7 ms·

The State of the Tau

108 points by mhartl 13y ago

thetwiceler 13y ago
Wow this is hilarious. Truly a "manifesto" to set a constant for 2pi. It definitely has a great point, and I totally agree that it's a better constant than pi, but it's not the worst historical accident of mathematics. The thing is, tau is really my go-to variable when I need a 2nd constant to compare with t. It is already used as a time constant, a dummy integration variable that substitutes for t... The manifesto has a pre-made counter-argument to this... But I'm not sure it's convincing. The thing is that we like to say some variation of e^(-tau) a lot, and we also like to say e^(2 pi i) a lot. And often we combine these (rotation and exponential decay), and we get e^(-tau+2pi i). And this would ruin tau notation! It would be e^(-tau'+tau i)...
- samatman 13y ago
  May I recommend ת ? Tau notation is pretty excellent stuff. There's even a gloss: ת in mainstream Kabbalah corresponds to Saturn, the god of time. Electrical engineers need i for something else, incidentally, and use j for imaginary numbers. Mathematical notation is intentionally flexible.
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- mhartl 13y ago
  I really like using tau as a time constant, too, and I'm sorry to lose it when there are pi's in the picture. It's tough picking good notation, though, and unfortunately no choice is conflict-free. Before writing The Tau Manifesto, I tried several other candidates before settling on tau. After using it for a while, tau started to "feel right" in ways that other choices didn't. Then, as noted in the manifesto, I found out that I wasn't alone: Since the original launch of The Tau Manifesto, I have learned that Peter Harremoës independently proposed using τ to “π Is Wrong!” author Bob Palais in 2010, John Fisher proposed τ in a Usenet post in 2004, and Joseph Lindenberg anticipated both the argument and the symbol more than twenty years before!
  - thetwiceler 13y ago
    Yeah, I've literally never seen lower-case pi used as a variable (thank goodness!). tau is a good choice because it looks like pi, and is possible for normal people to write (I would have had a fit if it were lowercase xi!). I like the explanation page for why tau a lot, and I think that page is much more important than actually using tau in real mathematical works. It's a great explanation of how 2*pi is the more natural quantity, and how it comes into play in the different functions. But it's just much easier to deal with factors of 2 than it is to explain alternate notation. In fact, I think a lot of making mathematical works easy to understand is using commonly accepted notation...
  - Radix_ 13y ago
    Why didn't you make the circle constant a circle in your manifesto? I'm not sure but it seems the function composition symbol could be safely reused without confusion or perhaps a larger circle with a drawn radius. Wouldn't a circle have avoided some of the Tau criticisms?
    - sesqu 13y ago
      The major problem is that a circle is too ambiguous a symbol; you'd need to add a notch or line or something to make it clear that you're referencing a constant. And when you do that, it becomes harder to write. I always favored a variation on ◷, something like ◯̶ or ◯̵ or ○̵. Unfortunately, combined glyphs are representationally very unstable.
- jessriedel 13y ago
  Yea, the thing about a circle constant (whether pi or tau) is that it's so ubiquitous you can't use the symbol for anything else. It's pretty rare to use lower case pi for anything in physics and math, and most of the exceptions including something like boldface or a vector hat to distinguish it. There are no available symbols. The only option is to introduce a new one, like ת suggested by samatman, or possibly by using a weird typeface (like \mathfrak in LaTeX). Incidentally, my friend has pointed out to me that sigma is a much better choice than tau if we're going with existing symbols. Who cares about tau sounding like a "t" for "turn", when sigma actually looks like someone trying to measure the circumference of a circle with the radius of the circle: σ.
  - anigbrowl 13y ago
    There are no available symbols. Why not just start in on Chinese characters? There's plenty of those and I'm sure Chinese mathematicians could suggest some likely candidates based on their long mathematical traditions. http://en.wikipedia.org/wiki/Chinese_mathematics http://en.wikipedia.org/wiki/Chinese_mathematics
  - manicbovine 13y ago
    \pi is very frequently used to denote a projection or some other mapping.
