τ as a circle constant

[studiot]

 

It seems to me that all along you've been trying to show that the entire discussion is pointless

 

Yes isn't it?

 

Changing to tau would certainly work since both pi and tau are only numbers.

 

Such a change would undoubtedly benefit some uses.

 

Equally certainly it would make some applications harder.

 

You have utterly failed to demonstrate that any net benefit would accrue from such a change. or even that the situation would not deteriorate.

 

Surely a properly researched and reasoned proposal would examine both sides dispassionately and weigh these in the balance, rather than predetermining the issue by only demonstrating the supporting evidence. That would be proper Science. The other is the Method of Politicians.

[Endercreeper01]

 

Yes isn't it?

 

Changing to tau would certainly work since both pi and tau are only numbers.

 

Such a change would undoubtedly benefit some uses.

 

Equally certainly it would make some applications harder.

 

You have utterly failed to demonstrate that any net benefit would accrue from such a change. or even that the situation would not deteriorate.

 

Surely a properly researched and reasoned proposal would examine both sides dispassionately and weigh these in the balance, rather than predetermining the issue by only demonstrating the supporting evidence. That would be proper Science. The other is the Method of Politicians

Well in many equations where pi is used, It is some even number multiple of pi and a single pi only comes from integration. And the circle constant should use the radius, like in all of our other equations, instead of the diameter.

[md65536]

Well in many equations where pi is used, It is some even number multiple of pi and a single pi only comes from integration. And the circle constant should use the radius, like in all of our other equations, instead of the diameter.

Not only that, but in some equations where a single pi shows up, there is a natural relationship with a half circle (see animation at "http://en.wikipedia.org/wiki/Euler's_identity" for example). And as studiot pointed out, "A non integer ratio or any other number makes less sense than unity as the basis for counting turns." -- we don't count turns by half-circles.

 

Equally certainly it would make some applications harder.

Do you have any examples? Of interest would be examples where converting from pi to tau would not only result in an extra step (as with dividing a diameter by two to obtain a radius), but where the occurrence of pi actually makes more intuitive sense than an occurrence of tau, and the use of tau would make a formula harder to understand? Edited September 2, 2013 by md65536

[studiot]

Hmm non even expressions of pi in formulae in simple math off the top of my head?

 

[math]E = a + b + c - \pi [/math]

[math]V = \frac{4}{3}\pi {r^3}[/math]

[math]A = \pi {r^2}[/math]

[math]\cos x = \sin \left( {x + \frac{\pi }{2}} \right)[/math]

 

[math]y(x,0) = f(x) = \frac{{8A}}{{{\pi ^2}}}\left( {\frac{1}{{{1^2}}}\sin \frac{{\pi x}}{l} - \frac{1}{{{3^2}}}\sin \frac{{3\pi x}}{l} + \frac{1}{{{5^2}}}\sin \frac{{5\pi x}}{l} - \frac{1}{{{7^2}}}\sin \frac{{7\pi x}}{l} + ..........} \right)[/math]

 

 

And the circle constant should use the radius,

And who are you to tell me that I have to use the radius?

However, just to humour you, I have used the radius in the above, where appropriate.

Edited September 2, 2013 by studiot

[Endercreeper01]

Hmm non even expressions of pi in formulae in simple math off the top of my head?

 

[math]E = a + b + c - \pi [/math]

 

[math]V = \frac{4}{3}\pi {r^3}[/math]

 

[math]A = \pi {r^2}[/math]

 

[math]\cos x = \sin \left( {x + \frac{\pi }{2}} \right)[/math]

 

[math]y(x,0) = f(x) = \frac{{8A}}{{{\pi ^2}}}\left( {\frac{1}{{{1^2}}}\sin \frac{{\pi x}}{l} - \frac{1}{{{3^2}}}\sin \frac{{3\pi x}}{l} + \frac{1}{{{5^2}}}\sin \frac{{5\pi x}}{l} - \frac{1}{{{7^2}}}\sin \frac{{7\pi x}}{l} + ..........} \right)[/math]

 

 

And who are you to tell me that I have to use the radius?

