The Pi Manifesto(thepimanifesto.com)
thepimanifesto.com
The Pi Manifesto
http://www.thepimanifesto.com/
18 comments
I really don't appreciate the argument that "it doesn't matter what the constant is". Of course it doesn't matter if you are only interested in the result of a calculation, but like you say, it does matter when you're trying to understand what the formulas express.
I find it surprising that this article, and many mathematicians asked to comment in the media, pull out the area formula as their prime counter-example, seemingly forgetting that using pi is only obscuring the mechanics behind it.
I find it surprising that this article, and many mathematicians asked to comment in the media, pull out the area formula as their prime counter-example, seemingly forgetting that using pi is only obscuring the mechanics behind it.
1. This is a good discussion to have. Those who are dismissive show, in my opinion, a lack of intellectual curiosity. Elegance for the sake of elegance is a worthwhile goal.
I consider myself to have intellectual curiosity but don't see this discussion to be particularly compelling. I don't think there is anything about intellectual curiosity which requires one to consider every question that's raised no matter what. There simply isn't enough time to consider everything, so prioritization is always necessary.
I agree with the sentiment expressed in your last sentence, but it's hard for me to imagine that one wouldn't give priority to something so fundamental as the circle constant (which touches so much in math, science, engineering, software, etc). That's doubly true considering the time investment for reading about the arguments is rather small, 15 minutes or so. And it's triply true considering that very many other people have vetted the argument for you, enough that mainstream media are writing articles on the topic.
So no, you don't have to consider every question that's raised "no matter what" to be considered intellectually curious, but this is hardly such a trivial, general case as that implies.
So no, you don't have to consider every question that's raised "no matter what" to be considered intellectually curious, but this is hardly such a trivial, general case as that implies.
I guess the pertinent question is whether this stimulates your intellectual curiosity because you find it to be an interesting question, or at least because it raises interesting questions, or whether it stimulates your curiosity because you think it's exceedingly important and making this change will have big consequences.
I think the dismissive types see people obsessing over it as the latter, and this is, at least in my opinion, way more than is called for. There's nothing wrong with considering the consequences of such a change just for some mathematical fun. (After all, who here hasn't drawn out a complete system of units based on furlongs, fortnights, etc. as the fundamental units, rather than mks or cgs? What's that? None of you? Oh....) But the original manifesto seems to be overstating its case severely in terms of the consequences of such a change, and I think that's what others are reacting to.
Ignoring the boring costs of switching the installed base of mathematicians, software, etc. from one constant to the other, the benefit of a switch seems to be that certain things become easier and other certain things become harder. At best, there's a small net gain. Hardly seems worth the discussion when considered in that light.
If you're considering it as an interesting exercise to extract meaning from equations, well, go for it! But that seems to fit more under the banner of "What would happen if we switched this?" rather than "It would provide enormous benefit if everybody switched this!" as the original manifesto seemed to be saying.
I think the dismissive types see people obsessing over it as the latter, and this is, at least in my opinion, way more than is called for. There's nothing wrong with considering the consequences of such a change just for some mathematical fun. (After all, who here hasn't drawn out a complete system of units based on furlongs, fortnights, etc. as the fundamental units, rather than mks or cgs? What's that? None of you? Oh....) But the original manifesto seems to be overstating its case severely in terms of the consequences of such a change, and I think that's what others are reacting to.
Ignoring the boring costs of switching the installed base of mathematicians, software, etc. from one constant to the other, the benefit of a switch seems to be that certain things become easier and other certain things become harder. At best, there's a small net gain. Hardly seems worth the discussion when considered in that light.
If you're considering it as an interesting exercise to extract meaning from equations, well, go for it! But that seems to fit more under the banner of "What would happen if we switched this?" rather than "It would provide enormous benefit if everybody switched this!" as the original manifesto seemed to be saying.
I won't belabor the point any further than this, but the argument really is that it would provide enormous benefit. It's not just a "what would happen?", and it's not that τ "extracts" meaning from equations. It makes the intrinsic meanings of equations drastically more clear. For instance, what does sin(x) mean? What does Euler's equation mean? They're both eminently simple concepts, but they remain obscured by π.
It's exactly like refactoring code: sure, it does the same thing, but now it's more compact, more concise, more clear, more elegant; the parts of the system all fit together better, and people coming onto the project will be able to learn it faster. If you don't care about those things, then you won't refactor your code, or see the point of τ.
It's exactly like refactoring code: sure, it does the same thing, but now it's more compact, more concise, more clear, more elegant; the parts of the system all fit together better, and people coming onto the project will be able to learn it faster. If you don't care about those things, then you won't refactor your code, or see the point of τ.
>They're both eminently simple concepts, but they remain obscured by π.
They are equally obscured by tau, or whatever other arc-length you might choose! The meaning does not depend on the definition of a circle but on the properties of the exponential function:
http://en.wikipedia.org/wiki/Euler%27s_formula#Proofs
As for sin(x), it is a function based on circles and arc lengths, but specific values of arc length do not enter the definition.
They are equally obscured by tau, or whatever other arc-length you might choose! The meaning does not depend on the definition of a circle but on the properties of the exponential function:
http://en.wikipedia.org/wiki/Euler%27s_formula#Proofs
As for sin(x), it is a function based on circles and arc lengths, but specific values of arc length do not enter the definition.