  - mhartl 13y ago
    σ has potential, but there's a pretty bad conflict with standard deviation. The circle constant shows up an awful lot in statistics. Who cares about tau sounding like a "t" for "turn" Pi comes from perimeter, phi comes from Phidias, etc. There's a long tradition of naming constants using a linguistic root (typically Greek). To my knowledge, there's no precedent for using a symbol based on its visual appearance to a geometric object. (Of course, you could always break with tradition.)
    - sesqu 13y ago
      To be completely fair, π turns up occasionally in statistics too, as a symbol for probability.
- jessriedel 13y ago
  What's the worse historical accident of mathematics?
  - thetwiceler 13y ago
    The use of infinitesimals in calculus! Things like integrating against "dx" or taking a derivative of a function "df/dx" make calculus seem like something it's not. Worse, it really introduces a whole new language whose rules for manipulation are unclear. Like if we have a function f(x,y), we often write the "total" derivative of f as df = (df/dx) dx + (df/dy) dy And the rules for manipulating a differential like this are quite strange (e.g., dx on top cannot cancel the dx on bottom). There's a language of differential forms that makes this notation a bit more rigorous, but I still think the notation is very misleading and confusing, especially for those who begin to learn calculus.
    - kryten 13y ago
      This. Notation in formal mathematics is just totally awful. You pick on a great example as well there.
    - currymesurprise 13y ago
      To be fair to mathematics, in your example the proper notation is df = (del f/del x) dx + (del f/del y) dy, which does not suggest the possibility of cancelation.
- mtdewcmu 13y ago
  If it succeeds, it's worthy of the Ig Nobel Prize.
sixbrx 13y ago
Very interesting that Euler even sometimes used π as C/r!
codeflo 13y ago
No love for pi/4? It has (IMHO) the most beautiful infinite series: pi/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - ...
- philthesong 13y ago
  Does pi/4 equal to tau/8?
  - jessriedel 13y ago
    Yes.
- gizmo686 13y ago
  If you re-arrange the terms, you can find that that series equals tau/4.
  - Retric 13y ago
    No, no you can't it's tau/8.
    - gizmo686 13y ago
      Its a conditionally convergent series; you can re-arrange the terms to add up to any real number you like (or diverge). [1]. [1]http://en.wikipedia.org/wiki/Conditional_convergence http://en.wikipedia.org/wiki/Conditional_convergence
      - Retric 13y ago
        You can construct a new series such that every term in the old one is there. However, if you try and rearrange the terms you end up with a backlog that coverages to the difference between the series. Which is why the actual series converges in the first place.
schiffern 13y ago
Best argument for 𝛕 – when we meet aliens, they'll laugh at us if we're still using C/d. "No no, we were transmitting on hydrogen * 𝛕!"
- gizmo686 13y ago
  I don't think so. The stupid thing about pi is that it is the only place in math that you ever talk about the diameter of the circle, everything else is expressed in terms of the radius. Admittedly, the radius does feel like the natural way to define a circle (all points r away from the center), however that is not the only way to define a circle, and we have no other intelligent species to use a reference for how objectively natural that way is.
  - schiffern 13y ago
    We're saying the same thing. I'm using that as an example of why it seems like radius is the right one. Of course we have no examples, but we do know that an alien intelligence would be free from the historical accidents of humanity. "Mathematics is the only truly universal language", as Sagan would say. Here's a similar argument: In mathematics, do we measure angles in radians or diameterans? Why is that? >that is not the only way to define a circle Ok, let's hear 'em! What rigorous definition of a circle is more simply expressed in terms of diameter? The only one I can think of is, "the smallest volume shape that can contain a length D line segment at any angle", but that seems contrived to me. I shutter at using the solution to optimization problem as a definition, except when there is no other choice. What's worst, it seems like a trivial derivative of the radial definition, just with extraneous concepts tacked on.
    - gizmo686 13y ago
