Happy Tau Day(tauday.com)
tauday.com
Happy Tau Day
http://tauday.com/
127 comments
Your Cauchy argument is actually misleading and runs right into my joking example elsewhere in this thread about needing to define a new constant Sigma=4pi to work better with steradians.
Pi doesn't always represent only a pure circle. Eg, in solid angles there are 4pi steradians over a sphere. Or 2tau. Or what I define as Sigma.
This extra factor of two when using Tau should be just as disconcerting to tau enthusiasts for steradians as pi is for radians.
Your example is talking about shooting particles in all directions over 3D space. This calls for a solid angle approach. Which means you should be integrating over steradians just as you'd use radians for an angular system. There are 4pi steradians over a sphere's full solid angle, the wall only covers 2pi steradians. Meanwhile the extra factor of two comes from the integration of decreasing infinitesimal wall cross sections over the azimuthal angle.
The fact that it comes out to 1/2 tau is merely happy coincidence. Ie, the geometry introduced an extra factor of two because it's just as easily 1/4 sigma, and for a solid angle system (shooting particles in all directions) you should be using steradians. Hence sigma.
Pi doesn't always represent only a pure circle. Eg, in solid angles there are 4pi steradians over a sphere. Or 2tau. Or what I define as Sigma.
This extra factor of two when using Tau should be just as disconcerting to tau enthusiasts for steradians as pi is for radians.
Your example is talking about shooting particles in all directions over 3D space. This calls for a solid angle approach. Which means you should be integrating over steradians just as you'd use radians for an angular system. There are 4pi steradians over a sphere's full solid angle, the wall only covers 2pi steradians. Meanwhile the extra factor of two comes from the integration of decreasing infinitesimal wall cross sections over the azimuthal angle.
The fact that it comes out to 1/2 tau is merely happy coincidence. Ie, the geometry introduced an extra factor of two because it's just as easily 1/4 sigma, and for a solid angle system (shooting particles in all directions) you should be using steradians. Hence sigma.
Surely the Cauchy distribution you're replying to doesn't have any steradians going on, right? It's a 1D distribution and the gun described would point towards a random point on a circle, not on a sphere.
Argh, you may be right.
I was actually thinking of a separate example from physics (years ago) calculating the classical flow of particles through an aperture. Where you assume particles have maxwell-Boltzmann distribution of speeds and consider them travelling in all directions over 3D and also the effective size of the aperture vs the azimuthal angle of travel. It's quite similar to this example actually but given the speeds you also must account for which velocity range [v,v+dv] as a function of angle will put a particle through the hole in time dt.
I was actually thinking of a separate example from physics (years ago) calculating the classical flow of particles through an aperture. Where you assume particles have maxwell-Boltzmann distribution of speeds and consider them travelling in all directions over 3D and also the effective size of the aperture vs the azimuthal angle of travel. It's quite similar to this example actually but given the speeds you also must account for which velocity range [v,v+dv] as a function of angle will put a particle through the hole in time dt.
As you already realized, my example was supposed to be in 2D, but I forgot to mention it. Sorry about that.
But your point about steradians is still spot on. If I had been talking about the 3D case, there would have been a 1/(2pi) factor in the front of the pdf. Using tau would not have helped anyone realize that this particular 2pi is most naturally thought of as 1/2 of the full 4pi steradians. Using tau is not a substitute for careful thinking.
And I am not even actually advocating switching to tau; I am not sure if it is worth the effort. But I still think that tau is a good way of conveying the idea that the current definition of pi as basically a historical accident. This can be enlightening even if one doesn't start using tau. As a more extreme example, I think it is useful to recognize that the decimal base (as opposed to binary, hex, etc.) is a historical accident, but I wouldn't advocate changing it.
But your point about steradians is still spot on. If I had been talking about the 3D case, there would have been a 1/(2pi) factor in the front of the pdf. Using tau would not have helped anyone realize that this particular 2pi is most naturally thought of as 1/2 of the full 4pi steradians. Using tau is not a substitute for careful thinking.
And I am not even actually advocating switching to tau; I am not sure if it is worth the effort. But I still think that tau is a good way of conveying the idea that the current definition of pi as basically a historical accident. This can be enlightening even if one doesn't start using tau. As a more extreme example, I think it is useful to recognize that the decimal base (as opposed to binary, hex, etc.) is a historical accident, but I wouldn't advocate changing it.
One argument in favor of tau is that in many formulas pi
often has the multiplier 2 in front of it.
Of course you can construct infinitely many formulas, but most of those are used a vanishingly small number of times. The question is what common formulas use pi or tau. How often would people actually have to write, read or say a factor?
In any formula where the 2 in 2π has been canceled by some irrelevant 2, the cancellation is not really an advantage; it obscures semantics.
Also, when counting formulas don't forget to expand "angular frequency" ω to the 2πf it denotes.
Also, when counting formulas don't forget to expand "angular frequency" ω to the 2πf it denotes.
If mathematics has a bikeshed, this is it.
I enjoy the ridiculousness of it all, but people who consider this anything other than a well-executed joke really should get a hold of themselves.
I enjoy the ridiculousness of it all, but people who consider this anything other than a well-executed joke really should get a hold of themselves.
You're right about the bikeshedness, but that doesn't mean there isn't a clearly better bike shed design. I also find that these sorts of inefficiencies/inelegances compound. One or two are trivial, but when you have 30 of them, suddenly the mental tax becomes noticeable.