However, just to humour you, I have used the radius in the above, where appropriate.

Most equations that use single pi involve integration to get the answer. Otherwise, you have to look really hard to find the factor of 2. Edited September 3, 2013 by Endercreeper01

[studiot]

 

Most equations that use single pi involve integration to get the answer. Otherwise, you have to look really hard to find the factor of 2.

 

 

Meaning what?

[Endercreeper01]

 

Meaning what?

The equations with i

Single pi have some factor of 2 or are made using integration

[studiot]

 

The equations with i

Single pi have some factor of 2 or are made using integration

 

 

 

This is utter nonsense.

 

You highlighted my entire post where I offered you a selection of snap formulae containing non even multiples of pi in response to a claim that the are all even multiples and wrote something unintelligible to it.

 

When asked for amplification you wrote the above.

 

Did you not recognise the formulae, they are all in common use.

 

1) is of great interest to navigators, surveyors and astronomers

 

2) is of interest to in many disciplines and sports as is is the volume of a ball

 

3) is similarly of multifold interest as the area of a circle

 

4) should be known to school pupils studying trigonometry

 

5) is of interest to those string musicians who also have an interest in the science of their art

Edited September 3, 2013 by studiot

[md65536]

[math]A = \pi {r^2}[/math]

This was already mentioned. You can derive the area of a circle, for example, by splitting it into infinitesimal triangles of height r and base dt, all around the circumference. The triangles have an area of 1/2 r dt, and getting rid of the factor of 1/2 makes it look simpler but it doesn't make it more understandable.

http://en.wikipedia.org/wiki/Area_of_a_disk#Triangle_proof

 

Or, integrating the area of rings of size 2pi t dt, and the factor of 1/2 come from integrating t dt. http://en.wikipedia.org/wiki/Area_of_a_disk#Onion_proof

There's a natural factor of 1/2 in the area of a circle.

 

The internal angles of a triangle add up to pi... or 180 degrees, half a turn. Getting rid of the 1/2 doesn't help here either.

 

Are there any examples where the occurrence of pi is more easily understood, or more "natural" than if it were expressed using tau? Where the use of tau would obfuscate compared to the use of pi?

Surely a properly researched and reasoned proposal would examine both sides dispassionately and weigh these in the balance, rather than predetermining the issue by only demonstrating the supporting evidence. That would be proper Science.

And is there any such contradicting evidence?

Pi has been established as a convention and is firmly entrenched and would be a major pain to switch away from, well beyond the caring of most people. That's a good argument and is probably the reason pi will win for the foreseeable future. But the argument that pi is inherently or naturally better hasn't been supported here, while examples have been given supporting tau.

Edited September 3, 2013 by md65536

[studiot]

md65536

 

Studiot said:

..............I offered you endercreeper a selection of snap formulae containing non even multiples of pi in response to a claim that the are all even multiples.............

 

So why did you take my comment out of context?

 

I made no other claim that each formula contained a noneven multiple of pi.

 

In response to your question about 'natural'

 

I challenge you to offer me a method of directly measuring the radius of a ball.

 

Directly measuring the diameter of a rod is easy, directly measuring the radius of a rod is much more difficult and error prone, but can be done.

 

When I pointed out earlier that for many objects in the real world the diameter is more natural than the radius, in that it is more accessible considerable contempt was offered in return.

 

Any formula involving an extra calculation such as 2piR v piD is both more costly in effort and more contains an extra step and possible source of error.

 

Of course piRsquared is easier than piDsquareduponfour.

 

So please do not misrepresent my words.

Edited September 3, 2013 by studiot

[Endercreeper01]

I challenge you to offer me a method of directly measuring the radius of a ball.

Using the radius is more natural because we define a circle by its radius, not diamater.

 

This is utter nonsense.

 

You highlighted my entire post where I offered you a selection of snap formulae containing non even multiples of pi in response to a claim that the are all even multiples and wrote something unintelligible to it.

 

When asked for amplification you wrote the above.

 

Did you not recognise the formulae, they are all in common use.

