I don't understand when the chords come into play - I don't understand the rules.
But seems like you could generate nice, non-repeating elevator music by hooking something up to an RNG or PRNG, and never hear the same song in the elevator twice.
The earliest example I've seen of 2π is in a 1763 letter from Thomas Bayes (the paper that appeared in the Royal Society proceedings directly after the one that's famous). He used c for the circumference of a circle whose radius is unity.
If you've ever used Stirling's approximation, this is the paper that first points out that it's a divergent series.
The most compelling argument for me in using tau, (and I have started trying to think in tau when it comes up) is the radians argument: one quarter of a circle is tau/4, or pi / 8, you pick.
I am certain my kids will have an easier time remembering tau/4, as I do myself.
The other compelling thing for me came from remembering just how many integrals from 0 to 2pi I wrote over my freshman complex analysis class. A lot. Less notation is always nice; having tau represent the entire circle just makes a lot of sense!
I'm a Tauist myself, but it should be pointed out the first one is a legitimately good argument. (Local introduction of a constant is easy, though.)
I don't expect to wake up one day and everybody suddenly agrees "Yes, tau is the winner!" I expect that either things will peter out, or tau will just gradually start showing up in real papers and stuff. Unfortunately, since K-12 mathematical curricula seem to have gotten stuck in 1920, switching the "official curricula" to tau is well down on my list of things that needs to happen to K-12 math education and at the current rate even if formal mathematics did just wake up tomorrow and decide tau was the way to go, it would be at least 50 years before that penetrated back down.
I've dozens of subtle little reasons, but I think that one shows it off best and is easiest to understand.
I was going to add that zero crossings for sine waves (I am into sound synthesis) are at integer multiples of pi, but that's just a funny way of stating the above.
I keep seeing this in my comments so I might as well respond for posterity, especially since this is not a sound mathematical argument for pi.
> zero crossings for sine waves are at integer multiples of pi
This is actually a strong argument for tau.
The sin wave measures the height of a circle at the angle given in radians. "Integer multiples of pi" don't immediately show you that there are two very different zeros: one going up, and one going down. Using tau shows you that explicitly: on half turns around the circle sin(tau/2), you're at 0 going down; on whole turns sin(tau) you're going up. You (literally) "come full circle" with integer multiples of tau—those 0s are equivalent.
The argument is the same with e^(i * pi * x). See Section 2.3 on tauday.com and the chart under "Eulerian Identities." Each integer increment corresponds to a rotation in the complex plane. The reason it's on the real line at 2, 4, 6 is because it takes two rotations to get back to the real line.
At 1 rotation (i), you're fully imaginary; at 2, fully real, but negative; 3, fully imaginary again, but negative; 4, you're back where you started, real and positive.
I'm sorry but your comment came off as condescending.
The graph I linked coincides with my needs, it was specially crafted to demonstrate that, not to show off some silly notion of elegance with no ounce of application. Sorry for not making that clear.
As for: Elegance is not just whether something is "pretty," as in, hey look, integers! It's also whether it has strong meaning. -- please try applying that value to your opinions about giving up pi for tau. I'm not the one guilty of that kind of thinking, you are.
> The mathematical world is as full of lonely pi's, as it is of 2*pi's. Now we need to move to tau/2 and tau, only to get a pi-manifesto in a couple of decades.
Again, I ask: have you read the Tau Manifesto? If not, I suggest you do that, as it makes the case there much more clearly than I could here.
The point being: for children learning the basics, the odd 2 or 1/2 here and there masks the underlying relationships. But don't take my word for it-- read the manifesto.
I read it a few years ago. I disagree. It's like saying removing a seldom used letter from the alphabet would mean kids learn to spell better.
The whole point of learning is to learn to read things and understand them. That means looking behind 'fluff' like 2, or 1/2, and understanding the concepts.
The right sentiment, but I don't fully agree with this.
There are a lot of people who aren't kids, aren't mathematicians, but have to deal with maths all the time. In fact, our own field of Computer Science has a lot of math.
Now, I can't speak as a real mathematician (though I know more maths than most people), but I can definitely speak as a CS major - learning to use radians was never natural for me, and it is the basis of most of calculus. Had I learned it using Tau, I'm guessing it wouldn't have even been an issue. It wouldn't even be something I have to "learn". It just makes sense - one half of a circle is one half Tau.
The mathematical world is as full of lonely pi's, as it is of 2*pi's. Now we need to move to tau/2 and tau, only to get a pi-manifesto in a couple of decades.
I like the compromise of using Tau and its fractions when it makes sense and using a single Pi when it's not so intuitively-connected with a circle. e.g. \int_{-\infty}^{\infty} e^{−x^2} dx = \sqrt{\pi}.
I'm not a big fan of introducing a new constant (though I believe \pi should have been 2\pi), but I love thinking of the integral you wrote down as \sqrt{\tau / 2} because then the answer practically tells you how to derive it!
How to derive the value of the integral: Square the integral to make it an integral in two variables, introduce polar coordinates, then change variables.