      The most natural, diameter based, definition I can think of is, for any given point on the circle, the distance to the farthest point is a constant, D. You probably need to add additional restrictions to make this rigorous (such as requiring the circle to be form a continuous, closed loop). You could also imagine rotating a line segment about its center. This has a nice sense of symmetry, as we are rotating about the center, not an arbitrary end. This also implies a generalization to off center rotations.
thomasahle 13y ago
Clearly the idea of pi and tau are both wrong, since they make the 2d circle the reference instead of the 1d circle. Notice the formula for the volume of an n-ball is proportional to pi^(n/2)r^n! Define tau properly, and could become (tau r)^n!! Also notice how many other cases outside of 2d geometry uses sqrt(pi) or sqrt(2pi). The gamma function, normal distribution, stirling approximation etc. are all full of it, and when you rarely need pi or tau, adding a ^2 is much nicer than always having to draw a big ugly sqrt. I don't mind tau especially, but it is seems the wrong battle to fight. Sqrt(pi) also has mythical powers: http://www.squarerootofpi.com/ http://www.squarerootofpi.com/
- merlincorey 13y ago
  I'm not against being open to things like ancient aliens, but I think the fact that squarerootofpi references that and astrology, etc puts it on completely different footing from tau, which the post we are discussing right now mentions being included in programming languages and scientific papers. [edit: obviously this says nothing about square root of half tau and its uses!] I put tau into my math-related code last year, and I have enjoyed the results (which are much the same as before, of course).
- saalweachter 13y ago
  That is not entirely correct. http://en.wikipedia.org/wiki/N-sphere http://en.wikipedia.org/wiki/N-sphere All N-sphere volume formulas have whole-powers of 'pi' (or 'tau') as factors
- mhartl 13y ago
  You should read http://tauday.com/tau-manifesto#sec-getting_to_the_bottom_of_pi http://tauday.com/tau-manifesto#sec-getting_to_the_bottom_of... which goes into great detail on surface areas and volumes of hyperspheres. Spoiler alert: it's actually pi/2 = tau/4 that emerges victorious.
gaze 13y ago
After conferring with some fellow mathematicians, nobody had heard of the "tau movement", and nobody cares. Tau is for people who want to feel like they're taking stances on things that matter in math without actually putting in any of the work to do math.
- B-Con 13y ago
  I think tau is really just motivated by people who like things to be "clean". You see them in all walks, wishing programming language syntax, math notation, spelling, grammar, etc, had been chosen to be cleaner or make more sense from the beginning. I don't have a strong opinion, but from what I've seen I think tau makes the most sense. These things are fun to think about, because it requires that you think about how pi is used in mathematics and contemplate how we organize and use the concepts related to pi. It's a good brain exercise. But it's hardly worth getting into a fight over.
  - dajohnson89 13y ago
    But how much cleaner is using tau, beyond grade-school geometry? Euler's equation is arguably the most beautiful relationship in mathematics: e^(ipi)= -1. I'm afraid tau would dirty it up. Edit: OK, e^(itau)=1, which is quite nice. :-/
    - epidemian 13y ago
      Not only that. If you think about multiplying any number z by e^(ia), where a is a real number, as rotating z by a radians in the complex plane, then i think the tau version of the Euler's equation conveys a more fundamental meaning: rotating any complex number by a whole turn yields that same number, it's the same as multiplying it by the multiplicative identity (i.e. doing nothing). e^(iτ) = 1 And, as rotating something by one whole turn is the same as rotating it by any number of whole turns, you can get a nice intuitive series of equivalences that are not as pretty if you where to use pi: 1 = e^0 = e^(iτ) = e^(i2τ) = ...
      - 13y ago
mkl 13y ago
Pi is just fine. It makes just as much sense as tau, has history on its side, is simpler for many formulas, etc. The Pi Manifesto has good counterarguments and heaps of examples: http://www.thepimanifesto.com/ http://www.thepimanifesto.com/
- mhartl 13y ago
  Many of the examples in The Pi Manifesto are wrong. See the rebuttal I posted last year: http://tauday.com/tau-manifesto#sec-the_pi_manifesto_a_rebuttal http://tauday.com/tau-manifesto#sec-the_pi_manifesto_a_rebut... Indeed, the author of The Pi Manifesto essentially rebutted his own work in March of last year: http://spikedmath.com/fact-005.html http://spikedmath.com/fact-005.html