> The meaning does not depend on the definition of a circle but on the properties of the exponential function
Actually, you can go backwards and say that its circular properties define the exponential function. Euler's formula describes the rotation of the unit vector through the imaginary plane.
> As for sin(x)...specific values of arc length do not enter the definition
It's not about the definition, it's about the meaning. Sin(x) is the height of the circle at x radians. And it's super awesome with tau: one tau is full circle, and one period.
Actually, you can go backwards and say that its circular properties define the exponential function. Euler's formula describes the rotation of the unit vector through the imaginary plane.
> As for sin(x)...specific values of arc length do not enter the definition
It's not about the definition, it's about the meaning. Sin(x) is the height of the circle at x radians. And it's super awesome with tau: one tau is full circle, and one period.
>And it's super awesome with tau: one tau is full circle, and one period.
Well, if you've already made the jump to understanding negative heights as going below the real line. This is a topic for an introductory course in geometry? When I learned about sine and cosine, it was first defined in terms of SOHCAHTOA!
Well, if you've already made the jump to understanding negative heights as going below the real line. This is a topic for an introductory course in geometry? When I learned about sine and cosine, it was first defined in terms of SOHCAHTOA!
The section of the Tau Manifesto on the area of the circle (which the author describes as pi's coup de grace) was the one that convinced me tau makes more sense.
Area is defined by integrals- good point.
The area of the traditional unit circle is π, which has strong ties to the definition of every trigonometric function, and the reason that radians of common fractions of the unit circle are expressed in terms of π is related to the integrals used to derive arc length.
Just as an exercise, try setting the area of the unit circle to 2π, and then see how meaningful your radian measurements are. How many radians are in a quarter arc of the circle with area 2π? :3
Just as an exercise, try setting the area of the unit circle to 2π, and then see how meaningful your radian measurements are. How many radians are in a quarter arc of the circle with area 2π? :3
> The area of the traditional unit circle is π
The area of the unit circle is 3.14(etc) units squared. The result is a number; it's how you get there that's important. You get there by integrating. The "right" equation for area is not πr^2 or τr^2/2; it's the integral that leads to either of those equations. The 1/2 in the τ version is meaningful because it is an artifact of the integration. The lack of the 1/2 in the π version shows why it's "wrong"—it's not as meaningful.
> try setting the area of the unit circle to 2π
This is a nonsensical statement. As is "traditional unit circle," but I let that one slide already.
> the reason that radians of common fractions of the unit circle are expressed in terms of π
The reason they're expressed in terms of π is because a circle constant is needed, and π was chosen. And it was chosen hastily.
Your arguments are unsound!
The area of the unit circle is 3.14(etc) units squared. The result is a number; it's how you get there that's important. You get there by integrating. The "right" equation for area is not πr^2 or τr^2/2; it's the integral that leads to either of those equations. The 1/2 in the τ version is meaningful because it is an artifact of the integration. The lack of the 1/2 in the π version shows why it's "wrong"—it's not as meaningful.
> try setting the area of the unit circle to 2π
This is a nonsensical statement. As is "traditional unit circle," but I let that one slide already.
> the reason that radians of common fractions of the unit circle are expressed in terms of π
The reason they're expressed in terms of π is because a circle constant is needed, and π was chosen. And it was chosen hastily.
Your arguments are unsound!
>And it was chosen hastily.
It was chosen here:
http://arxiv.org/abs/math/0506415
Elegance is found in arguments and proofs, not in results, and so any attempt to look at equations is really missing the point. If you believe Euler's methods might be simplified by using 2pi instead of pi, first consider a look at the methods themselves.
It was chosen here:
http://arxiv.org/abs/math/0506415
Elegance is found in arguments and proofs, not in results, and so any attempt to look at equations is really missing the point. If you believe Euler's methods might be simplified by using 2pi instead of pi, first consider a look at the methods themselves.
Your reasoning is circular (get it!). If the circle constant were τ, Euler would have found zeta(2)=τ^2/24 instead. The proof is the same.
The definition of the circle constant comes first.
The definition of the circle constant comes first.
>Your reasoning is circular (get it!). If the circle constant were τ, Euler would have found zeta(2)=τ^2/24 instead. The proof is the same.
>The definition of the circle constant comes first.
You didn't read the paper, did you? The "circle constant" wasn't even defined when it was written. He picked it out of thin air in that very paper in order to make his arguments more clear.
>The definition of the circle constant comes first.
You didn't read the paper, did you? The "circle constant" wasn't even defined when it was written. He picked it out of thin air in that very paper in order to make his arguments more clear.
I'll rephrase using his words. He wrote: "Namely, I have found for six times the sum of this series to be equal to the square of the perimeter of a circle whose diameter is 1."
The reason he uses π is due to his choice of diameter. Had he looked at the unit circle instead with a radius of 1, he would have written: "Namely, I have found for twenty-four times the sum of this series to be equal to the square of the perimeter of a circle whose radius is 1."
Again, the proof is the same, but he chose to use a unit diameter rather than radius. This is exactly equivalent to saying π=C/D instead of τ=C/r.
The reason he uses π is due to his choice of diameter. Had he looked at the unit circle instead with a radius of 1, he would have written: "Namely, I have found for twenty-four times the sum of this series to be equal to the square of the perimeter of a circle whose radius is 1."