 

1) is of great interest to navigators, surveyors and astronomers

 

2) is of interest to in many disciplines and sports as is is the volume of a ball

 

3) is similarly of multifold interest as the area of a circle

 

4) should be known to school pupils studying trigonometry

 

5) is of interest to those string musicians who also have an interest in the science of their art

Equation 1 is this:

Spherical Excess

 

The difference between the sum of the angles , , and of a spherical triangle and radians (),

It has single pi because it is the change in the sum of angles a, b, and c and pi radians, 2 and 3 are made using integration, in equation 4, pi/2 is also tau/4 (1/4 of a turn), and what is equation 5? Edited September 4, 2013 by Endercreeper01

[md65536]

So why did you take my comment out of context?

Which comment did I misrepresent?

[studiot]

 

 

Endercreeper

 

Using the radius is more natural because we define a circle by its radius, not diamater.

 

 

Except for the greater part of humanity that uses the diameter because they actually have to measure this quantity.

 

This challenge still stands, unanswered.

 

 

Studiot

 

 

In response to your question about 'natural'

 

I challenge you to offer me a method of directly measuring the radius of a ball.

 

 

Studiot

 

I made no other claim that each formula contained a noneven multiple of pi.

 

Yes, indeed equation 1 is spherical excess, but again the point I am making is that it contains an odd multiple of pi.

 

The last equation is the Fourier series for the initial displacement of a stretched string of length, l.

As such it applies to guitars, pianos, harpsichords etc.

[Endercreeper01]

Except for the greater part of humanity that uses the diameter because they actually have to measure this quantity.

Yes, but the circle is defined by the radius.

 

 

Yes, indeed equation 1 is spherical excess, but again the point I am making is that it contains an odd multiple of pi.

As I said:

 

It has single pi because it is the change in the sum of angles a, b, and c and pi radians

[md65536]

Except for the greater part of humanity that uses the diameter because they actually have to measure this quantity.

 

This challenge still stands, unanswered.

I don't get whether you're arguing that pi is the better constant to use, or just that it should be kept because so many people use it. (I guess there are many people measuring pipe diameters and multiplying by pi to get circumference, and they'd be confused if they had to divide by 2, or offended if the constant they were using wasn't the "official circle constant" used by mathematicians or something like that.)

 

I have no doubt that the practical importance of diameters led to the historical choice of pi instead of tau. Just for the sake of argument, imagine that tau had been chosen instead, and all along we'd been using 6.28 as the constant that everybody knows. Then imagine that someone came on these boards and said "We should be using half of that value instead, because more people deal with diameters than radius. We should change the value used in mathematics, because it is more practical and convenient for the greater number of people measuring pipe."

 

Would you support this argument and say that it has merit?

If yes, then I accept your argument but disagree.

If no, then I think your argument that pi is better in mathematics is false, and that catering to the people who measure diameter is not a valid justification, and that the real justification is that pi should be kept because it's already ubiquitous.

[studiot]

 

I don't get whether you're arguing that pi is the better constant to use, or just that it should be kept because so many people use it.

 

I am arguing for a balanced view.

I am also arguing for not mixing up separate issues and using one to bolster the argument for or against another.

 

I see three spearate issues here.

And you have only scratched the surface of the any of these.

 

1) Firstly, Pi is a particular number on the number line. Should we replace it with another number in formulae, equal to double its value, thus removing the need for a 2 in certain formulae.

 

2) Alternatively should we incorporate that 2 in another part of the formula?

For instance

[math]circumference = 2(\pi R)[/math]

The 2 could be applied to the pi or the R

[math] = (2\pi )®[/math]

or

[math] = (\pi )(2R) = \pi D[/math]

 

I see the diameter v radius issue as separate from the pi v 2pi issue.

Why should we specify a circle by its radius, not its diameter? We do not do this , sorry we cannot do this fro any other conic curve eg an ellipse so why is the circle special?

In fact it has been found convenient to have both diameter and radius available and to choose the most appropriate for the job in hand.

