The proponents of tau that cite the ubiquity of the 2pi factor in maths seem to forget that it's an exceedingly huge swath of knowledge and thus any notion of ubiquity might as well be moot, or totally dependent on the region of maths this person frequents. I wouldn't be surprised if there are some domains of maths where even natural numbers are used rarely or almost never (formal logic springs to mind). That's why it's possible to cherry-pick favourable examples which support either side of pi-tau debate.
But you might say, of course, that those fields of mathematics which tau makes "easier" are the ones important in early math education (especially below the university level). Hence adopting tau would make them substantially friendlier and more intuitive to many people. While the idea of "fixing" math concept to make them more bearable to laymen should not be dismissed automatically, I would like to point out that the question of tau vs. 2pi is by no means the only issue of this kind. Indeed, there are a couple of more "warts" in everyday maths that could also warrant "fixing". Consider:
* The direction in which positive and negative angles on two-dimensional, Cartesian plane are measured [1]. Counter-intuitively, the measure increases when going counterclockwise, while going clockwise decreases it.
* The main diagonal [2] of a matrix goes from upper-left to lower-right corner, which coincides with the shape of backslash character.
* The established order of indices for matrix' elements is row-column, so that A_xy refers to element in x-th row and y-th column of matrix A. This goes against the habit of specifying the horizontal coordinate before the vertical coordinate when talking about XY planes [3].
* Definition of convex [4] and concave [5] functions (for R->R ones) do not agree with the intuitive associations based on plots of those functions. Clearly, the convex one looks like a valley, and the concave one resembles a hill or mountain.
I'm sure there are many more examples of such unreasonable, counter-intuitive conventions, so we really have a lot of work ahead of us. So, anyone fancies writing the Slash Diagonal Manifesto?...
* Counter-clockwise angles comes from making charts and graphs with the independent variable going from left to right, and the dependent variable from bottom to top. This is far too ingrained in the way we teach and learn about mathematics to change now, much more than the choice of pi or tau (though we do make y go from top to bottom in many computer graphics contexts). There’s no reasonable way that we could integrate direction of reading on a clock with direction of reading typical charts with direction of reading text into a uniform system, and I don’t consider changing all the clocks, texts, and charts in the world to be a reasonable possibility.
* What does the shape of the slash character have to do with anything? A slash is a fraction bar. Fractions and matrix diagonals are entirely unrelated, though in both cases numbers are read from top left to bottom right, in accordance with our typical reading direction in western texts.
* Matrix index ordering is a pain in the butt and will be confusing whichever way they’re labeled. The logic behind the current system is to use the first index for the component that results when multiplying by a vector, and using the second index within that component. Picking the opposite convention would also end up confusing. Figuring out the proper ordering when dealing with non-commutative “number” systems in general is a pain, and I don’t think there’s any easy answer. We have matrices multiply column vectors on the left, because that’s typically how we notate operators acting on some input. But it means that composition is multiplication from left to right, which is a bit confusing. There’s no way to make the order be always left-to-right or always right-to-left. But much more importantly, matrices are a kind of painful abstraction to use in general. Mathematics education would be much improved in many ways if we used Geometric Algebra instead of matrix representations a lot more of the time. http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf
* It’s much easier to just call these “concave up” and “concave down”. Problem solved.
But you might say, of course, that those fields of mathematics which tau makes "easier" are the ones important in early math education (especially below the university level). Hence adopting tau would make them substantially friendlier and more intuitive to many people. While the idea of "fixing" math concept to make them more bearable to laymen should not be dismissed automatically, I would like to point out that the question of tau vs. 2pi is by no means the only issue of this kind. Indeed, there are a couple of more "warts" in everyday maths that could also warrant "fixing". Consider:
* The direction in which positive and negative angles on two-dimensional, Cartesian plane are measured [1]. Counter-intuitively, the measure increases when going counterclockwise, while going clockwise decreases it.
* The main diagonal [2] of a matrix goes from upper-left to lower-right corner, which coincides with the shape of backslash character.
* The established order of indices for matrix' elements is row-column, so that A_xy refers to element in x-th row and y-th column of matrix A. This goes against the habit of specifying the horizontal coordinate before the vertical coordinate when talking about XY planes [3].
* Definition of convex [4] and concave [5] functions (for R->R ones) do not agree with the intuitive associations based on plots of those functions. Clearly, the convex one looks like a valley, and the concave one resembles a hill or mountain.
I'm sure there are many more examples of such unreasonable, counter-intuitive conventions, so we really have a lot of work ahead of us. So, anyone fancies writing the Slash Diagonal Manifesto?...
[1] http://en.wikipedia.org/wiki/Angle#Positive_and_negative_ang... [2] http://en.wikipedia.org/wiki/Main_diagonal [3] http://en.wikipedia.org/wiki/Cartesian_coordinate_system#Car... [4] http://en.wikipedia.org/wiki/Convex_function [5] http://en.wikipedia.org/wiki/Concave_function