Furthermore, the cost falls mostly on the students, while the experts have already paid it and don't see the need to worry about it anymore.
Furthermore, the cost falls mostly on the students, while the experts have already paid it and don't see the need to worry about it anymore.
Yeah, but this bikeshed got decided hundreds of years ago.
Its only appealing to internet hipsters who want to brag about their organically grown free range tau.
Making the next generation of math students use "2 pi" in their formulas is going to be vastly easier than literally rewriting all the books.
Its only appealing to internet hipsters who want to brag about their organically grown free range tau.
Making the next generation of math students use "2 pi" in their formulas is going to be vastly easier than literally rewriting all the books.
As a practicing physicist, I found it illuminating and non-trivially useful to know this when it was first introduced to me years ago. This is true even though I will never use in a paper, and even if I only rarely write it in my notes. In my head, 2\pi is a single character, to the point that I often write \frac{1}{2} 2\pi.
I find it worthwhile to think about questions like this, whose answers, on some level, do not matter, but, on another level, can be felt to be more or less in line with insights about the underlying structure.
Examples of other questions in the same category:
- Whether the natural numbers should be defined so as to include zero or not (and, relatedly, whether counting should start on zero or on one).
- How to structure a database or a piece of computer code (including the details of how to formulate individual lines of code).
Examples of other questions in the same category:
- Whether the natural numbers should be defined so as to include zero or not (and, relatedly, whether counting should start on zero or on one).
- How to structure a database or a piece of computer code (including the details of how to formulate individual lines of code).
> If mathematics has a bikeshed, this is it.
In my experience, few professional mathematicians care about this. The question that Bromskloss (https://news.ycombinator.com/item?id=11993750) mentions, of whether or not 0 is a natural number, is likely to incur a far more passionate response. (At the risk of downvotes: it is.)
In my experience, few professional mathematicians care about this. The question that Bromskloss (https://news.ycombinator.com/item?id=11993750) mentions, of whether or not 0 is a natural number, is likely to incur a far more passionate response. (At the risk of downvotes: it is.)
In therms of practical consequences such as making mathematical expressions look different, it seems to me that 0 vs. 1 makes a smaller difference than does τ vs. π, because many statements about the set ℕ of natural numbers make no mention of what its first element is anyway. Often you can drag along any of the two definitions of ℕ through the argument, without changing it's conclusions. For example, if f:ℕ→ℝ (and ε∈ℝ, N∈ℕ, n∈ℕ), the limit L∈ℝ is given by (∀ε>0)(∃N)(∀n≥N)(|f(n)-L|<ε), regardless of whether one takes ℕ={0,1,2,…} or ℕ={1,2,3,…}.
Well, certainly there is a difference if one wishes to compute, say, `\sum_{n \in \mathbb N} 1/n!`. In that case, it's easy just to say what you mean; but, then again, so is it easy to say what you mean in any occurrence of `\pi` or `\tau`. I spoke only of the passion that the issue evokes, not its practical consequences.
> Well, certainly there is a difference if one wishes to compute, say, `\sum_{n \in \mathbb N} 1/n!`.
Certainly. To be clear, I don't mean to argue against anything you said. I just find it to be an interesting observation that statements quite often are true for both versions of ℕ.
Specifically, I suppose, the two versions work the same when one is concerned with only the most fundamental essence of the natural numbers, as captured by the Peano axioms. https://en.wikipedia.org/wiki/Peano_axioms#Formulation
Certainly. To be clear, I don't mean to argue against anything you said. I just find it to be an interesting observation that statements quite often are true for both versions of ℕ.
Specifically, I suppose, the two versions work the same when one is concerned with only the most fundamental essence of the natural numbers, as captured by the Peano axioms. https://en.wikipedia.org/wiki/Peano_axioms#Formulation
> Specifically, I suppose, the two versions work the same when one is concerned with only the most fundamental essence of the natural numbers, as captured by the Peano axioms.
I find this statement interesting and plausible, but difficult to formalise. The Peano axioms explicitly name the least element of the natural numbers, so in some sense they foreground, rather than effacing, the issue of whether it is 0 or 1; but, on the other hand, I suppose that one could argue that an argument using "only the most fundamental essence of the natural numbers" is one that, as it were, is allowed only to use a constant naming the least element, without pre-supposing any idea of what that name 'means' in any external axiomatic or intuitive sense.
I find this statement interesting and plausible, but difficult to formalise. The Peano axioms explicitly name the least element of the natural numbers, so in some sense they foreground, rather than effacing, the issue of whether it is 0 or 1; but, on the other hand, I suppose that one could argue that an argument using "only the most fundamental essence of the natural numbers" is one that, as it were, is allowed only to use a constant naming the least element, without pre-supposing any idea of what that name 'means' in any external axiomatic or intuitive sense.
I think I follow, and I agree. How I view the natural numbers on this fundamental level – and I would say it is the picture that the Peano axioms nail down – is as consisting of a first element and then an infinite chain of elements after it (equivalently: the natural numbers are a chain that extends infinitely in one direction but not in the other). As you say, it does not matter, then, if that first element is called "0", "1", or "17" (until we bestow additional structure on the numbers by introducing addition and multiplication operators).
I have to disagree.
I'm a programmer, but I'm not good at math. I know this. And yet, the Tau Day Manifesto explained the geometry of the circle to me more clearly and understandably than any of the math classes I've ever taken. And it's not just because it's a well-written work; the concept is genuinely simpler to understand.
