4 ms·

Shout out to another base-12 fan. If only we had 12 fingers it might have happened. Arbitrary divisions of angle aren't very interesting, since the radian is s

by rm445 10y ago

Shout out to another base-12 fan. If only we had 12 fingers it might have happened. Arbitrary divisions of angle aren't very interesting, since the radian is so fundamental. But there is an angle system using multiples of 100: gradians. 100 is a quarter-turn and the internal angles of a triangle add up to 200. A lot of electronic calculators still offer them (the DRG button = degrees, radians, gradians).

jacobolus 10y ago
Gradians are absolutely terrible. For everyday use, radians are also pretty impractical. I personally like just writing fractions of a full turn, but degrees/minutes/seconds aren’t too bad. There are two cases where linear divisions of arclength-measured angles make sense: (1) the angles to make regular polygons of low numbers of sides, (2) repeated binary divisions, whose cartesian coordinates can be computed easily because we can just add two vectors and renormalize using an inverse square root which we know how to efficiently compute, and which are then nice for fixed point representations using binary integers in computers. For the first case, I think rational fractions of a turn work best; for the second case, I like binary angles (“brads”) https://en.wikipedia.org/wiki/Binary_scaling#Binary_angles https://en.wikipedia.org/wiki/Binary_scaling#Binary_angles ; losing the ability to precisely represent a thirds of a full turn is just one of those trade offs you sometimes need to make. Otherwise, treating angle as a linear quantity with precise measurements (instead of some kind of approximations) is for most problems less useful than just using a cartesian coordinate representation for an angle (possibly keeping track of the coordinates squared if we want to stick to rational arithmetic.) Radians are only really useful for solving calculus problems by hand, or writing academic math/science papers where it’s important to stick to established conventions. For practical computation radians are almost always an inferior choice, more computationally expensive and less precise (they’re built into lots of existing libraries, so can often be convenient despite the inefficiency).
- runarberg 10y ago
  > I personally like just writing fractions of a full turn Well just multiply your fraction with τ = 2π and you get radians. E.g. a whole revolution is τ radians (or 360°), half a revolution is ½τ radians or (180°), etc.
- fanf2 10y ago
  Grads are part of how the metric system was constructed: the metre was originally 1/10000 the distance from the North Pole to the equator via Paris, so 1 grad of arc on a great circle is 100km. Compare the nautical mile which is 1 minute of arc.
pessimizer 10y ago
Little Twelvetoes https://www.youtube.com/watch?v=_uJsoZheTR4 https://www.youtube.com/watch?v=_uJsoZheTR4
rhaps0dy 10y ago
We have 12 non-thumb-finger visible divisions, though! http://www.garywallace.net/wp-content/uploads/NF-SCI-0001-P02.jpg http://www.garywallace.net/wp-content/uploads/NF-SCI-0001-P0... Each finger has three numbers, starting with the index and ending in the pinky, and you count pointing to the relevant division with your thumb.
protomyth 10y ago
One of the Native American tribes went with base-8 because they counted the areas between the fingers because you can haul bottles with those. I too wish base-12 were used more. Given how much a penny is worth, getting rid of decimal money and substituting a base-12 coin would solve a lot of pizza / dinner cost split problems.
deleted 10y ago
[deleted]

by rm445 10y ago

jacobolus 10y ago

Gradians are absolutely terrible. For everyday use, radians are also pretty impractical. I personally like just writing fractions of a full turn, but degrees/minutes/seconds aren’t too bad. There are two cases where linear divisions of arclength-measured angles make sense: (1) the angles to make regular polygons of low numbers of sides, (2) repeated binary divisions, whose cartesian coordinates can be computed easily because we can just add two vectors and renormalize using an inverse square root which we know how to efficiently compute, and which are then nice for fixed point representations using binary integers in computers. For the first case, I think rational fractions of a turn work best; for the second case, I like binary angles (“brads”) https://en.wikipedia.org/wiki/Binary_scaling#Binary_angles https://en.wikipedia.org/wiki/Binary_scaling#Binary_angles ; losing the ability to precisely represent a thirds of a full turn is just one of those trade offs you sometimes need to make. Otherwise, treating angle as a linear quantity with precise measurements (instead of some kind of approximations) is for most problems less useful than just using a cartesian coordinate representation for an angle (possibly keeping track of the coordinates squared if we want to stick to rational arithmetic.) Radians are only really useful for solving calculus problems by hand, or writing academic math/science papers where it’s important to stick to established conventions. For practical computation radians are almost always an inferior choice, more computationally expensive and less precise (they’re built into lots of existing libraries, so can often be convenient despite the inefficiency).

runarberg 10y ago
> I personally like just writing fractions of a full turn Well just multiply your fraction with τ = 2π and you get radians. E.g. a whole revolution is τ radians (or 360°), half a revolution is ½τ radians or (180°), etc.
fanf2 10y ago
Grads are part of how the metric system was constructed: the metre was originally 1/10000 the distance from the North Pole to the equator via Paris, so 1 grad of arc on a great circle is 100km. Compare the nautical mile which is 1 minute of arc.

pessimizer 10y ago

Little Twelvetoes https://www.youtube.com/watch?v=_uJsoZheTR4 https://www.youtube.com/watch?v=_uJsoZheTR4

rhaps0dy 10y ago

We have 12 non-thumb-finger visible divisions, though! http://www.garywallace.net/wp-content/uploads/NF-SCI-0001-P02.jpg http://www.garywallace.net/wp-content/uploads/NF-SCI-0001-P0... Each finger has three numbers, starting with the index and ending in the pinky, and you count pointing to the relevant division with your thumb.

protomyth 10y ago

One of the Native American tribes went with base-8 because they counted the areas between the fingers because you can haul bottles with those. I too wish base-12 were used more. Given how much a penny is worth, getting rid of decimal money and substituting a base-12 coin would solve a lot of pizza / dinner cost split problems.

deleted 10y ago

[deleted]