Again, the proof is the same, but he chose to use a unit diameter rather than radius. This is exactly equivalent to saying π=C/D instead of τ=C/r.
>Again, the proof is the same, but he chose to use a unit diameter rather than radius. This is exactly equivalent to saying π=C/D instead of τ=C/r.
So it does not make it more clear? This contradicts your original assertion.
So it does not make it more clear? This contradicts your original assertion.
Hardly. It's acknowledging that both choices are definitions.
One provides clarity and is related directly to the unit circle, the other is related to the circle with radius 1/2. Which is more intuitive?
One provides clarity and is related directly to the unit circle, the other is related to the circle with radius 1/2. Which is more intuitive?
>One provides clarity
How so? Neither is more intuitive. The unit circle is itself a definition you have grabbed. The notion of defining a circle by its radius comes to us from Euclid:
>"Let the following be postulated":
>1. "To draw a straight line from any point to any point."
>2. "To produce [extend] a finite straight line continuously in a straight line."
>3. "To describe a circle with any centre and distance [radius]."
>4. "That all right angles are equal to one another."
>5. The parallel postulate: "That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles."
In fact, pi/2 itself is sitting right there in the fourth axiom, and pi is in the fifth. 2pi is nowhere to be found.
How so? Neither is more intuitive. The unit circle is itself a definition you have grabbed. The notion of defining a circle by its radius comes to us from Euclid:
>"Let the following be postulated":
>1. "To draw a straight line from any point to any point."
>2. "To produce [extend] a finite straight line continuously in a straight line."
>3. "To describe a circle with any centre and distance [radius]."
>4. "That all right angles are equal to one another."
>5. The parallel postulate: "That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles."
In fact, pi/2 itself is sitting right there in the fourth axiom, and pi is in the fifth. 2pi is nowhere to be found.
It's tau/4 that's sitting in the fourth postulate, and tau is right there in the fifth (the sum of all four angles formed by the "straight line falling" on the side of the two straight lines' intersection).
Besides, Euclid would have been a tau advocate, as he defined circles with their radius, which is clearly superior to the diameter.
I can't help people chose the poorer constant for so long; I can only hope to help correct them.
Besides, Euclid would have been a tau advocate, as he defined circles with their radius, which is clearly superior to the diameter.
I can't help people chose the poorer constant for so long; I can only hope to help correct them.
>tau is right there in the fifth (the sum of all four angles formed by the "straight line falling" on the side of the two straight lines' intersection).
...yes, but that makes the postulate meaningless! You have to look at one side of the line in order for the postulate to have any relevance.
>Euclid would have been a tau advocate
Oh yeah? Well.. well... Ramanujan would have been a pi advocate! Ha!
>which is clearly superior to the diameter.
It is expedient in the process of mathematical argumentation. Looking at expedience, though, we see that using a constant 2pi introduces an untoward amount of fractions into just about every mathematical calculation -- see for example here:
http://en.wikipedia.org/wiki/Basel_problem#A_rigorous_proof_...
Irrespective of the definition of constants, which is long since forgotten at this point (how much of a pain is it to define a circle, starting from ZFC?), it is kind of disappointing to see you refusing to read the proofs which you claim to be clarifying -- most of them get uglier moving to tau, on a cursory examination of the seminal work Proofs from THE BOOK. Go on, mentally replace every instance of "2pi" with "tau" and "pi" with "tau/2" in, say, this paper:
http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Zeta/EZe...
...yes, but that makes the postulate meaningless! You have to look at one side of the line in order for the postulate to have any relevance.
>Euclid would have been a tau advocate
Oh yeah? Well.. well... Ramanujan would have been a pi advocate! Ha!
>which is clearly superior to the diameter.
It is expedient in the process of mathematical argumentation. Looking at expedience, though, we see that using a constant 2pi introduces an untoward amount of fractions into just about every mathematical calculation -- see for example here:
http://en.wikipedia.org/wiki/Basel_problem#A_rigorous_proof_...
Irrespective of the definition of constants, which is long since forgotten at this point (how much of a pain is it to define a circle, starting from ZFC?), it is kind of disappointing to see you refusing to read the proofs which you claim to be clarifying -- most of them get uglier moving to tau, on a cursory examination of the seminal work Proofs from THE BOOK. Go on, mentally replace every instance of "2pi" with "tau" and "pi" with "tau/2" in, say, this paper:
http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Zeta/EZe...
> hastily
Was two millennia not long enough? We could take another couple centuries, I guess. I don't think we're ever going to get a new answer for the ratio of the circumference of a circle to the diameter of that circle, though.
> The lack of the 1/2 in the π version shows why it's "wrong"—it's not as meaningful.
Frankly, I don't know what this means. I've read it several dozen times, and each time, it seems increasingly more inane. I can't help but wonder what you would say about the derivative (and antiderivative) of e^x; would you complain that it's not meaningful? Does it need more coefficients? More exponents?
Was two millennia not long enough? We could take another couple centuries, I guess. I don't think we're ever going to get a new answer for the ratio of the circumference of a circle to the diameter of that circle, though.
> The lack of the 1/2 in the π version shows why it's "wrong"—it's not as meaningful.