Pi really is a pedagogical disaster, and tau really does help.
I'm a programmer, but I'm not good at math. I know this. And yet, the Tau Day Manifesto explained the geometry of the circle to me more clearly and understandably than any of the math classes I've ever taken. And it's not just because it's a well-written work; the concept is genuinely simpler to understand.
Pi really is a pedagogical disaster, and tau really does help.
I find your interpretation rather benign. Saying, “pi is wrong, tau is right” is the equivalent of mathematical bigotry to me.
I feel very fortunate that the tau vs. pi argument was around when I went back to study math in earnest. I found tau much more intuitive and it helped me to visualize and learn the material more effectively.
I now use tau whenever I can. However, I don't find switching back and forth to accommodate pi loyalists that taxing. You don't really have to choose one exclusively.
I now use tau whenever I can. However, I don't find switching back and forth to accommodate pi loyalists that taxing. You don't really have to choose one exclusively.
It's just the same useless argument every year. Just use what you want. I've been taught pi since secondary school. I understand it well and can use it effectively. Never have I sat there and thought, "if only there was a shortcut to multiplying this by 2". There's whole sites, videos and movements to get tau popular. I just personally don't understand the point.
That's exactly the issue. You are thinking in terms of pi, so it might be hard to see from a different perspective.
It's like using a slightly off abstraction for a concept. At first you have to make a small effort to hold it in your head, then at some point it's committed and you can manipulate the concept directly.
It's like using a slightly off abstraction for a concept. At first you have to make a small effort to hold it in your head, then at some point it's committed and you can manipulate the concept directly.
Let x be a randomly chosen real number in [0,1]. What does it mean to "manipulate x's concept" or "think in terms of x"?
Do these phrases attach themselves to the real number, or to the expression language? If the latter, do you say two number-expressions are equivalent if applying some normalization function yields two equal expressions? Do you consider two number-expressions distinct if they evaluate to the same real number, but cannot directly be related to each other?
For example, let:
Do these phrases attach themselves to the real number, or to the expression language? If the latter, do you say two number-expressions are equivalent if applying some normalization function yields two equal expressions? Do you consider two number-expressions distinct if they evaluate to the same real number, but cannot directly be related to each other?
For example, let:
S = { (x, e^(ix) + 1) | x in R, x > 0 },
T = { x | (x,y) in S, y = 0 },
c1 = min(T),
c2 = 6 * sqrt(sum(n^-2, n > 0))
This isn't to save time multiplying by two. This is to uniform a hole slew of previously unrelated equations that can now be represented similarly.
I honestly can't tell if you're being sarcastic or not.
I don't understand your comment. Have you watched the video linked within this post?
This video is very good at pointing out many of the benefits of tao. [0]
[0] - https://www.youtube.com/watch?v=H69YH5TnNX
This video is very good at pointing out many of the benefits of tao. [0]
[0] - https://www.youtube.com/watch?v=H69YH5TnNX
You wrote "This is to uniform a hole slew of previously unrelated equations that can now be represented similarly."
You seem to be saying that introducing a new constant tau=2pi and redefining a slew of previously unrelated equations in terms of tau instead of pi is suddenly going to make them "uniformly represented".
I don't understand why they're not considered uniformly represented when defined just as consistently using pi?
You seem to be saying that introducing a new constant tau=2pi and redefining a slew of previously unrelated equations in terms of tau instead of pi is suddenly going to make them "uniformly represented".
I don't understand why they're not considered uniformly represented when defined just as consistently using pi?
Have you watched the linked video? If not I will not be able to explain what's going on as well as the presenter.
He is able to show that many common physical calculations for volume or area all relate to the same generic form.
He also is able to present many other cases where pie wins out. It is very much worth a watch.
He is able to show that many common physical calculations for volume or area all relate to the same generic form.
He also is able to present many other cases where pie wins out. It is very much worth a watch.
Getting a "This video does not exist." error from your link.
There is a trailing character missing,
https://www.youtube.com/watch?v=H69YH5TnNXI
https://www.youtube.com/watch?v=H69YH5TnNXI
Whoops! Sorry, my bad! Thanks for catching my mistake!
I use 1.07 pi; it should be the same, as it is only a matter of convention.
The only thing useful about it is you have an excuse to eat two pies....
The choice between Pi vs. Tau is completely arbitrary and should be based on minimizing the friction of use. Everyone knows Pi, few knows about Tau. I see no point in spending any more energy than that on this kind of nonsense decisions.
"Because that's the way we've always done is" is never a valid argument.
>> "Because that's the way we've always done is" is never a valid argument.
But it is. For the pi/tau thing, it's better if the whole world uses one set of formulas. For the guy who pointed to which side of the road we drive on, it's best if we all agree to drive on the same side. And for units, it's best if we all agree to measure things using the same units - otherwise probes crash into mars and such. All of those are arbitrary decisions and we're better of just sticking with them - except the units one, by using SI units a lot of formulas simplify and arbitrary constants go away. I'm not so sure tau/pi have a similar advantage. If you do change, you'll put the world in a weird transition phase until the change is widespread and that causes problems until it's resolve. Also, with mathematics you have existing and historical papers that use pi, so what to do about those? The legacy of pi would be around a long long time.
