A trick to eliminate 2π (sometimes)
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A trick to eliminate 2π (sometimes)
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Re: A trick to eliminate 2π (sometimes)
#2Interesting. It's probably not worth defining a new constant for exp(2π), but this is a further demonstration of the Tau Manifesto's argument that 2π is much more of a fundamental value than π.
Re: A trick to eliminate 2π (sometimes)
#3`Θ^i = 1` does make this really slick, imho.
I often wonder if someday when we meet alien intelligences, they'll have a completely different set of constants, derivable from our own but different. Θ=535.491... may be such an example.
Re: A trick to eliminate 2π (sometimes)
#4Clever!
Re: A trick to eliminate 2π (sometimes)
#5Interesting, but I wonder if you’d get all the same benefits by just measuring angles in units of turns instead of radians. That seems cleaner than the weird dbar differential stuff.
Re: A trick to eliminate 2π (sometimes)
#6Responding to this part in the article:
> I’m not entirely sure about the intuitive meaning of “taking the derivative and dividing by 2π”. Is there some sort of fundamental connection to periodic functions?
If you have a function f(x) where x is measured in radians, and there are 2pi radians per turn, then you can change variables.
Let t represent turns. One turn is 2*pi rad, and you want t = 1 when you've gone all the way around in x, so t = x/2pi.
By the chain rule,
df(x)/dx = df(t)/dt dt/dx = 1/2pi * df(t)/dt
So I think this might be the meaning you're looking for when you do the rescaling of the derivative.
You're using turns as units instead of radians. cos(x=2pi)=cos(t=1)=1, and so on.
Re: A trick to eliminate 2π (sometimes)
#7[deleted]
Re: A trick to eliminate 2π (sometimes)
#8Easy trick to eliminate difficulties arising from pi: fix them via legislation [0].
Re: A trick to eliminate 2π (sometimes)
#9Responding to this part in the article: > I’m not entirely sure about the intuitive meaning of “taking the derivative and dividing by 2π”. Is there some sort of fundamental connection to periodic functions? If you have a function f(x) where x is measured in radians, and there are 2pi radians per turn, then you can change variables. Let t represent turns. One turn is 2*pi rad, and you want t = 1 when you've gone all t…
FWIW they mention this in the article:
> I like this, because it kind of eliminates the need for radians: the x in usin(x) has the unit of “turns”. I think this is conceptually much simpler.
Re: A trick to eliminate 2π (sometimes)
#10Interesting. It's probably not worth defining a new constant for exp(2π), but this is a further demonstration of the Tau Manifesto's argument that 2π is much more of a fundamental value than π.
Fundamental sounds like a value judgment. Pi is transcendental. 2 isn't. That's really the distinction. Unless there were other finite factors in pi, that is the number that's always going to have to be approximated in computation.