A new section of The Tau Manifesto: Getting to the bottom of pi
tauday.com
tauday.com
pi is not a lie. It's not part of some conspiracy theory among math elites to mislead the masses. It's not about math bureaucrats refusing to admit they were wrong. Everything they told you about pi is true (or it isn't, but in that case it's not true of tau either). pi and tau are really just two different notations for the same concept. Only perhaps one is more elegant and convenient than the other, making it more fundamental in some mystical way (and perhaps preferable if you ever need to convince visiting aliens that you are intelligent).
For the former, you just need to know your meme:
http://www.urbandictionary.com/define.php?term=the+cake+is+a...
For the latter, you need to know that it comes from Bob Palais' original article.
For both, you might need to lighten up.
...imagine we lived in a world where we used the letter h to represent “one half” and had no separate notation for 2h. We would then observe that h is ubiquitous in mathematics. In fact, 2h is the multiplicative identity, so how can one doubt the importance of h? But this is crazy: 2h is the fundamental number, not h. Let us therefore introduce a separate symbol for 2h; call it 1. We then see that h=1/2, and there is no longer any reason to use h at all.
Color me converted as well.
[My analogy was not very good. The circumference is the only dimension of a circle stated in terms of diameter, not radius, and so substituting tau for pi is simply relating the radius of a circle to its circumference instead of its diameter. Circumference = tau * radius. This has the benefit of bringing the circumference formula in line with all the other circle equations.
When we look at the area of a circle, tau is not very intuitive, however. The formula for the area of a circle uses the square of the radius, so there's not an intuitive way to factor two into the equation without just making extra mental work for yourself.
For volume, the effect is somewhat neutral. Volume = 4/3 pi * radius^3 = 2/3 tau * r^3. 2/3 vs. 4/3 doesn't seem to add much insight into the equation.
Circumference = tau * r
Area = 1/2 tau * r^2
Volume = 2/3 tau * r^3
4th dimension circle = 3/4 tau * r^4? : P
Maybe there's something to this...]
EX: e^(i * tau / 2) = -1 wait what?
Just think of it like an integral (that's what it is, after all!), raise the power and divide by the power. Suddenly the origin of this formula is no longer obscured.
Regarding Euler's identity: e^(i * tau) = 1
Edit: Also that's is hardly a more fundamental equation because you could also say e^(1024 pi * i) = e^(4096 * tau * i) = 1.
The great thing about math is it's it works out either way. So, it really comes down to which is easier to deal with and students seem to have an easier time remembering and understanding 2pir and pi*r^2, but in the end it's tau day so that's what people want to argue about.
Are you speaking from empirical experience as a math teacher?
What makes 2 pi r easier to understand than tau r?
In fact, one can view a circle as being made up of a bunch of really thin triangles (this is how one usually derives the area of a circle), and this again is something that third-graders can appreciate--cut a paper circle into slices, and see if you can get the slices to look more and more like triangles as you cut them smaller. (I vaguely remember doing something like this in elementary school.)
A fourth dimension perfect circle looks like a sphere at first glance, but has the maximum possible internal surface area upon closer inspection.
2π is arguably more fundamental than π, but given the things above, I don't care much.
If you want to tackle something worthwhile, update theoretical physics lectures to 'modern' notation (where modern means 1960s).
That's one of the things tau can help solve. I suspect the main reason that people still use degrees so much is because using pi undermines the beauty and intuitiveness of radians.
I could have used 0.14 τ to make my point as well: If you want to avoid fractions in common cases, you'll need to multiply with an arbitrary number like, say 100 (which has obvious benefits in a base-10 system), but arguments could be made for something like 400 (in which case we end up with gon) or, say 360, which has the benefit that it's the established standard (even though it's not the most obvious choice).
Actual radians (ie not expressed as fractions of 2π) are pretty user-unfriendly in a base-10 system.
I think this is the source of your problem: you view an expression like "τ/8" or "π/4" as less an "actual" radian measure than the inexact ".785". That's a function of how a lot of us were taught math, I think, and I'm quite sure that it's only made worse by the fact that the use of π camouflages the fact that this is related to fractions of a circle. But your premise is faulty. Of course radians expressed in decimal are user-unfriendly, but radians expressed in decimal are no less "actual" than radians expressed as fractions of τ or π.
There are applications for which neither radians nor turn are a good fit. Similarly, oftentimes the 'messy' SI units are a far better fit than any 'more fundamental' system of natural units.
People can't be bothered to change, except radians are much more mathematically easy to use, so people who need to do advanced maths will use it.
However, laziness is also the reason why people don't change to tau. tau is just a multiple of pi. Confronted with such a small change, people will just think that they can't be bothered to change, especially since pi is already widely supported and implemented in any of the tools that we use, and tau is not.
There's no way that changing to tau will make more people use radians.
\[ \int_0^{2\pi}\int_0^\infty f(r, \theta) r dr d\theta \]
The upper limit of the $\theta$ integration is always 2$\pi$."
This statement is false. If $r = r(\theta)$, then it can be the case (e.g., for a "looped" limacon $r = \cos(\theta)$) that $\theta$ will not range from $[0,2\pi]$. The upper limit is not necessarily "always" $2\pi$.
The "epiphany" refers to the specific realization that there are three families of constants, and in particular that pi is not a member of the family of volume constants but rather is part of a useless family of its own.
In any case, you certainly deserve mention, so I've added you to the acknowledgments. Thanks!
So, up with tau!
Sure they do—or at least, they did. Physicists realized that h is "wrong", i.e., confusing and unnatural, because it's off by a factor of, um, 2 pi. They introduced h-bar precisely to rectify the problem. And the substitution worked: if you open up a standard book on quantum mechanics, the ratio of uses of h to h-bar is just about epsilon. (You'll also see lots of (2 pi)s.)
The difference between "robot" and "postlapsarian" is "just semantics", after all.