But it is. For the pi/tau thing, it's better if the whole world uses one set of formulas. For the guy who pointed to which side of the road we drive on, it's best if we all agree to drive on the same side. And for units, it's best if we all agree to measure things using the same units - otherwise probes crash into mars and such. All of those are arbitrary decisions and we're better of just sticking with them - except the units one, by using SI units a lot of formulas simplify and arbitrary constants go away. I'm not so sure tau/pi have a similar advantage. If you do change, you'll put the world in a weird transition phase until the change is widespread and that causes problems until it's resolve. Also, with mathematics you have existing and historical papers that use pi, so what to do about those? The legacy of pi would be around a long long time.
Often it is. It's a perfectly good answer to, for example, "why do we drive on the side of the road that we do?"
If the choice really is arbitrary then yes. If there was a compelling argument to driving one side or the other then it would cease to be a good answer IMHO.
https://en.wikipedia.org/wiki/Dagen_H
https://en.wikipedia.org/wiki/Dagen_H
How is "because we've always done it that way" an answer to "why do we drive on the side of the road that we do?"
It's more or less just a re-statement of the question.
And it's wrong: we haven't always driven on that side of the road.
We don't always drive on that side now; it depends on where in the world we find ourselves.
Left versus right is symmetric: there is no inherent advantage. Both choices have exactly the same advantages and disadvantages, just with "left" and "right" swapped.
Whether or not to include a factor of two isn't symmetric in this way.
It's more or less just a re-statement of the question.
And it's wrong: we haven't always driven on that side of the road.
We don't always drive on that side now; it depends on where in the world we find ourselves.
Left versus right is symmetric: there is no inherent advantage. Both choices have exactly the same advantages and disadvantages, just with "left" and "right" swapped.
Whether or not to include a factor of two isn't symmetric in this way.
I sometimes wonder if handedness or eye dominance have any effect on accident statistics.
Of course it's a valid argument - you have to consider the costs and benefits of switching whenever you propose a change in a broadly accepted convention.
For example, one could make an argument that oral and written communication in the United States would be more efficient if everyone was forced to communicate in Klingon. But the switching cost would be so astronomically high that any possible gains pale in comparison.
(I'm not saying the pi/tau cost/benefit is comparable to English/Klingon, I'm just pointing out that "that's the way we've always done it" is in fact a perfectly legitimate argument. The onus is on the person proposing a change to show that the change is worth it.)
For example, one could make an argument that oral and written communication in the United States would be more efficient if everyone was forced to communicate in Klingon. But the switching cost would be so astronomically high that any possible gains pale in comparison.
(I'm not saying the pi/tau cost/benefit is comparable to English/Klingon, I'm just pointing out that "that's the way we've always done it" is in fact a perfectly legitimate argument. The onus is on the person proposing a change to show that the change is worth it.)
To use your own argument, 2*pi is far, far more common to encounter in physics/maths ie. real world examples, therefore minimizing friction would be using Tau ;)
So to save confusion over a mere factor of two, you're proposing devoting an entire new Greek letter to just double a constant, and requiring students to understand pi anyway to make sense of hundreds of years of mathematical and scientific corpus?
It's just like that xkcd comic about introducing a new standard and now having N+1 standards. Except in this case the new standard offers only a factor of two.
It's just like that xkcd comic about introducing a new standard and now having N+1 standards. Except in this case the new standard offers only a factor of two.
Look, it's more about saying that the area of a right triangle is naturally represented (and explained) as half that of a rectangle sharing the height and width - but being divided in two equal halves by the hypotenuse.
You can argue that you're so used to right triangles that it's the square and the rectangle that should be considered special, and "twice the area" - but I don't think that makes much sense.
Pi is a useful constant, but it's chief role is in cycles/frequencies and circles. Not half-circles. It is perfectly ok to disagree - but I'm one of those people to find the concepts behind pi start making much more sense when thinking of tau as the constant, and half-tau (pi) as the special case.
Maybe it's because I always hate having logic and math concepts waved away as "that's the way it is" when there obviously is some pattern or explanation that's being hidden or lost. I'm no good at rote calculation, and with tau a lot of things that just look odd and "special" unify quite nicely.
It may very well be that for higher dimensions than two (or three) tau doesn't make much more sense than pi - but I found the "tau manifesto" examples plenty convincing.
It was easier for me to grasp, and I believe it would be a lot easier to teach.
You can argue that you're so used to right triangles that it's the square and the rectangle that should be considered special, and "twice the area" - but I don't think that makes much sense.
Pi is a useful constant, but it's chief role is in cycles/frequencies and circles. Not half-circles. It is perfectly ok to disagree - but I'm one of those people to find the concepts behind pi start making much more sense when thinking of tau as the constant, and half-tau (pi) as the special case.
Maybe it's because I always hate having logic and math concepts waved away as "that's the way it is" when there obviously is some pattern or explanation that's being hidden or lost. I'm no good at rote calculation, and with tau a lot of things that just look odd and "special" unify quite nicely.
It may very well be that for higher dimensions than two (or three) tau doesn't make much more sense than pi - but I found the "tau manifesto" examples plenty convincing.
It was easier for me to grasp, and I believe it would be a lot easier to teach.
It's not about the factor, it's about making the connection to the circle obvious.
[deleted]
Relax, the whole thing is a joke, including my comment above.
Ha, yeah I know. On rereading my comment, it does come across as somewhat defensive but I definitely wasn't meaning it to be that way. Actually I was smiling as I wrote it.
See my other comments on this thread about sigma, for instance.
