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When I read this article a while back the part that impressed me was how in intuitive radians become: 2/3 * NewPi = (2/3's around the circle in radians).

When I read this article a while back the part that impressed me was how in intuitive radians become: 2/3 * NewPi = (2/3's around the circle in radians). It is a much more intuitive way to think about angles. Ask a third grader, how much cherry pie is missing. "About a two-thirds" they will say. They don't mention PI, and right now no one does. This is the way people think about angles naturally. NewPi makes this more intuitive by allowing you describe angle as a number between 0 and 1 (which is usually the way to go, see splines, animation, etc). 0.25NewPi just makes sense. It is a fourth of a circle, and this way of thinking would help kids understand radians instantly. Probably the only drawback is when doing wicked tricks on a snowboarding game such as SSX. Doing a 1080 just sounds cooler then a 3, but which is more intuitive?=)

amalcon 16y ago
You can do this with regular pi; you only need to use diameter-ans instead of radians. You can do it without a pi at all; there's an actual (used) unit called the "cycle". 1 Hertz is defined as 1 cycle/second. The only problem with these units is that, like with degrees, d/dx sin(x) != cos(x). It messes with calculus.
- jasonkeene 16y ago
  Hah exactly my thoughts!
- aston 16y ago
  Not true. Calculus would be the same (a radian is a radian). sin(pi + x) != cos(x) in the new system, though.
  - zbanks 16y ago
    Chain rule: Let x be angle in radians, y be angle in diameter-ians (2pi) x = 2y d/dx sin(x) = cos(x) d/dy sin(y) = d/dx sin(2x) d/dx (2x) = 2 cos(2x) = 2 cos(y) It's the same reason why you use radians for angles in calculus: degrees mess it up. http://en.wikipedia.org/wiki/Trigonometric_functions#The_significance_of_radians http://en.wikipedia.org/wiki/Trigonometric_functions#The_sig...
    - jacobolus 16y ago
      What? sin(x) is still sin(x). Just because we have a different value for pi doesn’t mean we have a different sin function. sin(6.28...) = sin(6.28...), etc., etc. Crucially, we still have exp(x + iy) = exp(x) * (cos(y) + i sin(y)), because we haven’t changed the definitions of any of these functions. If we did change the definitions of sin and cos, that’d really be a bummer, you’re right. We still measure angles in radians. No diameter-ians in sight. Our old x or new y (notice, those have the same value) is just a different fraction of newpi than it is of oldpi, is all. It amazes me that amalcon’s being voted up and aston is being voted down. People clearly aren’t thinking it through for themselves.
      - amalcon 16y ago
        Just because we have a different value for pi doesn’t mean we have a different sin function. Which is exactly what I just got done saying that I'm not saying. Reading comprehension, much? To be entirely clear: All I was saying is that the reason we use radians in the first place (instead of, say, cycles) is that it fixes calculus. It has little to do with 2pi. It only relates to the comment it was said in reply to.
        jacobolus 16y ago
        > Reading comprehension, much? Yes, I’ve re-read your original post 4 times, and your intended meaning is quite confusing, because you’re talking about a different way of changing our notation than the link is, but without clearly stating that, and your notation change, which you criticize, is something of a non-sequitur in context of the parent comment and the article, as far as I can tell. Thus, you seemed to be implying† that the new definition of pi results in messing up calculus. To clear things up: “We could measure angles in any arbitrary units we want, but using radians makes calculus work, and if we’re using radians, the circumference is 2 pi of them, which is why pi as a unit is not ideal, and newpi = 2*pi would be better. If we wanted we could have an angle of pi ‘diametrans’ in a complete circle instead, using our existing definition of pi ~ 3.14, but that would be stupid, because it would break all kinds of symmetries in calculus.” †: This is apparently a misinterpretation though (mine and also aston’s, who wrote “a radian is a radian”), and you don’t actually mean to be implying that.
  - amalcon 16y ago
    Er, I'm not saying that changing pi would mess up calculus. I'm saying that changing the unit we use for angles (in such a way as to make the existing pi "right") would mess up calculus.
  - sophacles 16y ago
    Sorry, but sin(pi/2 + X) == cos(x) would make more sense to me -- a cos is half a cycle ahead of sin, so the pi/2 term just works with that.
- nitrogen 16y ago
  There's also radians/sec, which comes up in network analysis. Radians correspond directly with things in the real world, like analog filters (cutoff frequency omega of a single-pole filter in rad/s=1/RC). Radians/sec are used because the math is a lot simpler. Switching to Hz puts pi into the equations, and switching to three-legged pi (thri?) would put 1/2 in the equations.
makmanalp 16y ago
... or the 0.5 pir^2 and 0.5 gt^2 link, it's really eye opening to people who did not quite /get/ the math. Math, in educational sense, (not in the purist proofs sense) is all about seeing the links between things. Geometry and calculus are fundamentally related. We're still stuck teaching kids "sohcahtoa".
- jacobolus 16y ago
  Yup, we’d be better off if we had a name for cosine that clearly implied "x coordinate", and a better name for sine that implied "y coordinate", (or at the very least, were shorter to spell and easier to say) and just did away with tangent, secant, etc. altogether, at least for basic pedagogy (1/sin(x) is almost always way clearer than csc(x)). Then we could make up a good name for the 2-argument arctangent function that inverts the function f(x) = cos(x), sin(x), and call it something much more intuitive than arctan2. Everyone would be much better off.

