They'll also laught at the west for seeing chess as the ultimate intellectual game instead of Go.
While the Baroque rules of Chess could only have been created by humans, the rules of Go are so elegant, organic, and rigorously logical that if intelligent life forms exist elsewhere in the universe, they almost certainly play Go. - Edward Lasker
Because even at high level math that's pretty annoying. As much as I agree with the philosophical force behind \tau = 2\pi, the conventional force behind rarely having to define or question the definition of \pi is bigger.
Have you tried it? It’s really not too bad, unless you’re in some context where τ is expected to mean something else – and those are relatively rare – or unless you’re simultaneously flouting other conventions.
Yes, it's honestly super trivial to make the substitution. It's a really minor translation cost on the myriad of formulae that are written involving pi.
But if I see "define tau = 2 pi" at the beginning of a paper I'm going to have a hard time taking it seriously. It's got useful pedantic purposes, but frankly it's so trivial at the level of good maths that the only reason anyone would write that is political. And then it's a minor headache.
And then mathematical curtesy dictates that you omit excess.
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Similarly, why do we write numbers in base 10? It's an arbitrary convention and in many contexts other bases might be strictly beautiful (conceded for argument's sake). I'll just write at the top of my paper that all numerals are written in octal for the sake of beauty.
The context of this discussion was graders marking people down on homework assignments, not published journals.
You’re right that it’s political. Changing any kind of convention is always political. [By the way, I believe you mean pedagogical rather than pedantic.]
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The use of octal or some other number notation system instead of decimal causes a dramatically larger conversion difficulty for unfamiliar readers than the use of τ does. It’s not a comparable distinction.
By your logic equating all such choices, no one should ever use a minority notation for anything, even if they think it has substantial benefits.
Thankfully, not everyone agrees, and sometimes our notation improves (A couple examples I particularly like are Knuth’s use of the Iverson Bracket in writing sums, and [] and {} notation for Stirling numbers.)
As a matter of fact, there are contexts where writing numbers in octal is not just "beautiful", but useful, and is actually done. See asciitable.com and the Unix chmod utility (chmod 755 script.sh) for some common examples off the top of my head. And, of course, programmers and computer scientists deal with numbers in binary and hexadecimal all the time, and you do find numbers written in such bases in academic and other papers.
Also, the question "How many factors of the prime p are in n factorial?" is easy to answer if n is written in base p (0 * [ones digit] + 1 * [p digit] + (p+1) * [p^2 digit] + (p^2+p+1) * [p^3 digit] + ...), and one can take this in the other direction--"Find the smallest n for which p^k divides n"--without much difficulty. This is how I'm planning to answer a particular Project Euler problem, and if I were writing up my solution, I would write some numbers in base p.
Methinks you picked a really bad example to illustrate your point.
The things I find harder to do in bases other than 10 are: recognize numbers (primes/squares/cubes/triangular/Fibonacci/pentagonal/etc), factor them, do arithmetic (though I'm fairly accustomed to doing it in binary), and judge magnitudes. If a paper dealt with these things, and there were no apparent benefits to using octal or whatever but the paper used it anyway, then I'd probably be annoyed. This is probably what you meant to refer to with your example.
Now, you said the numbers would be written in octal "for the sake of beauty". I don't know why someone would find it "beautiful" to write numbers in octal unless they turned out simpler for some reason--e.g. if all the numbers in question turned out to only have the digits 0 and 3. Which would either be a coincidence--and in that case I think a mathematician would find it disappointing rather than beautiful, to see a pattern which turns out not to be robust or to reflect any underlying truth--or it would be the result of some underlying truth, in which case it likely would be better to write it in base 8. The optimal strategy in two-player Nim is best understood and implemented when the numbers in each pile are written in base 2, because you need to compute XORs.
Many of the arguments for tau are based around beauty, yes. This is because many of the arguments for pi are based around beauty. However, there are also usability arguments, which seem to address what you're concerned with. I don't think you can disagree: that it is easier to know of the 3rd and 6th roots of unity as cis(τn/3) and cis(τn/6), instead of cis(2πn/3) and cis(πn/3)--or cis(2πn/6)); easier to reason that a wheel that makes 12 rotations in 5 seconds spins at a rate of 12τ/5 radians per second, rather than 24π/5 radians per second; and easier to remember formulas and facts involving τ... ok, this one seems disputable, but I think I can actually argue for it.