Frankly, I don't know what this means. I've read it several dozen times, and each time, it seems increasingly more inane. I can't help but wonder what you would say about the derivative (and antiderivative) of e^x; would you complain that it's not meaningful? Does it need more coefficients? More exponents?
I'm going to give you the benefit of the doubt and assume that you're not trolling but are frustrated. In which case I must also assume that you haven't really spent the time to understand the argument at tauday.com. It is not that the ratio of the circumference to the diameter (called π) will change, but that the ratio of the circumference to the radius (now called τ) is more useful.
Which shouldn't come as too much of a surprise because the radius is the smallest amount of information that determines what a circle is, as well as the basis for how we define radians.
Which shouldn't come as too much of a surprise because the radius is the smallest amount of information that determines what a circle is, as well as the basis for how we define radians.
Your assumptions are invalid.
The radius is not the smallest amount of information which determines a circle. The radius and origin can determine a circle. So can the diameter and origin, or the circumference and origin, or the area and origin. Don't forget your geometry.
The radius is not the smallest amount of information which determines a circle. The radius and origin can determine a circle. So can the diameter and origin, or the circumference and origin, or the area and origin. Don't forget your geometry.
I said smallest amount of information. The diameter, circumference, and area are all functions of the radius.
Sure, you can write any in terms of any of the others, but the radius is the smallest: both in terms of absolute value as well as dimensionality.
Sure, you can write any in terms of any of the others, but the radius is the smallest: both in terms of absolute value as well as dimensionality.
Why does smallest in terms of absolute value matter? In any event, I can define the circle via the center and r/2 or center and r/4, etc... I don't see any value in caring about the absolute value.
As for dimensionality, how are you using the word? No matter what, we need 3 numbers to define a circle in R^2. Two numbers define its position and one number defines it size. I don't see what you're getting at. If you mean that because radius is a measure of length (one dimension) and area is a measure of area (two dimensions) then I have two questions: 1) Why does that matter? Its still just a single number. 2) Even if it does matter, why is radius more fundamental than diameter.
As for dimensionality, how are you using the word? No matter what, we need 3 numbers to define a circle in R^2. Two numbers define its position and one number defines it size. I don't see what you're getting at. If you mean that because radius is a measure of length (one dimension) and area is a measure of area (two dimensions) then I have two questions: 1) Why does that matter? Its still just a single number. 2) Even if it does matter, why is radius more fundamental than diameter.
The circumference of the traditional unit circle is τ, which has strong ties to the definition of every trigonometric function, and the reason that radians of common fractions of the unit circle are expressed in terms of τ is related to the periods of sine and cosine.
…it's a poor argument that supports your opponent after switching key nouns.
…it's a poor argument that supports your opponent after switching key nouns.
Why do you refer to the area? Why not the diameter or the circumference? If you care about the area, why not use the unit circle with area 1?
A quarter arc of a circle is π/2 radians, regardless of area.
A quarter arc of a circle is π/2 radians, regardless of area.
(To answer your question, the unit circle, with radius one, has area π.)
A quarter arc of the unit circle is π/2. A quarter arc of the circle with area 2π, though... Here, let's do it in τ for fun. :3
So, the circle with area τ has radius sqrt(2) (from A = πr^2), diameter 2sqrt(2), and circumference τsqrt(2) (from C = Dπ), thus a quarter arc of that circle is τsqrt(2)/4. That's not a pretty number at all to work with.
Maybe keeping the unit circle at its current size is a better idea.
A quarter arc of the unit circle is π/2. A quarter arc of the circle with area 2π, though... Here, let's do it in τ for fun. :3
So, the circle with area τ has radius sqrt(2) (from A = πr^2), diameter 2sqrt(2), and circumference τsqrt(2) (from C = Dπ), thus a quarter arc of that circle is τsqrt(2)/4. That's not a pretty number at all to work with.
Maybe keeping the unit circle at its current size is a better idea.
A quarter arc of the unit circle is a quarter τ (τ/4). A half arc of the unit circle is a half τ. A 5/19ths arc of the unit circle is 5/19ths τ.
How much simpler could it get?
Defining the unit circle as a circle with area τ is a contradictory statement and is semantically invalid.
How much simpler could it get?
Defining the unit circle as a circle with area τ is a contradictory statement and is semantically invalid.
I just want to ask all of the Pi/apathetic people-- how long did it take you to understand radians? For me, it was a week before I was comfortable naming any angle in radians in a reasonable amount of time (this is after a week of drilling).
This is just my point of view, but calculating radians was a significant roadblock into making quick trigonometric calculations. In fact, I'd have to say it was the biggest roadblock. This has nothing to do with how "clean" it looks or how I "feel" about how it's presented.
That said, I don't think it's worth it to make the switch because of all the hassle. I'm just curious about all of the hostility towards Tauists.
tldr; it has nothing to do with any mathematical formula looking "cleaner," but everything to do with teaching math effectively.
This is just my point of view, but calculating radians was a significant roadblock into making quick trigonometric calculations. In fact, I'd have to say it was the biggest roadblock. This has nothing to do with how "clean" it looks or how I "feel" about how it's presented.
That said, I don't think it's worth it to make the switch because of all the hassle. I'm just curious about all of the hostility towards Tauists.
tldr; it has nothing to do with any mathematical formula looking "cleaner," but everything to do with teaching math effectively.