amalcon 16y ago

You can do this with regular pi; you only need to use diameter-ans instead of radians. You can do it without a pi at all; there's an actual (used) unit called the "cycle". 1 Hertz is defined as 1 cycle/second. The only problem with these units is that, like with degrees, d/dx sin(x) != cos(x). It messes with calculus.

jasonkeene 16y ago
Hah exactly my thoughts!
aston 16y ago
Not true. Calculus would be the same (a radian is a radian). sin(pi + x) != cos(x) in the new system, though.
- zbanks 16y ago
  Chain rule: Let x be angle in radians, y be angle in diameter-ians (2pi) x = 2y d/dx sin(x) = cos(x) d/dy sin(y) = d/dx sin(2x) d/dx (2x) = 2 cos(2x) = 2 cos(y) It's the same reason why you use radians for angles in calculus: degrees mess it up. http://en.wikipedia.org/wiki/Trigonometric_functions#The_significance_of_radians http://en.wikipedia.org/wiki/Trigonometric_functions#The_sig...
  - jacobolus 16y ago
    What? sin(x) is still sin(x). Just because we have a different value for pi doesn’t mean we have a different sin function. sin(6.28...) = sin(6.28...), etc., etc. Crucially, we still have exp(x + iy) = exp(x) * (cos(y) + i sin(y)), because we haven’t changed the definitions of any of these functions. If we did change the definitions of sin and cos, that’d really be a bummer, you’re right. We still measure angles in radians. No diameter-ians in sight. Our old x or new y (notice, those have the same value) is just a different fraction of newpi than it is of oldpi, is all. It amazes me that amalcon’s being voted up and aston is being voted down. People clearly aren’t thinking it through for themselves.
    - amalcon 16y ago
      Just because we have a different value for pi doesn’t mean we have a different sin function. Which is exactly what I just got done saying that I'm not saying. Reading comprehension, much? To be entirely clear: All I was saying is that the reason we use radians in the first place (instead of, say, cycles) is that it fixes calculus. It has little to do with 2pi. It only relates to the comment it was said in reply to.
      - jacobolus 16y ago
        > Reading comprehension, much? Yes, I’ve re-read your original post 4 times, and your intended meaning is quite confusing, because you’re talking about a different way of changing our notation than the link is, but without clearly stating that, and your notation change, which you criticize, is something of a non-sequitur in context of the parent comment and the article, as far as I can tell. Thus, you seemed to be implying† that the new definition of pi results in messing up calculus. To clear things up: “We could measure angles in any arbitrary units we want, but using radians makes calculus work, and if we’re using radians, the circumference is 2 pi of them, which is why pi as a unit is not ideal, and newpi = 2*pi would be better. If we wanted we could have an angle of pi ‘diametrans’ in a complete circle instead, using our existing definition of pi ~ 3.14, but that would be stupid, because it would break all kinds of symmetries in calculus.” †: This is apparently a misinterpretation though (mine and also aston’s, who wrote “a radian is a radian”), and you don’t actually mean to be implying that.
- amalcon 16y ago
  Er, I'm not saying that changing pi would mess up calculus. I'm saying that changing the unit we use for angles (in such a way as to make the existing pi "right") would mess up calculus.
- sophacles 16y ago
  Sorry, but sin(pi/2 + X) == cos(x) would make more sense to me -- a cos is half a cycle ahead of sin, so the pi/2 term just works with that.
nitrogen 16y ago
There's also radians/sec, which comes up in network analysis. Radians correspond directly with things in the real world, like analog filters (cutoff frequency omega of a single-pole filter in rad/s=1/RC). Radians/sec are used because the math is a lot simpler. Switching to Hz puts pi into the equations, and switching to three-legged pi (thri?) would put 1/2 in the equations.

makmanalp 16y ago

... or the 0.5 pir^2 and 0.5 gt^2 link, it's really eye opening to people who did not quite /get/ the math. Math, in educational sense, (not in the purist proofs sense) is all about seeing the links between things. Geometry and calculus are fundamentally related. We're still stuck teaching kids "sohcahtoa".

jacobolus 16y ago
Yup, we’d be better off if we had a name for cosine that clearly implied "x coordinate", and a better name for sine that implied "y coordinate", (or at the very least, were shorter to spell and easier to say) and just did away with tangent, secant, etc. altogether, at least for basic pedagogy (1/sin(x) is almost always way clearer than csc(x)). Then we could make up a good name for the 2-argument arctangent function that inverts the function f(x) = cos(x), sin(x), and call it something much more intuitive than arctan2. Everyone would be much better off.