If the only formulas that existed had "2π" printed right on them, then you could accustom yourself to treating "2π" as an atomic concept, and it wouldn't make much difference if we wrote them with "2π" or "τ". However, instead, we have a bunch of formulas with 2π, a fair number of formulas with [some factor]π (e.g. "Coulomb's constant k = 1/(4πε)" and "volume of sphere = 4/3πr^3"), and a couple of formulas with π by itself. This is three patterns to recognize and remember, as opposed to two: τ and [some factor]τ. [1] And your attempts to interpret 2π as a thing in itself will be confounded by your need to interpret π as a thing in itself, both when you think about the formulas with [some factor]π, and when you manipulate expressions and do arithmetic. (We saw above the results of plugging k=6 into 2πn/k; note that this kills the 2π abstraction even though the formula has 2π.)
Basically, your understanding of the circle constant will be fragmented into π and 2π. This was true for me even before I read "The Tau Manifesto", perhaps (not sure) before "π is wrong". If you think in terms of π, you're bound to notice the 2π in the underlying pattern when it comes up; you try to put everything in terms of 2π, and you become disappointed when it fails to simplify some expressions (the 1/(4πε) becomes 1/(2(2π)ε)), and frustrated when arithmetic demands that the 2 be cancelled (obscuring the underlying 2π). You'll probably try for a while; give up, somewhat dissatisfied; and then forget about trying to make sense of the situation. Whereas with τ, you never ever need to think about whether or not this 3π/4 is better represented or thought of as 3/8(2π), whether you should cancel out factors of 2, whether "2π 4π 8π 16π" is really π * 2^n or 2π * 2^(n-1); the cases where the circle constant stands by itself appear without any special arithmetic tricks, and the cases where it's stuck with baggage are immediately plain.
There's probably a reason physicists came up with an entire symbol to represent Planck's constant divided by 2π. It just sucks when your atomic concept isn't an atomic symbol and is likely to get broken up or partially destroyed by arithmetic.
As I've written this, I've become convinced that things really would have felt much better and nicer had I been using τ. So it is important, it makes a significant difference; and this, plus external support (today's event, upvotes, friends' approval, and Vi Hart's video) make me more confident that we will succeed in changing it.
[1] As a side note, what makes the formulas with τ beautiful--or any formulas in general--is the same thing that makes them easier to remember. Fewer different patterns to deal with. Running the set of all formulas in your head through a compression utility would probably produce a smaller output.
Any pedagogical institution flunking students for labeling and using constants when solving math problems should be burned down and the ground upon where it stood strewn with salt so that life never again will prevail there.
I taught science and technology at a secondary school for a while and through the process I had to get certified to teach in the state of Virginia. Part of the certification process is passing the math praxis. If I remember correctly, all teachers in the state of Virginia must pass the math praxis I. I was surprised (actually dismayed) at the large number of my colleagues who thought it was challenging or had to take it more than once before passing. Here are some sample questions: http://www.studyguidezone.com/praxis_math.htm
My wife (who also got certified to teach in Virginia) just told me that many of her colleagues couldn't pass this simple math test and several had to take it multiple times before passing. She said that some teachers simply couldn't pass it so chose to teach in North Carolina for a while and then move back to Virginia (because after so many years the teaching certification from another state was automatic).
tl;dr: The tech leaders of tomorrow will not be from America.
I love this video. Vi Hart says, "No! You're making excuses for pi." With this beautifully succinct exclamation, Vi cuts through the pi smokescreen and puts her finger squarely on the problem. This phrase will, I think, become a rallying cry for tauists everywhere. As the author of The Tau Manifesto, I am proud to have Vi on Team Tau!
There are lots of things we do for conventional reasons, such as having electrons exhibit negative charge.
If you're doing any actual complicated math or physics, the last thing you care about is having a different constant floating around in your terms.
What it is, however, is a great learning tool - This thing called pi, maybe we could get away with, or even be better off calling it 2pi. - Can get lots of people thinking about math and possibly learn something cool like trig. But when used in a psuedo intellectual way, it 'really grinds my gears'.
Perhaps you're right about people doing "actual" complicated maths simply not caring about a multiplicative constant. However, a lot of people do most or all of the complicated maths they're ever going to do in their lives when they're in university. During this period, the correct symbol makes every formula and equation simpler and easier to learn. Crucially, it's also during this period that people's understanding of complicated maths is most important, as they are judged by letter grades upon which many of life's opportunities depend.
How often do people actually make this error? I make all sorts of errors of that class when doing math by hand, but I've never once been off by a factor of two because the formula calls for 2pi.
It's not about errors but understanding. A lot of math involving trig is abstract enough to be confusing to most people. Tau makes is [slightly] less so. For example, understanding that sin represents the y value of a point on a unit circle is easier when 1 tau is a full circle rather than 2 pi.
For the sake of completeness, cos is the x value, because of the identity sin(x)^2 + cos(x)^2 = 1. You should find a good unit circle trigonometry picture if your class isn't giving it to you.