The teaching argument (which by the way this article did not address) is for me the most powerful one. All the other arguments look secondary to me.
> That said, I don't think it's worth it to make the switch because of all the hassle.
It would be no hassle for pupils. I bet that it would be easier and faster to teach Tau first, then mention that Pi is half Tau. In my opinion, teaching should have the priority. I don't really care if the rest of the world use Pi, but I'll teach Tau first.
> That said, I don't think it's worth it to make the switch because of all the hassle.
It would be no hassle for pupils. I bet that it would be easier and faster to teach Tau first, then mention that Pi is half Tau. In my opinion, teaching should have the priority. I don't really care if the rest of the world use Pi, but I'll teach Tau first.
I agree with teaching Tau first. Personally, it took me at least a week to begin to understand radians, and almost a month to understand it pretty well. Tau is more intuitive than Pi in this case.
It only took me about a day, but I suspect that's because I was already used to cycles at the time (1 cycle=2pi radians). Getting used to cycles took me about a week.
To my mind, the problem is not with pi; the problem is with degrees. Everyone learns about degrees first, and then must "un-learn" these artificial numbers and begin thinking in fractions of a circle (with an extra constant thrown in there one way or the other, in the case of radians). If we began labeling globes, protractors, and the like in radians (or fractional cycles), this problem would go away.
To my mind, the problem is not with pi; the problem is with degrees. Everyone learns about degrees first, and then must "un-learn" these artificial numbers and begin thinking in fractions of a circle (with an extra constant thrown in there one way or the other, in the case of radians). If we began labeling globes, protractors, and the like in radians (or fractional cycles), this problem would go away.
"the problem is with degrees"
I partly disagree. For kids, defining something using irrational numbers would probably be very confusing. Using integers is much easier.
So, why 360? Because we talk about right angles a lot and we want to have a third, a half, etc... and (I'm guessing) they wanted it to be a multiple of 10.
I think the point is to be able to teach geometry to kids without worrying about them getting confused by fractions and/or irrational numbers.
I partly disagree. For kids, defining something using irrational numbers would probably be very confusing. Using integers is much easier.
So, why 360? Because we talk about right angles a lot and we want to have a third, a half, etc... and (I'm guessing) they wanted it to be a multiple of 10.
I think the point is to be able to teach geometry to kids without worrying about them getting confused by fractions and/or irrational numbers.
So just use cycles instead of degrees. Simple fractions or a circle. Fractions are already taught to young children as "how many pieces a circle[1] can be divided into." It would seem to kill two birds with one stone.
I think most kids who even study pre-Calc could handle applying a conversion of "2pi" from there.
[1] Where circle="cake","pie","pizza",etc.
I think most kids who even study pre-Calc could handle applying a conversion of "2pi" from there.
[1] Where circle="cake","pie","pizza",etc.
Good point. I forgot we learned about "slices of a pizza" that early.
By the way, the history of the degree is somewhat interesting:
http://en.wikipedia.org/wiki/Degree_(angle)
By the way, the history of the degree is somewhat interesting:
http://en.wikipedia.org/wiki/Degree_(angle)
I don't think in radians, I think in cycles (radians/tau). I used to think in degrees, and could never get the hang of radians. Unfortunately, I cannot imagine the mathematical world moving to cycles, ever. The most I can hope for is that teachers introduce others to a single intermediary constant (tau) rather than two (2,pi).
The problem with using cycles as the primary unit of angle is that the wonderful trigonometric derivative symmetry only occurs when the functions are calibrated for radians.
I've never understood the particular argument of "single intermediary constant (tau) rather than two (2,pi)". 2pi is one constant that contains multiple glyphs, just as 1/2 is one constant, just as tau/2 would be one constant. That it is derived logically from other constants does not make it two separate constants. Even multi-digit numbers are derived logically from their component glyphs (12 = 10+2).
I've never understood the particular argument of "single intermediary constant (tau) rather than two (2,pi)". 2pi is one constant that contains multiple glyphs, just as 1/2 is one constant, just as tau/2 would be one constant. That it is derived logically from other constants does not make it two separate constants. Even multi-digit numbers are derived logically from their component glyphs (12 = 10+2).
The two-glyph thing is a matter of description length and parametrization. 10+2 is an expression, whereas 10 is also a constant that is the base of our number system. Thus, 10 carries more meaning than 12, even though both are constant - 10 is potentially the aforementioned parameter, but what is the 2? With sufficient study, 12 becomes a number of important constants as well (such as the number of inches in a foot, or the integral of a centered quadratic function), but not ones considered fundamental.
Essentially, it's about the difference between a concept and a measurement, or the difference between (x+y)/x and 1+x/y. 2pi is an expression, pi is the concept.
Essentially, it's about the difference between a concept and a measurement, or the difference between (x+y)/x and 1+x/y. 2pi is an expression, pi is the concept.
I've read this a few times, and I don't think I really understand. Perhaps you could explain further?
10+2 is an expression, whereas 10 is also a constant that is the base of our number system.
There is no notion of an "expression" as distinct from a "number" (or "function" if it involves a variable) in any branch of math apart from computer science[1]. In algebraic terms, (12) and (10+2) and (6x2) and (0xC) and (2^4-2) and "twelve" are all literally the same thing. Well, technically they are all equivalent notations for the same abstract concept.