See my other comments on this thread about sigma, for instance.
While I appreciate the actual methematical point of the site (and assume there is a small sense of deadpan humor intended), discussions as to which day should be "celebrated" are pointless since the day's importance to the general public has nothing to do with the value of pi or tau, but instead the fact that pi is a homonym of pie and march 14th happens to be the same numerically as the first few digits.
As far as I know, that's the extent of the popularity - you can make pies on Pi Day. Again, I'm assuming the Tau day thing is a bit of deadpan, but otherwise it seems to me to be that simple.
As far as I know, that's the extent of the popularity - you can make pies on Pi Day. Again, I'm assuming the Tau day thing is a bit of deadpan, but otherwise it seems to me to be that simple.
> Tau is completely arbitrary and should be based on minimizing the friction of use.
Isn't that what the proponents of Tau also argue for?
Isn't that what the proponents of Tau also argue for?
Like how we should stick with GMT?
While choice between tau and pi is arbitrary from mathematical point, things like pedagogy, notational simplicity and aesthetics matter.
The reason why I think π is probably better choice is because small multiples of constant are cognitively easier to process than fractions. 2π is easier to write and see as single object than τ/2. All we need to do is to make slight cognitive adjustment and think and teach 2π as a number instead of 2×π.
The reason why I think π is probably better choice is because small multiples of constant are cognitively easier to process than fractions. 2π is easier to write and see as single object than τ/2. All we need to do is to make slight cognitive adjustment and think and teach 2π as a number instead of 2×π.
My favorite example of why τ is the true circle constant is actually the equation A = 1/2 τ r^2. It fits the usual quadratic form and shows how circumference and area relate: c = dA/dr = τ r and so A = ∫ c dr = ∫ τ r dr = 1/2 τ r^2. At first glance those 1/2s seem superfluous and like it would be nicer to just have a 2 in the other terms, but the 1/2 shows that differentiation and integration with respect to r are happening, where the "1/2" and the "^2" anhialate each other on differentiation and then re-form on integration.
Reminds me of Taylor series. The factorial terms eluded me for so long until I realized it was a cancelling factor for accumulated derivation of polynomials. Now there's this link in my mind between powers / derivation / factorials.
What polynomials are being derived, and from what premises? Are you thinking of how to derive series solutions to differential equations or something? I'm unclear on what you're getting at here.
(Or, oh: did you mean "differentiation" when you said "derivation"? That would fit what you've said. Sorry for the pedantic post; I'll just leave this here in case anyone else is confused.)
(Or, oh: did you mean "differentiation" when you said "derivation"? That would fit what you've said. Sorry for the pedantic post; I'll just leave this here in case anyone else is confused.)
Sorry for the fuzzy vocabulary. TBH I wasn't even convinced I understood taylor series correctly. A quick search on wikipedia hint at that derivation is used sometimes; derivative; differentiation etc etc.
About the point:
About the point:
d(n)(x^n.dx)
= n d(n-1)(x^(n-1).dx)
= (n * (n-1 * (...)))
= n! * x
hence the 1/n! term.Treating 2π as a single entity is probably the best option for now. Not sure if formulas like 1/2 (2π) r^2 will ever catch on, but in more complicated formulas it's probably better to not simplify things like (2π)^4 / 6 to 2^3 π^4 / 3.
The idea of 2π as symbolic entity in its own right, should be easier to accept than τ.
Introduce a single symbol that joins 2 and π, a bit like the Jupiter symbol (http://unicode-table.com/en/2643/), and pronounced 'two-pi'. Like τ it also starts with a T, for turn.
That would be totally backward compatible in reading, writing and speech.
Introduce a single symbol that joins 2 and π, a bit like the Jupiter symbol (http://unicode-table.com/en/2643/), and pronounced 'two-pi'. Like τ it also starts with a T, for turn.
That would be totally backward compatible in reading, writing and speech.
A ligature is what you're suggesting.
https://en.m.wikipedia.org/wiki/Typographic_ligature
https://en.m.wikipedia.org/wiki/Typographic_ligature
> 2π is easier to write and see as single object than τ/2
The point is that you shouldn't have to write 2π in the first place. So notational simplicity would mandate the use of τ, not π.
Noting that you will have to write τ/2 for a half turn is moot, in your case you'll also have to write π/2 for a quarter turn anyway.
The point is that you shouldn't have to write 2π in the first place. So notational simplicity would mandate the use of τ, not π.
Noting that you will have to write τ/2 for a half turn is moot, in your case you'll also have to write π/2 for a quarter turn anyway.
It gets worse!
When using solid angles over a sphere, you're still going to need to use 2Tau steradians to cover a sphere. This factor of 2 will continue to cause confusion for the Tau fans.
Therefore we must also define a new constant Sigma = 4pi so we can cleanly and easily deal with steradians.
Anybody up for writing the Sigma Manifesto?
</sarcasm>
When using solid angles over a sphere, you're still going to need to use 2Tau steradians to cover a sphere. This factor of 2 will continue to cause confusion for the Tau fans.
Therefore we must also define a new constant Sigma = 4pi so we can cleanly and easily deal with steradians.
Anybody up for writing the Sigma Manifesto?
</sarcasm>
Does it help the pedagogy and understanding? Does it help form equations that match the form of other equations in some mathematically-meaningful way?
Then, yes, I'd support it just fine. So the sarcasm fails.