 

3) Thirdly, if we did introduce a new constant why choose tau? This symbol is already heavily overworked in many different disciplines, unlike pi which has only a few alternative uses (continuous product and Buckingham's theorem.)

[Endercreeper01]

 

I am arguing for a balanced view.

I am also arguing for not mixing up separate issues and using one to bolster the argument for or against another.

 

I see three spearate issues here.

And you have only scratched the surface of the any of these.

 

1) Firstly, Pi is a particular number on the number line. Should we replace it with another number in formulae, equal to double its value, thus removing the need for a 2 in certain formulae.

 

2) Alternatively should we incorporate that 2 in another part of the formula?

For instance

[math]circumference = 2(\pi R)[/math]

The 2 could be applied to the pi or the R

[math] = (2\pi )®[/math]

or

[math] = (\pi )(2R) = \pi D[/math]

 

I see the diameter v radius issue as separate from the pi v 2pi issue.

Why should we specify a circle by its radius, not its diameter? We do not do this , sorry we cannot do this fro any other conic curve eg an ellipse so why is the circle special?

In fact it has been found convenient to have both diameter and radius available and to choose the most appropriate for the job in hand.

 

3) Thirdly, if we did introduce a new constant why choose tau? This symbol is already heavily overworked in many different disciplines, unlike pi which has only a few alternative uses (continuous product and Buckingham's theorem.)

I dont care what symbol we use, but for argument 2, then radius is better then diameter because a circle is defined by its radius.

[md65536]

I am arguing for a balanced view.

[...]

I see three spearate issues here.

And you have only scratched the surface of the any of these.

That's fine. Plus I agree with you on issue 3).

 

However none of your 3 issues addresses the real issue brought up in OP's link. You have missed the main issue and ignored it as silly or irrelevant.

 

Also, it is fine to ask for a balanced view, but nobody is bringing forth the other side of the view. It is unfair to call for a balanced view, implying that there is a good argument supporting your side, and then expect someone else to present it. That "good argument in support of pi over tau" may not exist.

 

But okay, some points in support of pi have been mentioned, including that it is the established constant and it has a more natural relationship to diameter, which was probably how circles were more commonly defined when the ratio was first investigated.

 

However, my personal main issue with your arguments is that you're supporting the easy side of an argument with bad reasoning. That we should keep pi as it is, is the default side of the argument, easy to justify. You're trying to shut down the harder side of the argument---the side which requires going out on a limb a bit and trying to convince people of a major change of mind---using bad reasoning. "The world works on diameters" is bad reasoning. "This is a silly waste of time" is bad reasoning. The arguments seem designed to end critical thought about an issue that you don't want to bother thinking about, and perhaps assume that anyone who does is a detrimental crackpot???

[Endercreeper01]

I agree with md - you are using bad reasoning to try and end critical thought:

You have utterly failed to demonstrate that any net benefit would accrue from such a change.

And who are you to tell me that I have to use the radius?

However, just to humour youI have used the radius in the above, where appropriate.

This is utter nonsense.

You highlighted my entire post where I offered you a selection of snap formulae containing non even multiples of pi in response to a claim that the are all even multiples and wrote something unintelligible to it.

Except for the greater part of humanity that uses the diameter because they actually have to measure this quantity.

This challenge still stands, unanswered.

mad indeed.

[md65536]

I think I need to exit this thread, because I'm arguing obsessively and hypocritically. My main point is just a personal pet peeve with the forums, that many people who are justifiably harshly critical of arguments that challenge accepted or assumed answers, often tend to abandon that harshly critical thinking when arguing for such accepted or assumed answers.

 

I think I've been a pain in the butt, sorry studiot.

 

 

I still think that the value of tau has a more natural relation to circles, and the understanding of the meaning of tau or pi becomes clearer when that relationship is examined, but despite acknowledging merit in a choice of the value of tau over pi, I don't think it's worth either changing everything nor having 2 "standard" ratios.

 

I think the best option for people who think the value of tau is clearer or more illustrative than pi, is to define it in terms of pi (instead of treating it as a standard), before using it in their work.

