too late to change it, so why fight against history?
One of the best quotes I've ever heard is: "The standard you walk past is the standard you accept."
There are several problems with simply letting things remain as they are. I'm sure you've heard them already, but allow me to reiterate them for the benefit of the non-mathematicians here:
1. Pedagogical: I never "got" radians, or had a really firm grasp on trigonometry until I had heard of tau-vs-pi. Suddenly, everything just clicked. Sure, once you know it, it's a trivial mental substitution to replace every instance of pi with tau/2, but... only if you know to do this. I'm not the only one. Persisting on using pi is doing millions of students a disservice. A tau is one turn. A pi is half a turn! Wat? Do you buy bread half a loaf at a time and tell everyone that this is fine because you just need to by two half-loafs and it all works out?
2. Theoretical: Something that opened my eyes about physical theories is that it's easy to confuse oneself by cancelling small constants... such as when making the tau -> 2 pi substitution. The '2' gets cancelled all over the place, making other integer constants "wrong", and not always in an obvious way. They then get hand-waved away because they stop making sense, and so important things can be missed.
3. Boundary pushing: Most endeavours are limited by physics. Mathematics is unique in that it is limited only by human brain capacity. Ergo, the cutting edge of mathematics is bounded by how far a single human brain can go, but no further. Progress is achievable only through efficiency of thought, via shortcuts, elegance, abstraction, consistency, simplicity, and other similar means. Unnecessary complexity at the bottom is unnecessary and should be dropped to make room at the top.
I could go on and on, but better articles have been written on the topic.
The point is, if you see someone turning right three times to turn left, you immediately, indistinctively think less of them.
You're saying that no-no-no-no... the triple turners are just fine the way the are, and nobody should listen to those strange people in the minority who insist on turning directly towards the intended destination. Sure, it's faster, but we've never done it that way...
> Pi is wrong, and you know it, it's just that you (likely) feel that it is too late to change it, so why fight against history?
I'm also a mathematician, and pi is just fine.
> 1. Pedagogical: I never "got" radians, or had a really firm grasp on trigonometry until I had heard of tau-vs-pi.
The way I remember it, I learned trigonometry via triangles, whose internal angles sum to tau/2. This is absurd! Why don't they sum to a whole thing instead of a half thing?
> 3. Boundary pushing:
By the time one gets to the cutting-edge, of all the things that are holding one back a missing or extra factor of two is certainly not one of them.
There you go, arguing a trivial point at length instead of doing interesting math. This is why tau is not worth anyone’s time. There’s no one weird trick to understanding math.
> Ergo, the cutting edge of mathematics is bounded by how far a single human brain can go, but no further. Progress is achievable only through efficiency of thought, via shortcuts, elegance, abstraction, consistency, simplicity, and other similar means.
I think this is perhaps the strongest pro-tau argument. How many intuitive insights may have been missed or delayed, because a 2 was factored out of 2pi?
Perhaps a softer stance on the way to tauism is to encourage folks to keep the 2 and the pi together as a single pseudo-symbol 2pi. So you'd never factor out the 2 (A = 1/2(2pi)r^2) nor would you distribute over it ((2pi)^2 would never become 4pi^2). I don't think that would work well in the educational space - now there is this extra weird rule about reductions - but it might help in other fields. At that point I guess you might as well use tau though.
> I think this is perhaps the strongest pro-tau argument. How many intuitive insights may have been missed or delayed, because a 2 was factored out of 2pi?
About as many as were missed or delayed because a 1/2 was lost from 1/2tau.
The problem with pi vs tau is that it is pretty much impossible to convince everybody that one of them is the best and also that there are no other better choices.
So, in the absence of unanimity, the incumbent pi wins by default.
As an example of the diversity of opinions in this matter, while I agree that tau is better than pi, I believe that an even better choice than either of them is pi/2.
Any choice of the constant is equivalent to a choice of the measurement unit for the plane angle. Choosing tau is equivalent to choosing the cycle as the measurement unit for the plane angle ("cycle" was actually a standard name for this unit of plane angle, even if it is now out of fashion, e.g. in the old literature there are many references to cycles per second, cycles per meter etc.), while choosing (pi/2) as the constant, is equivalent to choosing the right angle as the measurement unit for the plane angle.
There are several arguments why the right angle is a more appropriate unit than the cycle, besides the fact that the right angle was already the unit of plane angle used by Euclid.
Even in a plane, but much more so in 3D space, a rotation of 1 cycle is ambiguous, it is not associated with a certain axis of rotation, like a rotation of 1 right angle. Given a rotation of 1 right angle, a rotation of an arbitrary angle is just the rotation corresponding to a multiple of that right angle, which can be non-ambiguously defined, like you expect for any physical quantity, to have a value that is a multiple of its unit.
Also in the trigonometric functions, the right angle is the most convenient unit for the arguments (making their reductions equivalent with taking the fractional part) and I am dismayed that the floating-point standard has chosen to recommend the functions sinPi, cosPi and the like, to be included in the standard libraries of the programming languages, instead of the much more convenient functions where the angles are measured in right angles, i.e. sin((pi/2)*x) and so on (such functions do not have the argument reduction problems that plague the functions with arguments in radians; the former are more convenient for large angles, while the latter are more convenient for very small angles).
Could you explain why a cycle is ambiguous in 3D? It seems like for any choice of coordinates, one cycle would always entail a return to an original point. But I'm probably misunderstanding.
With respect: It's not bikeshedding.
Pi is wrong, and you know it, it's just that you (likely) feel that it is too late to change it, so why fight against history?
One of the best quotes I've ever heard is: "The standard you walk past is the standard you accept."
There are several problems with simply letting things remain as they are. I'm sure you've heard them already, but allow me to reiterate them for the benefit of the non-mathematicians here:
1. Pedagogical: I never "got" radians, or had a really firm grasp on trigonometry until I had heard of tau-vs-pi. Suddenly, everything just clicked. Sure, once you know it, it's a trivial mental substitution to replace every instance of pi with tau/2, but... only if you know to do this. I'm not the only one. Persisting on using pi is doing millions of students a disservice. A tau is one turn. A pi is half a turn! Wat? Do you buy bread half a loaf at a time and tell everyone that this is fine because you just need to by two half-loafs and it all works out?
2. Theoretical: Something that opened my eyes about physical theories is that it's easy to confuse oneself by cancelling small constants... such as when making the tau -> 2 pi substitution. The '2' gets cancelled all over the place, making other integer constants "wrong", and not always in an obvious way. They then get hand-waved away because they stop making sense, and so important things can be missed.
3. Boundary pushing: Most endeavours are limited by physics. Mathematics is unique in that it is limited only by human brain capacity. Ergo, the cutting edge of mathematics is bounded by how far a single human brain can go, but no further. Progress is achievable only through efficiency of thought, via shortcuts, elegance, abstraction, consistency, simplicity, and other similar means. Unnecessary complexity at the bottom is unnecessary and should be dropped to make room at the top.
I could go on and on, but better articles have been written on the topic.
The point is, if you see someone turning right three times to turn left, you immediately, indistinctively think less of them.
You're saying that no-no-no-no... the triple turners are just fine the way the are, and nobody should listen to those strange people in the minority who insist on turning directly towards the intended destination. Sure, it's faster, but we've never done it that way...