One of the other things I don't see mentioned very often in this discussion is that mathematics evolves. We almost never get it right the first time. The original Maxwell's Equations were 20 equations, rather hairy ones at that, now expressed in 4 with superior notation. Derivative and integral notation did not spring fully-formed from Newton or Leibniz, it has evolved. Matrix notation evolved. Number notation has evolved.
The idea that pi itself may have to evolve because it wasn't quite right is perfectly natural and normal. What's bizarre is the idea that it must be held to be perfect, that criticism is all-but-morally wrong, and that anybody even talking about it is crazy.
I blame our terrible math system, for teaching people that math consists entirely of edicts handed down from, I presume, aliens, or possibly some form of diety, since apparently it can't be humans as we're not allowed to touch the Holy Notation. But that's not how it works in reality, and there's nothing bizarre about the idea that we might want to change pi; what's bizarre is the idea that such a thing is blasphemy.
Then, yes, I'd support it just fine. So the sarcasm fails.
One of the other things I don't see mentioned very often in this discussion is that mathematics evolves. We almost never get it right the first time. The original Maxwell's Equations were 20 equations, rather hairy ones at that, now expressed in 4 with superior notation. Derivative and integral notation did not spring fully-formed from Newton or Leibniz, it has evolved. Matrix notation evolved. Number notation has evolved.
The idea that pi itself may have to evolve because it wasn't quite right is perfectly natural and normal. What's bizarre is the idea that it must be held to be perfect, that criticism is all-but-morally wrong, and that anybody even talking about it is crazy.
I blame our terrible math system, for teaching people that math consists entirely of edicts handed down from, I presume, aliens, or possibly some form of diety, since apparently it can't be humans as we're not allowed to touch the Holy Notation. But that's not how it works in reality, and there's nothing bizarre about the idea that we might want to change pi; what's bizarre is the idea that such a thing is blasphemy.
>Does it help the pedagogy and understanding? Does it help form equations that match the form of other equations in some mathematically-meaningful way?
Absolutely! In 3D systems. Particularly using spherical coordinates.
It's ironic you chose Maxwell's Equations as your example because Sigma would get rid of those pesky factors of 4pi appearing in them when written in Gaussian units!
Besides, Maxwell's Equations are really best represented as just one equation when expressed in covariant form.
Absolutely! In 3D systems. Particularly using spherical coordinates.
It's ironic you chose Maxwell's Equations as your example because Sigma would get rid of those pesky factors of 4pi appearing in them when written in Gaussian units!
Besides, Maxwell's Equations are really best represented as just one equation when expressed in covariant form.
Maybe we could subscript tau with dimension n.
Mathematically, tau is clearly superior, since it represents a full circle. The question is whether the gain is big enough compared to the trouble of introducing a new constant. It's not like pi is going to disappear any time soon.
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I just wish instead of celebrating these days on arbitrary month/day combinations, that we'd instead use some physics or math based use of them. Pi day should be when we've gone half way around the Sun and Tau, New Years Day.
Multiply the current dates by 2 and to a very near approximation (~1% off) that's what you get :P
As today tauday is tuesday,
the Angle programming language now knows tau:
⦠ assert tau / 2 = pi
True
https://github.com/pannous/angle
the Angle programming language now knows tau:
⦠ assert tau / 2 = pi
True
https://github.com/pannous/angle
I'm in favor of keeping pi because of one simple equation:
[1] https://en.wikipedia.org/wiki/Euler%27s_identity
e^{i\pi} + 1 = 0
e^{i\pi} + 1 = 0
e^{i\pi} = -1
i\pi = \ln{-1}
\ln{x} + i\pi = \ln{x} + \ln{-1}
\ln{x} + i\pi = \ln{-x}
[1] https://en.wikipedia.org/wiki/Euler%27s_identity
This simultaneously illustrates how e^(180') is halfway round a unit. Tau (literally/numerically?) > Pi
On to more important matters, Gamma function or Pi function?
http://mathoverflow.net/questions/20960/why-is-the-gamma-fun...
Obviously, the Pi function is the right one since Pi(n) = n! for all integers n.
http://mathoverflow.net/questions/20960/why-is-the-gamma-fun...
Obviously, the Pi function is the right one since Pi(n) = n! for all integers n.
Using tau would uglify euler's equation, wouldn't it?
You be the judge of that. Compare
e^(iπ) = -1
e^(iτ) = 1.
e^(iπ) + 1 = 0
Writing it equal to 0 isn't a hack, it's a common method of understanding a function. You factor polynomials by setting them equal to 0, for example.
In the case of Euler's identity, what we're really asking is "what values of x make e^(ix) + 1 = 0 true?" and the answer is "every multiple of π". Using τ instead hides half of the answers.
In the case of Euler's identity, what we're really asking is "what values of x make e^(ix) + 1 = 0 true?" and the answer is "every multiple of π". Using τ instead hides half of the answers.
You mean "every multiple of 2π, shifted by π". The value -1 only comes around once per revolution of the unit circle!
The solutions to
The solutions to
e^(ix) + 1 = 0
{ π, 3π, 5π, ... }
e^(ix) - 1 = 0
{ 0, 2π, 4π, 3π, ... }
{ 0, τ, 2τ, 3τ, ... }
(x - r0)(x - r1)...(x - rn) = 0
e^(ix) - 1 = 0Poor phrasing and math-before-coffee on my part. The point I was going for is that e^(ix) has real values for each multiple of π. That is mathematically interesting, and is obscured if you use τ instead.