108 points by mhartl 13y ago

Wow this is hilarious. Truly a "manifesto" to set a constant for 2pi. It definitely has a great point, and I totally agree that it's a better constant than pi, but it's not the worst historical accident of mathematics. The thing is, tau is really my go-to variable when I need a 2nd constant to compare with t. It is already used as a time constant, a dummy integration variable that substitutes for t... The manifesto has a pre-made counter-argument to this... But I'm not sure it's convincing. The thing is that we like to say some variation of e^(-tau) a lot, and we also like to say e^(2 pi i) a lot. And often we combine these (rotation and exponential decay), and we get e^(-tau+2pi i). And this would ruin tau notation! It would be e^(-tau'+tau i)...

samatman 13y ago
May I recommend ת ? Tau notation is pretty excellent stuff. There's even a gloss: ת in mainstream Kabbalah corresponds to Saturn, the god of time. Electrical engineers need i for something else, incidentally, and use j for imaginary numbers. Mathematical notation is intentionally flexible.
- deleted 13y ago
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- deleted 13y ago
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mhartl 13y ago
I really like using tau as a time constant, too, and I'm sorry to lose it when there are pi's in the picture. It's tough picking good notation, though, and unfortunately no choice is conflict-free. Before writing The Tau Manifesto, I tried several other candidates before settling on tau. After using it for a while, tau started to "feel right" in ways that other choices didn't. Then, as noted in the manifesto, I found out that I wasn't alone: Since the original launch of The Tau Manifesto, I have learned that Peter Harremoës independently proposed using τ to “π Is Wrong!” author Bob Palais in 2010, John Fisher proposed τ in a Usenet post in 2004, and Joseph Lindenberg anticipated both the argument and the symbol more than twenty years before!
- thetwiceler 13y ago
  Yeah, I've literally never seen lower-case pi used as a variable (thank goodness!). tau is a good choice because it looks like pi, and is possible for normal people to write (I would have had a fit if it were lowercase xi!). I like the explanation page for why tau a lot, and I think that page is much more important than actually using tau in real mathematical works. It's a great explanation of how 2*pi is the more natural quantity, and how it comes into play in the different functions. But it's just much easier to deal with factors of 2 than it is to explain alternate notation. In fact, I think a lot of making mathematical works easy to understand is using commonly accepted notation...
- Radix_ 13y ago
  Why didn't you make the circle constant a circle in your manifesto? I'm not sure but it seems the function composition symbol could be safely reused without confusion or perhaps a larger circle with a drawn radius. Wouldn't a circle have avoided some of the Tau criticisms?
  - sesqu 13y ago
    The major problem is that a circle is too ambiguous a symbol; you'd need to add a notch or line or something to make it clear that you're referencing a constant. And when you do that, it becomes harder to write. I always favored a variation on ◷, something like ◯̶ or ◯̵ or ○̵. Unfortunately, combined glyphs are representationally very unstable.
jessriedel 13y ago
Yea, the thing about a circle constant (whether pi or tau) is that it's so ubiquitous you can't use the symbol for anything else. It's pretty rare to use lower case pi for anything in physics and math, and most of the exceptions including something like boldface or a vector hat to distinguish it. There are no available symbols. The only option is to introduce a new one, like ת suggested by samatman, or possibly by using a weird typeface (like \mathfrak in LaTeX). Incidentally, my friend has pointed out to me that sigma is a much better choice than tau if we're going with existing symbols. Who cares about tau sounding like a "t" for "turn", when sigma actually looks like someone trying to measure the circumference of a circle with the radius of the circle: σ.
- anigbrowl 13y ago
  There are no available symbols. Why not just start in on Chinese characters? There's plenty of those and I'm sure Chinese mathematicians could suggest some likely candidates based on their long mathematical traditions. http://en.wikipedia.org/wiki/Chinese_mathematics http://en.wikipedia.org/wiki/Chinese_mathematics
- manicbovine 13y ago
  \pi is very frequently used to denote a projection or some other mapping.
- mhartl 13y ago
  σ has potential, but there's a pretty bad conflict with standard deviation. The circle constant shows up an awful lot in statistics. Who cares about tau sounding like a "t" for "turn" Pi comes from perimeter, phi comes from Phidias, etc. There's a long tradition of naming constants using a linguistic root (typically Greek). To my knowledge, there's no precedent for using a symbol based on its visual appearance to a geometric object. (Of course, you could always break with tradition.)
  - sesqu 13y ago
    To be completely fair, π turns up occasionally in statistics too, as a symbol for probability.
jessriedel 13y ago
What's the worse historical accident of mathematics?
- thetwiceler 13y ago
  The use of infinitesimals in calculus! Things like integrating against "dx" or taking a derivative of a function "df/dx" make calculus seem like something it's not. Worse, it really introduces a whole new language whose rules for manipulation are unclear. Like if we have a function f(x,y), we often write the "total" derivative of f as df = (df/dx) dx + (df/dy) dy And the rules for manipulating a differential like this are quite strange (e.g., dx on top cannot cancel the dx on bottom). There's a language of differential forms that makes this notation a bit more rigorous, but I still think the notation is very misleading and confusing, especially for those who begin to learn calculus.
  - kryten 13y ago
    This. Notation in formal mathematics is just totally awful. You pick on a great example as well there.
  - currymesurprise 13y ago
    To be fair to mathematics, in your example the proper notation is df = (del f/del x) dx + (del f/del y) dy, which does not suggest the possibility of cancelation.
mtdewcmu 13y ago
If it succeeds, it's worthy of the Ig Nobel Prize.