Yes, it's one of my favorite proofs. (I think I like it more than Euler's formula, especially since many calc teachers will look at e^{-x^2} and say it's un-indefinite-integrable without a second thought at what else it can do.)
But I'm not quite sure how you seeing it as \sqrt{\tau/2} helps you see the proof more easily. Because if you see \tau (ignoring the 1/2) you think "It has to do with circles or polar form." as per my rule of thumb?
I'm sorry I wasn't more clear, but your interpretation is what I meant. Per your rule of thumb, seeing \tau should suggest that it has to do with circles or polar coordinates, and the square root points to how to get the polar coordinates.
If it's not broken, don't fix it. I don't see pi broken as it has been used for _centuries_. Why should we change it all of a sudden? To signify that we're into a new era? The Tau era?
Why did we move from Roman Numerals or other systems of writing down numbers, to the current numerals we use today? Simple - they make lots of things easier.
Now, no one is claiming that Tau vs. Pi is even close to the same level of importance. But it makes some things just that much easier.
"The fact is that making things easier doesn't necessarily mean improvement."
Untrue, and if you're a programmer you ought to know better. Making something that works the exact same way, only easier, is definitely an improvement. Now it takes less of your finite mental reserves to accomplish a task, and you can now go further in the same amount of time. Making something that abstracts away some things and makes the rest easier is often an improvement, when the advantage of being easier outweighs the loss of control.
Or are you still programming in raw machine language?
Programmers live in such a rich ecosystem of things that are improvements merely because they are easier that it is easy to take that process for granted without understanding it. How many orders of magnitude less effective would I be in machine language? Certainly more than one, almost certainly more than two (working on a network + manual memory management = security fail).
Okay, I failed to properly address "improvement". By "improvement" I mean the time required to develop something new. Of course everything improves, simply because it's built on top of the base of an older version.
A simple example, it took months to write a simple browser with a little less features than the browsers available during the Win 9x era. Does it take weeks now? I doubt so.
Ok, to put it more simply, what I meant was improvement in the time required not as in the thing that was produced. Uh, get what I mean?
Ooooo, bad choice of example. It takes a day or so now, since WebKit is embeddable.
You're trying to separate something that can't be separated. You can't separate "making things take less time" from "making better things possible", because the bound in all cases is time. If you have to spend less time on X, then you've got more to spend on Y; if you can't spend less time on X you'll never reach Y. It's the rare improvement that can only improve quality, but doesn't in any way permit you to instead trade that for time.
WebKit is embeddable, sure, but if you're to write a browser that functions similarly to today's browser, it still takes months. You're gonna need to do bookmarks, tabs, history. Things have been made easier, surely but things are still as complicated as ever.
However this is drawn too far from the topic of Tau, what I'm trying to imply is that science and maths has work centuries with Pi, we've found many great theories with it as well. But in the modern era (quantum), Tau seems irrelevant (okay, I haven't studied too much about quantum, but I've read on articles and there's no mention on Pi either, correct me if I'm wrong).
What's the use of making things simple at the same time confusing people? Things have always worked out, and it should continue to work on.
The Greek letter tau is already used to refer to the period of an oscillation, the time constant of a decay interval (these are intimately related), plus plenty of other stuff. There really aren't any Greek letters left that aren't used for a million things already. tau-as-time-constant is the standard use for the thing, and the confusion with torque and natural temperature is bad enough as it is.
Yes, pi shows up as the prime counting function, but there it's a function, which clears up the otherwise ambiguous notation. Furthermore these abuses of notation are generally considered a bad thing, something we try to avoid.
As for the intuitiveness of such deep results as Stirling's formula and the even values of the Riemann zeta: this is to concern oneself with the upholstery on the Space Shuttle.
If you want a new pi symbol, might I suggest the variant pi described here: http://en.wikipedia.org/wiki/Pi_%28letter%29 -- though I'm afraid this is all a waste of time and energy.
It is not an exercise worth ignoring: I had also once thought that pi should be replaced.
Slightly different motivation, though, as it seemed to me that pi/2 was really more interesting, and anyway it is really much easier to write multiplication than fractions when you run into a discrepancy. Here the roots of the sine are at even multiples and the roots of the cosine are at odd multiples, and e^(i pi/2) = i is more interesting than the previous incarnations of the Euler formula.
The thing is that the important parts of these formulae are the way they are derived and the structure they represent, not their appearance on a sheet of paper. If you can understand what relates the zeta function at even numbers to the ratio of a circle's radius and circumference, it really does not matter if you choose to represent this relation with a drawing of an elephant.
You got a good case with pi/2, it's truly useful, as it represents the angle between 3d axes, the boundaries of the tangent and inverse trigonometric functions, the monotonic regions of cosine and sine etc. It will even make spherical coordinates easier to visualize mentally. (It will also make a two-digit number, 16, a common resident in lots of formulas). I vouch for it, and i call for a move to name it, confusingly, pi-bar
(Yes, we know that Tau doesn't really sound like anything, but this was fun and better than I expected.)