Yes; similarly, that a sinusoidal crosses zero twice in each period is obscured by a focus on the period itself.
You mean "every multiple of 2π". The value -1 only comes around once per revolution of the unit circle!
e^(i tau) - 1 = 0
Seems pretty comparable to me: this one elegantly demonstrates the full periodicity of complex exponentials, while the original elegantly demonstrates how complex exponentials can produce pure negative numbers.
Seems pretty comparable to me: this one elegantly demonstrates the full periodicity of complex exponentials, while the original elegantly demonstrates how complex exponentials can produce pure negative numbers.
Angular frequency (ω = 2πf) is related to this:
https://en.wikipedia.org/wiki/Angular_frequency
Note how convenient ω is when squared, in the expression of angular acceleration, eliminating a ridiculous 4 factor.
https://en.wikipedia.org/wiki/Angular_frequency
Note how convenient ω is when squared, in the expression of angular acceleration, eliminating a ridiculous 4 factor.
This ties into "ħ" vs. "h" as well:
E = ħω
E = hf
I don't mean this as an argument for anything; I just felt like mentioning it.
E = ħω
E = hf
I don't mean this as an argument for anything; I just felt like mentioning it.
OT, but Michael Hartl's RoR tutorials are amazing and helped me learn so quickly. Worth checking out[0] if you're looking to learn
[0]: https://www.railstutorial.org/
[0]: https://www.railstutorial.org/
i totally suck at math, however i think i know what is a circle, i can draw one with a compas. then i think i know what is the diameter of this circle, i can draw it by drawing two more circles and a line with a ruler. because a mathematician told me that the product of this diameter by a number is the circumference of the first circle i believe him, but if another mathematician ask me to draw two more circles and another line to define the same circumference... i will probably believe that the first one suck less than the second one at maths! (sorry for my english!).
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Pi is better because 'a slice of pie' links nicely to radius and makes it easy for kids to remember. QED.
Exactly. It's much a "stickier"/"easier" mnemonic to tell a(n English-speaking) grade-school student "the area of a circle is 'pi r squared'" and let them giggle at the incongruity of the fact that no, silly, pie is round!
It builds a connection to a thing they already know, and it's "silly"/"absurd" enough to be "sticky" in terms of memorization.
It builds a connection to a thing they already know, and it's "silly"/"absurd" enough to be "sticky" in terms of memorization.
Wow, they've really redesigned that site's look. Not sure I like it... the figures especially look bad.
What's a nice infinite series summing to tau that isn't merely 2S(n) where S(n) sums to pi?
The simplest formula that I can think of is
pi/4 = tau/8 = atan(1) = 1 - 1/3 + 1/5 - 1/7 + ...
pi^2 / 6 = tau^2 / 24 = 1 + 1/(2^2) + 1/(3^2) + 1/(4^2) + ...
pi^2 / 12 = tau^2 / 48 = 1 - 1(/2^2) + 1/(3^2) - 1/(4^2) + ...
Ha. I've said a lot of continental-philosophy stuff that gets downvoted by the techie culture, but this is amazing.
Just checking: Do you mean that your comment amounts to continental philosophy?
I'm saying this comment should be far less contentious than my comments that somehow cite Heidegger or Deleuze.
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Okay, tau is a little better.
Meanwhile, millions of lines of code are written in languages with no type system to speak of, millions of Americans use imperial measurements, billions worldwide speak languages that are inefficient and ambiguous, and many many people aren't even educated enough to know about pi or tau.
We have much more damaging problems than multiplying by 2. Given the gigantic amount of effort it would take to fix this one, I think that effort could be better spent.
Meanwhile, millions of lines of code are written in languages with no type system to speak of, millions of Americans use imperial measurements, billions worldwide speak languages that are inefficient and ambiguous, and many many people aren't even educated enough to know about pi or tau.
We have much more damaging problems than multiplying by 2. Given the gigantic amount of effort it would take to fix this one, I think that effort could be better spent.
> millions of Americans use imperial measurements
There are two types of countries: those using 'metric' units, and those who have been to the Moon.
There's absolutely nothing fundamentally wrong with standard units (indeed, they are better for concrete manipulation). One can do science and engineering just as well with grains as with grams, with cups as with litres, with inches as with centimetres. They could do with a bit more regularisation (e.g. to make a fluid ounce a cube exactly 1, 1¼, sqrt(2) or 2 inches — each has advantages — and adjust the ounce accordingly), but they're each completely acceptable for their purpose.
The advantage of the standard units is that they are easy to physically convert: cutting a yard into feet or a foot into inches is each (⅓ in one case; ⅓, ½, ½ in the other); dividing a gallon into cups is easy (½ × 4); dividing a pound into ounces is easy (½ x 4).
The number of times a person converts between barleycorns and parsecs in a lifetime is approximately zero, while the number of times he takes one physical quantity of a unit and breaks it into smaller quantities is pretty common; optimising for the former is IMHO kinda foolish.
There are two types of countries: those using 'metric' units, and those who have been to the Moon.
There's absolutely nothing fundamentally wrong with standard units (indeed, they are better for concrete manipulation). One can do science and engineering just as well with grains as with grams, with cups as with litres, with inches as with centimetres. They could do with a bit more regularisation (e.g. to make a fluid ounce a cube exactly 1, 1¼, sqrt(2) or 2 inches — each has advantages — and adjust the ounce accordingly), but they're each completely acceptable for their purpose.