Thus, 10 carries more meaning than 12
Even if I accept this (which I'm not convinced I do), it's beside the point: 10 and 12 are not equal. Unlike with pi and a hypothetical tau, using one where the other is called for would be an error.
[1]There is the notion of the limit, which is subtly different: limits do care how a function behaves at other points. One could make the case that this makes a limit into a sort of expression, but to be honest I think that only obscures the idea.
10+2 is an expression, whereas 10 is also a constant that is the base of our number system.
There is no notion of an "expression" as distinct from a "number" (or "function" if it involves a variable) in any branch of math apart from computer science[1]. In algebraic terms, (12) and (10+2) and (6x2) and (0xC) and (2^4-2) and "twelve" are all literally the same thing. Well, technically they are all equivalent notations for the same abstract concept.
Thus, 10 carries more meaning than 12
Even if I accept this (which I'm not convinced I do), it's beside the point: 10 and 12 are not equal. Unlike with pi and a hypothetical tau, using one where the other is called for would be an error.
[1]There is the notion of the limit, which is subtly different: limits do care how a function behaves at other points. One could make the case that this makes a limit into a sort of expression, but to be honest I think that only obscures the idea.
Okay, I'll try to grasp at my thoughts again.
Algebraically, 12 is not the same as 10+2. 12 is an element of, say, ℤ, while 10+2 is one of <+,ℤ²>. To make them interchangeable, we need to establish an equivalence relation. Given that relation, we then have the opportunity to express useful, non-obvious equivalences using transitivity.
Now that we have X=10+2=12, we need to choose which one represent the equivalence class of X. 12 is certainly shorter, but a seemingly magic constant. 10+2 implies that in other number systems, X=b+1+1 may be also true. If the scribe subscribes to the principle of MDL, we can speculate that this is the reason he chose the longer version, and if that is accurate, we have gained more information. If we chose 9+3, we would arguably lose information, since this expression is (hypothetically) misleading.
This is all to say that expressions are more informative than their equivalence classes, since they have been hand-picked to be representative.
To represent the equivalence class of 6.28... with 2*3.14... implies that the equivalence class of 3.14... is more important, and that the prototype likely involves two separate instances of the concept of π. This is misleading.
Algebraically, 12 is not the same as 10+2. 12 is an element of, say, ℤ, while 10+2 is one of <+,ℤ²>. To make them interchangeable, we need to establish an equivalence relation. Given that relation, we then have the opportunity to express useful, non-obvious equivalences using transitivity.
Now that we have X=10+2=12, we need to choose which one represent the equivalence class of X. 12 is certainly shorter, but a seemingly magic constant. 10+2 implies that in other number systems, X=b+1+1 may be also true. If the scribe subscribes to the principle of MDL, we can speculate that this is the reason he chose the longer version, and if that is accurate, we have gained more information. If we chose 9+3, we would arguably lose information, since this expression is (hypothetically) misleading.
This is all to say that expressions are more informative than their equivalence classes, since they have been hand-picked to be representative.
To represent the equivalence class of 6.28... with 2*3.14... implies that the equivalence class of 3.14... is more important, and that the prototype likely involves two separate instances of the concept of π. This is misleading.
Damnit, this is just as bad as the tau manifesto. The point is that it doesn't matter what the bloody constant is, we don't need any -more- goddamn manifestos. Call it an arc constant or an angle constant or whatever you want.
However, there is one massive abuse of terminology that is driving me insane, which is the use of the phrase "Quadratic forms". E = 1/2 k x^2 is not a quadratic form. A quadratic form is a homogeneous polynomial of degree two, and it's a topic discussed in number theory:
http://en.wikipedia.org/wiki/Quadratic_form
The vast majority of people will never encounter a legitimate quadratic form. Call it a quadratic equation or whatever -- it's not a quadratic form. Both the tau manifesto and the pi manifesto got this bit manifestly wrong.
However, there is one massive abuse of terminology that is driving me insane, which is the use of the phrase "Quadratic forms". E = 1/2 k x^2 is not a quadratic form. A quadratic form is a homogeneous polynomial of degree two, and it's a topic discussed in number theory:
http://en.wikipedia.org/wiki/Quadratic_form
The vast majority of people will never encounter a legitimate quadratic form. Call it a quadratic equation or whatever -- it's not a quadratic form. Both the tau manifesto and the pi manifesto got this bit manifestly wrong.
I don't see why E = 1/2 k x^2 isn't a quadratic form. It's a quadratic form in a vector space of dimension 1 (here, the real numbers), which matrix is just (1/2 k).
In fact here, we're dealing with a (degenerated) conic.
What's your point? Should the author have written E(x) = ...?
Besides, tau is pointless because a tau pie isn't nearly as fun as a pi pie.
What's your point? Should the author have written E(x) = ...?
Besides, tau is pointless because a tau pie isn't nearly as fun as a pi pie.
At best, it's a cubic form, since the spring constant is also a variable. Similarly 1/2 a t^2 is a cubic form. pi r^2 however is a quadratic form.
Nope, check the introduction section of your wikipedia link.
It's like saying that f(x) = ax is a function of two variables. Indeed it is, but it's commonly accepted in mathematics to use it as a function of one variable (x), with a parameter a.
