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> Of course you can work in other units, but you'll need to insert the appropriate scaling factors all over the place.

You probably take out more scaling factors than you introduce.

> Euler's formula (e^ix = cosx + isinx) is the simplest when working with radians.

Euler’s still simple:

e^(2 i pi y) = cosy + isiny

Or if you start noticing c = e^(2 pi) showing up all over the place:

c^iy = cosy + isiny

> How do you do the same with "turns" on a sphere?… You can't in any meaningful way.

Why not do the same thing? One steradian is 1/(4 pi) of a sphere’s solid angle. What if one “steturn” or whatever just covered a full solid angle? And similarly for higher dimensions?

Neither definition seems more natural to me, especially being used to all the factors of 2 and pi that pop over all over the place in the status quo.