sixbrx 13y ago

Very interesting that Euler even sometimes used π as C/r!

codeflo 13y ago

No love for pi/4? It has (IMHO) the most beautiful infinite series: pi/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - ...

philthesong 13y ago
Does pi/4 equal to tau/8?
- jessriedel 13y ago
  Yes.
gizmo686 13y ago
If you re-arrange the terms, you can find that that series equals tau/4.
- Retric 13y ago
  No, no you can't it's tau/8.
  - gizmo686 13y ago
    Its a conditionally convergent series; you can re-arrange the terms to add up to any real number you like (or diverge). [1]. [1]http://en.wikipedia.org/wiki/Conditional_convergence http://en.wikipedia.org/wiki/Conditional_convergence
    - Retric 13y ago
      You can construct a new series such that every term in the old one is there. However, if you try and rearrange the terms you end up with a backlog that coverages to the difference between the series. Which is why the actual series converges in the first place.

schiffern 13y ago

Best argument for 𝛕 – when we meet aliens, they'll laugh at us if we're still using C/d. "No no, we were transmitting on hydrogen * 𝛕!"

gizmo686 13y ago
I don't think so. The stupid thing about pi is that it is the only place in math that you ever talk about the diameter of the circle, everything else is expressed in terms of the radius. Admittedly, the radius does feel like the natural way to define a circle (all points r away from the center), however that is not the only way to define a circle, and we have no other intelligent species to use a reference for how objectively natural that way is.
- schiffern 13y ago
  We're saying the same thing. I'm using that as an example of why it seems like radius is the right one. Of course we have no examples, but we do know that an alien intelligence would be free from the historical accidents of humanity. "Mathematics is the only truly universal language", as Sagan would say. Here's a similar argument: In mathematics, do we measure angles in radians or diameterans? Why is that? >that is not the only way to define a circle Ok, let's hear 'em! What rigorous definition of a circle is more simply expressed in terms of diameter? The only one I can think of is, "the smallest volume shape that can contain a length D line segment at any angle", but that seems contrived to me. I shutter at using the solution to optimization problem as a definition, except when there is no other choice. What's worst, it seems like a trivial derivative of the radial definition, just with extraneous concepts tacked on.
  - gizmo686 13y ago
    The most natural, diameter based, definition I can think of is, for any given point on the circle, the distance to the farthest point is a constant, D. You probably need to add additional restrictions to make this rigorous (such as requiring the circle to be form a continuous, closed loop). You could also imagine rotating a line segment about its center. This has a nice sense of symmetry, as we are rotating about the center, not an arbitrary end. This also implies a generalization to off center rotations.

