The natural unit of measurement for angles is the turn; a whole turn, or a whole number of turns, brings you back to where you started, a half turn reverses you, and so on. The angle subtended, at the circle's centre, by an arc of a circle is just the ratio of the arc's length to the total circumference. It is conventional to measure angles in radians; an arc's radian angle is its length divided by the circle's radius, which gives a larger answer, by a factor of 2.π than the turn angle. This convention ultimately comes back to the form of the functions sin and cos when expressed in terms of the standard exponential function, exp, which is equal to its own derivative; this grows exponentially along the real axis but, in the complex plane, varies sinusoidally along the imaginary axis.

We thus have conventional exp defined by exp' = exp (equal to its own derivative, everywhere on the complex plane) from which one can obtain

- sin(θ) = (exp(i.θ) - exp(-i.θ)) / 2
- cos(θ) = (exp(i.θ) + exp(-i.θ)) / 2

in which θ is an angle measured in radians and sin(), cos() are
defined in terms of a right-angle triangle whose hypotenuse has length 1 and one
of the corners exhibits the given angle, θ, between the hypotenuse and one
of the other sides (if necessary, as an external angle); sin() is the length of
the side in question, cos() is the length of the opposite side (with signs
thrown in if you had to use an external angle). The cute thing about exp is that
its power series is just exp(x) = sum(i in [..., 1, 0]: x**i / i! :), giving sin
and cos similar (though *slightly* less tidy) power series.

Switching to measuring angles in turns, take t = θ / 2 / π, so θ radians is t turns, and define

- Sin(t) = sin(2.π.t) = sin(θ) = (exp(2.i.π.t) - exp(-2.i.π.t))/2
- Cos(t) = cos(2.π.t) = sin(θ) = (exp(2.i.π.t) + exp(-2.i.π.t))/2

this suggests defining E = exp(2.π) = 535.49 and Exp = (t-> exp(2.π.t)) as turn-based equivalents for e = exp(1) and exp. The power series of Exp isn't quite as tidy as that of exp: Exp(t) = sum(n in [..., 1, 0]: (x/2/π)**n / n! :). Similar factors of 2.π sneak into the formulae for Sin(t) and Cos(t). The differential relations become: Cos' = -2.π.Sin, Sin' = 2.π.Cos, Exp' = 2.π.Exp.

Now, exp(i.π) = -1 and exp(2.i.π) = 1 so we have Exp(i) = 1 and Exp(i/2) = -1; indeed, t-> Exp(i.t) traverses the unit circle, cycling with period 1 so that Exp(i.(n+t)) = Exp(i.t) for all n.

So, now, substitute these functions into the places occupied by the conventional ones to discover how physics looks when repetition is measured in cycles ...

- exp(-Energy/k/Temperature) -> Exp(-Energy/(2.π.k)/Temperature)
- exp(i.x·ω) -> Exp(i.x·(ω/2/π))
- exp(i.t.Energy / hbar) -> Exp(i.t.Energy / h)
- ...

$Id: turn.html,v 1.2 2009-08-09 13:40:04 eddy Exp $