The advantage of the standard units is that they are easy to physically convert: cutting a yard into feet or a foot into inches is each (⅓ in one case; ⅓, ½, ½ in the other); dividing a gallon into cups is easy (½ × 4); dividing a pound into ounces is easy (½ x 4).
The number of times a person converts between barleycorns and parsecs in a lifetime is approximately zero, while the number of times he takes one physical quantity of a unit and breaks it into smaller quantities is pretty common; optimising for the former is IMHO kinda foolish.
> There are two types of countries: those using 'metric' units, and those who have been to the Moon.
What about Liberia and Myanmar?
What about Liberia and Myanmar?
But I think the standard unit system is one of many reasons engineers rely on formulas for 100% of their calculations. When you have that many conversion factors just to get between energy, force, distance, power, etc., you lose out on the universality of physics. At that point you find someone who has already done all the unit conversions and isolated it in a nice factor out front.
Also, IMO stuff in decimal is almost always easier to compare than fractional. When someone asks me for a size up from an 8mm wrench, I know to grab the 9mm. When someone asks me for a size up from 5/16", it'll take me a bit to get to 11/32".
Also, IMO stuff in decimal is almost always easier to compare than fractional. When someone asks me for a size up from an 8mm wrench, I know to grab the 9mm. When someone asks me for a size up from 5/16", it'll take me a bit to get to 11/32".
> Also, IMO stuff in decimal is almost always easier to compare than fractional. When someone asks me for a size up from an 8mm wrench, I know to grab the 9mm. When someone asks me for a size up from 5/16", it'll take me a bit to get to 11/32".
A lot of that is just due to how we teach fractions — but decimal notation really is nice. I wish that we used duodecimal instead: all the advantages of decimal, but with a far better base.
A lot of that is just due to how we teach fractions — but decimal notation really is nice. I wish that we used duodecimal instead: all the advantages of decimal, but with a far better base.
This would be entirely missing the point of my argument.
I understand this perspective if arguments were a limited resource and that by not fighting about tau we would win other battles (which is not true). The reality is that there is no such thing as time "better spent" withoit enforcing some sort of oppressive regieme and ways to curb imagination. I believe the tau fight is one of the rare good fights.
Arguments are a limited resource. I have 45 minutes on the train and I'm using it to argue about tau, and I don't have the time or energy for more. And when society is divided on a lot of issues, people run into outrage fatigue and become apathetic.
And to say that there is no such thing as time better spent unless you enforce an oppressive regime is a perfect solution fallacy. Sure, people will always do inefficient things, but if I could persuade even one person to behave more efficiently, that's a pretty significant gain.
I'd also like to note that if you're arguing for tau over the more popular pi based on its efficiency, you probably shouldn't argue against efficency.
And to say that there is no such thing as time better spent unless you enforce an oppressive regime is a perfect solution fallacy. Sure, people will always do inefficient things, but if I could persuade even one person to behave more efficiently, that's a pretty significant gain.
I'd also like to note that if you're arguing for tau over the more popular pi based on its efficiency, you probably shouldn't argue against efficency.
Ooooo.... I feel your passion! Great points.
Internet conversations will only take things so far, after conversation generally action is necessary. Whats nice about hacker news is that with many actions we can take action with functional programming or framework xyz immediately. With something like tau, publicity is its problem. So few people know about it that any sort of exposure benefits it. To say that untyped systems are worse than typed requires minimal exposure, simply more experience to understand why
Internet conversations will only take things so far, after conversation generally action is necessary. Whats nice about hacker news is that with many actions we can take action with functional programming or framework xyz immediately. With something like tau, publicity is its problem. So few people know about it that any sort of exposure benefits it. To say that untyped systems are worse than typed requires minimal exposure, simply more experience to understand why
> "6/28 in the American calendar system"
it's enough to ignore this day by all people except Americans, so don't worry about others :)
it's enough to ignore this day by all people except Americans, so don't worry about others :)
One argument in favor of tau is that in many formulas pi often has the multiplier 2 in front of it. If these formulas are written in terms of tau, they may become slightly easier to memorize and manipulate. Perhaps so, but I don't really care about this. It’s not a big difference. Besides, there are also lots of formulas that are easier to memorize and manipulate using pi instead of tau. The probability density function of the standard Cauchy distribution f(x) = 1/pi * 1/(1 + x^2) is one example.
However, to get a deep understanding of mathematics, I want to understand the connections between different parts of mathematics. If I see a mathematical formula with the constant pi in it, I ask myself: "How is this connected to circles?" The idea of tau taught me that that 2pi is the natural state of affairs. If I see pi by itself, I need to ask myself: "How is this connected to half-circles? Or has the multiplier 2 been cancelled away?”
So, why does the pdf of the standard Cauchy distribution above contain pi instead of 2pi? What is the standard Cauchy distribution anyway? Take a gun that shoots particles in random directions, and place it one unit distance away from an infinitely long wall. Standard Cauchy distribution is where the particles will hit the wall. The particle will only hit the wall if it is shot in a direction towards it. This corresponds to 180 degrees - and there you have it: the connection to half-circles. Of course, you also have to work out the technical details. But on an intuitive level, when I see pi in the pdf of the standard Cauchy distribution I don’t think about how the missing multiplier makes it easier to remember a bunch of symbols; I think of particles hitting a wall.