thomasahle 13y ago

Clearly the idea of pi and tau are both wrong, since they make the 2d circle the reference instead of the 1d circle. Notice the formula for the volume of an n-ball is proportional to pi^(n/2)r^n! Define tau properly, and could become (tau r)^n!! Also notice how many other cases outside of 2d geometry uses sqrt(pi) or sqrt(2pi). The gamma function, normal distribution, stirling approximation etc. are all full of it, and when you rarely need pi or tau, adding a ^2 is much nicer than always having to draw a big ugly sqrt. I don't mind tau especially, but it is seems the wrong battle to fight. Sqrt(pi) also has mythical powers: http://www.squarerootofpi.com/ http://www.squarerootofpi.com/

merlincorey 13y ago
I'm not against being open to things like ancient aliens, but I think the fact that squarerootofpi references that and astrology, etc puts it on completely different footing from tau, which the post we are discussing right now mentions being included in programming languages and scientific papers. [edit: obviously this says nothing about square root of half tau and its uses!] I put tau into my math-related code last year, and I have enjoyed the results (which are much the same as before, of course).
saalweachter 13y ago
That is not entirely correct. http://en.wikipedia.org/wiki/N-sphere http://en.wikipedia.org/wiki/N-sphere All N-sphere volume formulas have whole-powers of 'pi' (or 'tau') as factors
mhartl 13y ago
You should read http://tauday.com/tau-manifesto#sec-getting_to_the_bottom_of_pi http://tauday.com/tau-manifesto#sec-getting_to_the_bottom_of... which goes into great detail on surface areas and volumes of hyperspheres. Spoiler alert: it's actually pi/2 = tau/4 that emerges victorious.

gaze 13y ago

After conferring with some fellow mathematicians, nobody had heard of the "tau movement", and nobody cares. Tau is for people who want to feel like they're taking stances on things that matter in math without actually putting in any of the work to do math.

B-Con 13y ago
I think tau is really just motivated by people who like things to be "clean". You see them in all walks, wishing programming language syntax, math notation, spelling, grammar, etc, had been chosen to be cleaner or make more sense from the beginning. I don't have a strong opinion, but from what I've seen I think tau makes the most sense. These things are fun to think about, because it requires that you think about how pi is used in mathematics and contemplate how we organize and use the concepts related to pi. It's a good brain exercise. But it's hardly worth getting into a fight over.
- dajohnson89 13y ago
  But how much cleaner is using tau, beyond grade-school geometry? Euler's equation is arguably the most beautiful relationship in mathematics: e^(ipi)= -1. I'm afraid tau would dirty it up. Edit: OK, e^(itau)=1, which is quite nice. :-/
  - epidemian 13y ago
    Not only that. If you think about multiplying any number z by e^(ia), where a is a real number, as rotating z by a radians in the complex plane, then i think the tau version of the Euler's equation conveys a more fundamental meaning: rotating any complex number by a whole turn yields that same number, it's the same as multiplying it by the multiplicative identity (i.e. doing nothing). e^(iτ) = 1 And, as rotating something by one whole turn is the same as rotating it by any number of whole turns, you can get a nice intuitive series of equivalences that are not as pretty if you where to use pi: 1 = e^0 = e^(iτ) = e^(i2τ) = ...
    - 13y ago

mkl 13y ago

Pi is just fine. It makes just as much sense as tau, has history on its side, is simpler for many formulas, etc. The Pi Manifesto has good counterarguments and heaps of examples: http://www.thepimanifesto.com/ http://www.thepimanifesto.com/

mhartl 13y ago
Many of the examples in The Pi Manifesto are wrong. See the rebuttal I posted last year: http://tauday.com/tau-manifesto#sec-the_pi_manifesto_a_rebuttal http://tauday.com/tau-manifesto#sec-the_pi_manifesto_a_rebut... Indeed, the author of The Pi Manifesto essentially rebutted his own work in March of last year: http://spikedmath.com/fact-005.html http://spikedmath.com/fact-005.html