It's like saying that f(x) = ax is a function of two variables. Indeed it is, but it's commonly accepted in mathematics to use it as a function of one variable (x), with a parameter a.
Unless you take every spring in the Universe to have the same spring constant (nonsense), k must also be treated as a variable. The energy relation for an individual spring is a quadratic form, but the energy relation for all springs is either a cubic form or a set of quadratic forms (or a bijection from the real line to the set of unary quadratic forms, if you want to be really technical and boring).
f(x) = a * x is an example of a function of x which only becomes a function when a value is assigned to the variable 'a'. In other words, it is a type of function.
>Indeed it is, but it's commonly accepted in mathematics to use it as a function of one variable (x), with a parameter a.
In introductory calculus, sure; in those areas of mathematics where the term 'quadratic form' is actually used, things have to be defined more precisely.
f(x) = a * x is an example of a function of x which only becomes a function when a value is assigned to the variable 'a'. In other words, it is a type of function.
>Indeed it is, but it's commonly accepted in mathematics to use it as a function of one variable (x), with a parameter a.
In introductory calculus, sure; in those areas of mathematics where the term 'quadratic form' is actually used, things have to be defined more precisely.
Ah, I think I see your point. You would be happier with E = 1/2k x^2 is a family of quadratic forms, indexed by the real number k, I guess. A bit nitpicky to me (as long as k != 0), but fair.
Does this help: f_a (x) = a * x? Is this a function of x in your opinion?
There's my cue to drag in the lambda calculus:
\g = (\a\x ...)
\f = (g a)
I disagree with change for change's sake. This whole tau thing is born of some idealist that thinks things only make sense his/her own way.
The only thing I didn't see in the article is that the symbol visually looks like a T, so when you see it in a formula, you have to really look at it to know what's going on.
The only thing I didn't see in the article is that the symbol visually looks like a T, so when you see it in a formula, you have to really look at it to know what's going on.
I would add that this stupid tau thing speaks to the conspiratorialist instinct common to many HN readers. You see, the self-evident truth of tau's superiority has been masterfully obscured by powerful dark forces in an attempt to protect their crude economic self interest, and if you don't agree, you're obviously either part of the conspiracy or one of the sheeple that's been snowed by it.
I have had that thought, but find it incredibly hard to entertain. Do you genuinely believe it, or is it just something that explains why others' opinions might differ?
I should have put quotes around everything following the first sentence and attritubted it to a hypothetical (and straw-man) tinfoil hat-wearing tau advocate.
The practical problem with tau is, as was pointed out in the article, tau is already used for other things. Shear stress, torque, time constants, you name it.
pi is a notational freak in that it represents something so fundamental that few dare tread upon the usage---pi truly is a globally reserved name. To a lesser extent, the same is true of e, but even a number as important as i doesn't enjoy this property: electrical engineers use j for sqrt(-1) because i is current.
So let's say we all start using tau. Then I decide I'm going to do some basic rotational mechanics, and now I have two taus, one for torque and one for 2pi. OK, that's a no-go. How about we just redefine pi=2pi? Well... how do we know whether someone means pi=~6.28, or pi=~3.14?
It's just no good. Tau is not a viable candidate name for the constant equal to 2pi. Find another character in another language. How about Pei (Hebrew)?
pi is a notational freak in that it represents something so fundamental that few dare tread upon the usage---pi truly is a globally reserved name. To a lesser extent, the same is true of e, but even a number as important as i doesn't enjoy this property: electrical engineers use j for sqrt(-1) because i is current.
So let's say we all start using tau. Then I decide I'm going to do some basic rotational mechanics, and now I have two taus, one for torque and one for 2pi. OK, that's a no-go. How about we just redefine pi=2pi? Well... how do we know whether someone means pi=~6.28, or pi=~3.14?
It's just no good. Tau is not a viable candidate name for the constant equal to 2pi. Find another character in another language. How about Pei (Hebrew)?
Pi is not globally reserved. It is commonly used in, for example, statistics to refer to multinomial probabilities, and wikipedia tells me it's also the name of the prime-counting function and parallax.
I should note that this was annoying to me when dealing with IRT, in which some models have 2pi in the normalizing constants. Not to say tau isn't used in statistics already as well.
There just aren't enough greek letters. I have wondered about what the symbol for tau should be, and I keep thinking of a circle including a radius line that extends slightly outside the circle, but tau is much easier to write, especially when using ascii.
I should note that this was annoying to me when dealing with IRT, in which some models have 2pi in the normalizing constants. Not to say tau isn't used in statistics already as well.
There just aren't enough greek letters. I have wondered about what the symbol for tau should be, and I keep thinking of a circle including a radius line that extends slightly outside the circle, but tau is much easier to write, especially when using ascii.
Here is what is wrong with the "anti-tauist" rants:
They take a bunch of people who already learned the subject and presume that those people are experts a teaching said subject. These people always assume that the way they learned is best, because "dammit, it was good enough for me". They just can't see any other way.
Sadly, this ignores all of the other people, who may be capable of understanding and properly using the subject if presented in a different way.
For pedagogical purposes, Tau is worth a shot, if it helps some people get to the point that they realize "for the math the constant doesn't matter".