[Amaton]

Except for the greater part of humanity that uses the diameter because they actually have to measure this quantity.

 

I feel like this is the problematic crux of this discussion.

 

The OP's question can be viewed in two lights. Which is the more practical constant for most common uses? Pi, obviously (as I would assume the majority to vote). Now, which is the more fundamentally natural? I'm not sure, and I believe this is still quite debatable. The usefulness in application deal is wholly irrelevant in regards to the abstract naturalness of the circle constant. What would a pure mathematician consider to be the more natural, the more beautiful?

 

For the latter question, I'd have to favor [math]2\pi[/math]. A circle is defined as the set of all points of distance [math]r[/math] from a certain point. That variable is the radius, and I simply prefer a constant that forms a united conversion factor between that variable and the circle's circumference.

 

Anyway, I don't really trust the famous Eulerian identity [math]e^{i\pi}+1=0[/math]. It's really not anything special if one thinks about it. The identities [math]e^{i\tau}=1[/math] and [math]e^{i\pi}=-1[/math] are much more representative of the complex exponential. The argument-output relationship for the complex exponential is after all just angular measures and their respective revolutions in the complex unit circle.

 

All in all, pi's good for physical measurements, but when it comes down to the abstract and beautiful mechanism of pure mathematics, [math]\tau[/math] seems to take the cake. And I reference Michael Hartl's wonderful Tau Manifesto, from beginning to end, as unchallenged corroboration. Still, for conventional reasons, I will most likely keep using [math]\pi[/math] in my everyday math, for school, whatever. I don't care if plumbers or engineers keep using it; it's convenient and totally practical.

[Endercreeper01]

And also check out this link: www.tauday.org

[mearo]

The only thing I see as debatable is should we change the standard. i.e. should we teach kids that a circle has τ radians instead of 2π radians.

 

Other than that it is just a non-issue. It's not as if we are trying to redefine 2π→π or anything...

π and τ are two different symbols, so there's no reason we can't even use them both interchangeably.. You don't have to pick one.

It's like getting into a heated argument over whether we should use h or h-bar. Just use which ever one you prefer in any specific situation, or even together if you're feeling crazy enough (like using τπ instead of 2π²)

 

As far as the question "should we change the school standard?" goes, I'd ask why this one tiny little change should be more important than any other standard that seems off due to historical reasons. For example, we still talk about "current" as positive charge flow, although talking about anything moving in a current would be much more accurate if it had been defined as negative (or even better, if electrons had just been defined as positive). But it works, so we don't bother changing it.

As long as everyone knows what's going on and they are free to define new things, or redefine things as they see fit in any work or papers they might write.

If you want to define current as negative charge flow nobody will stop you as long as you put 1 line in your paper saying you're doing so, same with τ.

Edited September 15, 2013 by mearo

[Endercreeper01]

The only thing I see as debatable is should we change the standard. i.e. should we teach kids that a circle has τ radians instead of 2π radians.

 

Other than that it is just a non-issue. It's not as if we are trying to redefine 2π→π or anything...

π and τ are two different symbols, so there's no reason we can't even use them both interchangeably.. You don't have to pick one.

It's like getting into a heated argument over whether we should use h or h-bar. Just use which ever one you prefer in any specific situation, or even together if you're feeling crazy enough (like using τπ instead of 2π²)

 

As far as the question "should we change the school standard?" goes, I'd ask why this one tiny little change should be more important than any other standard that seems off due to historical reasons. For example, we still talk about "current" as positive charge flow, although talking about anything moving in a current would be much more accurate if it had been defined as negative (or even better, if electrons had just been defined as positive). But it works, so we don't bother changing it.

As long as everyone knows what's going on and they are free to define new things, or redefine things as they see fit in any work or papers they might write.

If you want to define current as negative charge flow nobody will stop you as long as you put 1 line in your paper saying you're doing so, same with τ.

We are arguing if pi or tau is the better circle constant

[thenanonymouse]

The guy at www.tauday.org does make some good points...

Edited September 16, 2013 by thenanonymouse