What I find especially ironic is that while I completely understood that I could get the x and y values using rcos(t) and rsin(t) and used it a lot when I worked for a video game company, it's only more recently that the idea "clicked". I always thought about sin and cos in relation to triangles and not circles.
That's only a problem when you need to use radians, which is only necessary for calculus. You can do all the explanations with "cycles" (1 cycle = 2*pi radians), and it's actually simpler.
It's not simpler, it's identical. 1 "cycle" = 1 tau = 2 pi.
And in any case, I think using degrees to teach trig is a terrible idea and only causes more confusion later. sin90 = 1 makes no sense, and the fact that that's my first thought when thinking about sin only causes me problems. If I had instead been taught radians first, a lot of stuff would be significantly easier.
1 cycle = 1 "tau" radians. It's important to include units. I don't see how keeping irrational "magic numbers" out of the equation could make it anything but simpler. As an added benefit, it would serve as a transition between degrees and radians, by introducing the idea that there are multiple measures for an angle without simultaneously introducing a unit that has irrational values in every useful situation.
I agree that degrees are terrible, but sin(1/4)=1 makes a lot of sense. Probably even more than sin(1/4 tau)=1. The only reason to use radians instead of cycles is that changing the units breaks the wonderful trig derivative symmetry.
Radians are pure numbers (which is why I left them out). That said, I see what you mean now and I agree. sin(1/4 of a circle) makes more sense than sin(1/4 tau) which makes more sense than sin(1/4 360).
Basic teaching theory states that experts don't think like novices, and further, they are likely to not remember the misunderstandings and difficulties they encountered as novices, because the novice problems are what they now consider simple. This is a being studied a lot in the world of education.
I call it expert idiocy when the expert refuses to accept that his understanding is actually a pretty advanced state of thinking, and not immediately obvious to the beginner. It is this form of idiocy that leads to people feeling that "only freaks can get math" or "I'm not smart enough for physics" or "computer geniuses can only do basic tasks".
Perhaps you are correct. I don't remember the order in which I learned various concepts but I probably learned about degrees before pi and certainly before trig.
Since radians involve irrational numbers I can understand it being more difficult to learn than degrees. However degrees are a completely arbitrary unit that are used for historical reasons.
Getting stuck thinking in degrees hindered my ability to understand trigonometry, and I don't think I'm alone. It is my belief that teaching trig using radians would be less confusing than teaching trig using degrees first, then radians (as was my experience). With degrees there is no direct correlation between 90 degrees and the values of the trig functions which leads to people simply memorizing values. The same is true (to a lesser extent) when using 2 pi.
True, but you are now setting up a false dilemma. The point is not that arbitrary units are hard, nor that any 2 are relatively harder. It is that a consistent unit that is not arbitrary is easier than the arbitrary ones, and usually involves quite a few useful "symetries" (by which I don't mean real symetries, but conceptual linkages).
As for why people think degrees is easier: it is simply because common usage of degrees makes the concept familiar to learners. I totally agree that teaching trig in terms of radians first would be much better.
Electrons exhibiting negative charge is something that would be difficult to fix because the fix would result in a lot of confusion. Using tau as the circle constant doesn't cause nearly as many problems. I don't see it as a joke; we should do it.
It strikes me as more analogous to using j = sqrt(-1) so that it doesn't conflict with current: it works fine, but you're never going to get everyone to do it because people want the papers to stay consistent.
Actually, in this case, there's a particular discipline that's especially inconvenienced (physicists, due to the torque conflict). Contrast with j: there's a particular discipline that benefits, and they use j=sqrt(-1) all the time.
If you're doing complicated math or physics, the last thing you want is to be off by a multiple of 2 when you're done. It's happened to me enough times, I almost wish I were still doing physics so I could use tau instead of pi.
For electron charge, I have, at times for intermediate calculations, used 'e' in place of the negative sign from electrons, and then substituted -1 at the end. Being off by a multiple of -1 is just as annoying as a multiple of 2.
Even more annoying since it is even easier. Losing a negative sign is the most common mistake I make working through a complicated problem. Your idea of using something less likely to be overlooked when transcribing intermediate steps in solving a problem strikes me as a really good idea. Thanks.
I don't see it as a joke. Why not strive for the conceptually cleanest possible conventions?
It's pretty plain to me that tau is more deserving of the status of 'conceptual entity' than pi is. It's the number of radians in a cycle. Practically every time it occurs in physics, pi represents 'half the number of radians in a cycle'. Kind of crufty if you ask me.
I'm not so sure it is a joke. The author clearly states it's more for neophytes. And the convention can be explained of using 2π after showing what it all means by explaining Tau
If nothing else, we should at least get a non-separated printing character that looks like 2π connected up, similar to latin dipthongs.