Just like anything else: try to teach broadly, and let the experts do the adjusting, not make the novice bend to the expert's will or be damned.
They take a bunch of people who already learned the subject and presume that those people are experts a teaching said subject. These people always assume that the way they learned is best, because "dammit, it was good enough for me". They just can't see any other way.
Sadly, this ignores all of the other people, who may be capable of understanding and properly using the subject if presented in a different way.
For pedagogical purposes, Tau is worth a shot, if it helps some people get to the point that they realize "for the math the constant doesn't matter".
Just like anything else: try to teach broadly, and let the experts do the adjusting, not make the novice bend to the expert's will or be damned.
I stopped reading early, when he's claiming pi is better for the area of a circle, because that revealed that the author hasn't really thought about this very much.
If you look at the equations for the volumes of spheres in n dimensions (with 2D being just one of them), tau shows a clean pattern. pi leaves you with a mess.
If you look at the equations for the volumes of spheres in n dimensions (with 2D being just one of them), tau shows a clean pattern. pi leaves you with a mess.
When you ask hard-core Tauists what the area of a unit circle is, do they actually answer "tau over two"? Or do they just say "pi"?
By the way, this whole discussion reminds me of what W.V.O. Quine called "mathematosis".
By the way, this whole discussion reminds me of what W.V.O. Quine called "mathematosis".
It's τ/2. The area of a circle is τr²/2. You may be familiar with the idea of x²/2 from calculus: it's an integral, which can be used to compute areas.
Yes, that certainly is the integral of x * dx. So voila, there's your 1/2. And tau relates the triangle to the circle. Nice trick.
Mind you, I don't have a dog in this hunt so I'm not all up to speed on it. All I know for sure is that tau = 2 * pi, so I won't be terribly upset if I see either usage. I generally favor the use of notations which better reveal an underlying concept, but I don't like it when people get all high and mighty about things.
Mind you, I don't have a dog in this hunt so I'm not all up to speed on it. All I know for sure is that tau = 2 * pi, so I won't be terribly upset if I see either usage. I generally favor the use of notations which better reveal an underlying concept, but I don't like it when people get all high and mighty about things.
As the tau manifesto discusses, "half tau" is the more meaningful answer: the area of a circle is equal to the area of a triangle whose base is the circumference and whose height is the radius.
It's the integral of tau * r * dr from 0 to 1, which is tau/2.
personally, i'd answer "half tau".
half tau r^2 is a fine equation for an area.
half tau r^2 is a fine equation for an area.
:) Still, two syllables instead of one.
I suppose if I found myself writing "2 * pi" or "4 * pi^2" a lot, I might throw in a dash of tau here and there for brevity. Otherwise it's all the same to me.
I suppose if I found myself writing "2 * pi" or "4 * pi^2" a lot, I might throw in a dash of tau here and there for brevity. Otherwise it's all the same to me.
Though in fairness to tau, one can always choose a pithy question whose answer is shorter either way, to wit:
What is the area of a unit circle? Pi.
How many radians are in a circle? Tau.
What is the area of a unit circle? Pi.
How many radians are in a circle? Tau.
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I've considered refactoring my pathfinding code - 2.0 is this magic constant that is floating around all over my code.
Inertia has prevented me.
Inertia has prevented me.
Off topic, but if you're using floating-point for pathfinding code, you're going to have different results depending on your distance from the origin.
All this will ever lead to is Indiana passing a new law that Tau is exactly 6.
https://secure.wikimedia.org/wikipedia/en/wiki/Indiana_Pi_Bi...
https://secure.wikimedia.org/wikipedia/en/wiki/Indiana_Pi_Bi...
For me this debate is the monumental evidence that when people get obsessed over something their intellectual openness shrinks to a very small dividend of pi, sorry I meant tau :)
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TLDR:
- The area of a unit circle is Pi.
- The Tau Manifesto is full of selective bias. They pinpoint formulas that contain 2π while ignoring other formulas that do not.
- The area of a unit circle is Pi.
- The Tau Manifesto is full of selective bias. They pinpoint formulas that contain 2π while ignoring other formulas that do not.
And I open my eggs on the little end...
Given the nature of the debate, I think it's funny that the HN URL here ends in "404".
Is the authors replacement of "Tauists" with "Taoists" and "Tau" with "Tao" an indication that he is repressing his subconscious belief that Tau is the way?
2. From a pragmatist point of view, you're right, it doesn't matter. You continue reading and writing PHP and using π. They both get the job done. You don't have to participate any further.
3. There will be 2s floating around some equations forever, whether using π or τ. That's not the point. The point is not "cleanliness" or even teaching efficacy. The point is elegance and that comes from meaning. What does the equation say? Equation cleanliness and ease of understanding are both worthwhile side effects, but it's meaning that's important.
4. Going from π to τ would be nontrivial, and would involve confusion of its own. That makes it not worth it to some people, and that's a valid opinion.
5. This article suffers from more selection bias than the Tau Manifesto. The radius is the undisputed king of the circle; it defines it. The area of a circle is not, after all, π * (D/2)^2. But it's not about prettiness, it's about meaning! Area is a property defined by the integral, which has a natural meaning and result with τ. The result may be a little equation that's pretty or not depending on your point of view, but it's just a shortcut.
6. The other examples in the article similarly fall apart when meaning is considered.