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> We name things based on what people working in a field find most convenient for their work My point is that I with kore than a decade of working theoretical

> We name things based on what people working in a field find most convenient for their work My point is that I with kore than a decade of working theoretical physics I have NEVER seen anyone, student, professor or otherwise not use revolutions as their unit of concern. However they do it by squeezing together 2Pi (or very few by using tau) as a pseudo-unit. Imagine if we stopped using meters and instead used a unit we called meterians which was 1/sqrt(2) meter, yet everyone always in every single equation or table listing wrote out “72 sqrt(2) meterians” would that not be stupid? As for redefining sin and cos, there really is no need you almost always see equations using cos(x pi) and the “annoying factors” you get in derivations still pop out, just not from the function itself, but instead from the factor you’ve put in place so you can keep dealing with rotations or half rotations while pretending to use radians. As for concerns about other shapes it’s literally the case that the opposite of what you proclaim is true. If you rotate any other shape than a circle then there is no connection between radians as a length along the perimeter and rotation. A unit square requires 8 perimeter length units to rotate. By any kid even before they learn to multiply will be able to show you exactly that rotating a square once completely is one rotation. The fact that you might personally say “no that’s not intuitive! We should say that it’s 1 times (8 radial perimeter units) just shows that you’ve disconnected the math from what’s intuitive. And changing how we deal with trigonometric formulas to drop the constant reference to pi, even though we’re multiplying it in and deciding net it away, doesn’t change anything, the fact that it shows up everywhere in formulas is just a concealed even of mathematical robustness, if we decided instead to measure angles in units of 1/200*(4pi^2)/tau as is done in land measurements doesn’t make the math simpler or harder, it just changes the scaling factor. But there is the benefit that people immediately and intuitively understand the measurements written down without need of a calculator, though that would also be the case with simple fractions and direct percentages.

rsj_hn 5y ago
> My point is that I with kore than a decade of working theoretical physics I have NEVER seen anyone, student, professor or otherwise not use revolutions as their unit of concern. I think this is false. Everytime someone uses a trig identity or takes a derivate, they are using arc-length as their unit of concern. You may not have realized it, but that is what was happening. To speak merely of "rotation" and be disconnected from any kind of length or position in coordinate space doesn't allow you to do anything useful, and you have a degree of freedom in choosing what is a full revolution and now you have to do bookkeeping with coordinate transformations that ugly up all your formulas and and add extra steps. The radian is the natural unit to use: 1 unit of distance of arclength. You are not better off with a fake degree of freedom and then transforming back and forth whenever you need to work with any kind of function relating that angle to a length or rate of change. Moroever when you start generalizing to other curves -- for example, hyperbola and working with hyperbolic sines and cosines, it no longer makes sense to speak of rotations per se as these need not be periodic at all, but it makes a lot of sense to speak of arclength. Arclength works on any curve. It is just distance, it is canonical. It is the natural unit that measures how far you have travelled along a curve (once you select an orientation), thus allowing for a consistent approach and unification of all these identities.

rsj_hn 5y ago

> My point is that I with kore than a decade of working theoretical physics I have NEVER seen anyone, student, professor or otherwise not use revolutions as their unit of concern. I think this is false. Everytime someone uses a trig identity or takes a derivate, they are using arc-length as their unit of concern. You may not have realized it, but that is what was happening. To speak merely of "rotation" and be disconnected from any kind of length or position in coordinate space doesn't allow you to do anything useful, and you have a degree of freedom in choosing what is a full revolution and now you have to do bookkeeping with coordinate transformations that ugly up all your formulas and and add extra steps. The radian is the natural unit to use: 1 unit of distance of arclength. You are not better off with a fake degree of freedom and then transforming back and forth whenever you need to work with any kind of function relating that angle to a length or rate of change. Moroever when you start generalizing to other curves -- for example, hyperbola and working with hyperbolic sines and cosines, it no longer makes sense to speak of rotations per se as these need not be periodic at all, but it makes a lot of sense to speak of arclength. Arclength works on any curve. It is just distance, it is canonical. It is the natural unit that measures how far you have travelled along a curve (once you select an orientation), thus allowing for a consistent approach and unification of all these identities.