When you're programming something complicated, arbitrary ugliness in the APIs you're using brings you more trouble than when you're programming something simple. Right? So what's different about complicated math or physics?
That's how I realized pi was wrong, back before the manifesto came out -- from looking at math the same way I'd learned to look at programs.
Not to mention that they seem to have chosen one of the worst possible symbols: the one that happens to represent torque. Further, it looks a lot like t, which is often used to represent turns (== cycles == 2pi radians).
There are alphabets out there other than the Latin and Greek ones, with plenty of untapped symbols.
This is something I can't quite wrap my head around. Why would I want the symbol for a unit to be similar to a constant that I will very often use in the same calculations? That has a good chance of ruining my dimensional analysis if I finish up a calculation later. A bathroom break could convert 2 * pi radians into 1 cycle * radian, or 1 cycle into a unitless 2 * pi.
While not technically required, it's pretty convenient to treat radians as their own "angle" dimension. This lets you distinguish an angular velocity from a simple periodic repetition. The latter happen to rarely (never?) occur naturally, but I run into artificial ones often enough.
This is not a joke any more than the dvorak keyboard layout is a joke or the metric system is a joke. It's an objectively better approach whose only drawback is the overwhelming force of tradition.
You cannot compare use of the metric system to use of Dvorak or tau=2pi. While Dvorak and tau may be objectively better, the real world gains to be had by switching to either are slim in reality. I already type upwards of 90 words a minute on Qwerty and I doubt that switching to Dvorak would yield much of a gain to my typing. The problem behind pi is that it is never explained correctly, and this can be worked around by fixing pedagogy without having to introduce yet another physical constant. If the United States switched to the metric system, however, doing business with foreign economies would immediately become easier and we would no longer have to memorize long tables of arbitrary conversion factors. It makes much more sense objectively AND pragmatically to use units that are derived from physical quantities, rather than legends about kings' feet and so forth. With the other two examples, the pragmatic improvement isn't there -- the improvements in quality of life/quality of thought that we would experience simply aren't worth the trouble of switching.
But in analogous equations (energy of motion .5mv^2, energy of spring, .5kx^2), there is a 1/2. Basically, anytime you integrate a linear equation, you get a factor of 1/2. Using pi as the circle constant hides this fact.
I like that it hides the fraction. I'm a computer scientist. I don't like fractions.
Instead of celebrating tau, we should be celebrating integers. Robust, fast, compact, reliable, easy to understand. With pi I can use more integers. Thank you pi!
Well, as a computer scientist, I'd hope you use pi * r * r, because there's no reason to compute tau/2 every time, and pow() is needlessly inefficient for integer powers (on all languages I know. ~6 times slower in python). I'd expect you to make the same optimizations with the analogous equations, but that shouldn't have any bearing on which constant is better for learning, etc.
Tell that to my 12-year old nephew who memorized π to 128 decimal places in celebration of today! He'll be heart-broken to find out he memorized an inconsequential number.
I saw the first submission, but I wanted to use a different headline, so I appended a question mark to the URL and submitted a new link. It's always a good sign when people submit your stuff before you do, but in this case I had a clear idea of what I wanted the headline to be. If I recall correctly, essentially the same thing happened on Tau Day itself, with the same result.
Is 22π equal to 2 times 2π or is it equal to 11 times 2π? That is, is 22π equal to 2(2π) or (22)π? If 2π is a single symbol, then 22π would be equal to 2(2π), which would be really confusing.
What I was trying to say is that there is no need to introduce another symbol τ as we can simply refer to "the circle constant" as 2π. I am not suggesting to treat 2π as a single symbol in formulas.
The hard part is getting people to agree that 2pi is the true "circle constant" and that pi was a historical accident; once you've gotten over that hurdle agreeing that it should have its own symbol is a no-brainer.
- change electric theory so that it's a flow of negative charge and not positive.
- get the US to use SI units instead of Imperial or whatever they think they're using.
but that doesn't make it a better formula. Or if it did, then the following formula would be infinitely superior to all of the above (and it doesn't directly mention either π or τ):
e^(τi)-1=0. This is obviously better since it not only contains the fundamental constants e, tau, and the beautiful numbers 0 and 1, it also includes a minus sign, emphasizing the fact that we are working over a field with an additive inverse. ;)
You think you've got issues : my sci fi series TAU4 is getting all your emails, as Google alerts!
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Tau is only convenient in geometry. In calculus, Pi rules the day because the natural unit is radians. Since I like calculus more than geometry, I won't be accepting Tau anytime soon.
Read the Tau Manifesto (linked int he article). He covers that and shows why Tau is easier to use there, too. Dude was a physics prof at Caltech; I think calculus is sort of important to him, too.
