## Tau or Pi?

Started by SBR*, November 10, 2012, 07:05:05 am

#### Ryex

##### November 11, 2012, 06:37:09 pm #20
Quote from: winkio on November 10, 2012, 11:27:52 am
Torque is defined as a force acting at a radius from a center of mass, orthogonal (perpendicular) to that radius.  Yes, torque is defined based on radius, and can't be defined based on diameter.  There is no such thing as a diameter to a center of mass, it literally doesn't exist.  In fact, as you go further along in your math career, you will find that diameters are only a part of science and engineering equations, and are dropped entirely from the mathematics curriculum, because they are not an independent value, but are a function of the radius, and can be undefined in many cases.

Also, reading that I cant see it as an against Tau because it actually FOR Tau. Pi uses the diameter of a circle in it's ratio, so why use it in a application where the diameter doesn't exist?
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#### winkio

##### November 11, 2012, 07:23:32 pm #21
Sorry, for that comment (and the entire post accompanying it), I was thinking that tau had to do with replacing radians.  If we are using radians regardless of tau or pi, then that argument does not apply.  It's all about which value leads to more simplicity, and my argument for pi is in these two posts:

Quote from: winkio on November 10, 2012, 04:49:27 pm
From a simplicity standpoint, keeping track of pi/6 as the smallest easy angle (30 degrees) is much easier than keeping track of tau/12.  In rotational domains, angles are usually limited to between -pi and pi, not 0 and pi/2 regardless.  Plus, for higher dimensional spheres, you get 4pi steraidans in a sphere, 2(pi)^2 hypesteradians in a 3-sphere, etc., and there is no simple way to manage those constants.  From a practical standpoint, area calculation has been the most used application of pi, so it makes sense for this formula to be the simplest.

Quote from: winkio on November 10, 2012, 08:13:32 pm
I think the most likely reason is that visually, -pi/4 is a lot more intuitive than going the long way around for 7pi/4.  Also, the positive and negatives can represent clockwise and counterclockwise motions, which is especially useful for angular motion.

Another point is that you can only define an inverse of a trigonometric function over an interval of pi, so pi represents the domain of unique values for a trigonometric function.

There are arguments for both sides, but in all honesty the idea of tau is a solution in search of a problem.

#### Ryex

##### November 11, 2012, 07:36:06 pm #22
Quote from: winkio on November 11, 2012, 07:23:32 pm
There are arguments for both sides, but in all honesty the idea of tau is a solution in search of a problem.

a perfectly valid point, the math works no matter which you use

I just wanted to make sure that people really understood the argument.

In the end use which ever makes more sense to you, Calc 2 made a lot more sense however when I started thinking of 2 Pi as a whole unit (I didn't know about Tau at the time)
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#### Blizzard

##### November 12, 2012, 03:46:08 am #23 Last Edit: November 12, 2012, 03:55:05 am by Blizzard
Quote from: Ryex on November 11, 2012, 06:21:14 pm
so why is the fundamental circle constant described in terms of the diameter and not the radius? This is the point of using Tau

Because it's not. The fundamental formula is 2 Pi = C / r. Just because somebody derived the formula Pi = C / d from Pi = C / (2r), doesn't make it fundamental.

Quote from: Ryex on November 11, 2012, 06:21:14 pm
frankly it not that big of a deal. all you do is say 2 Pi = Tau and replace in functions as Tau is only a ratio, a scale value, nothing changes numerically.

Same here. I really don't understand why people are making such a big fuss about it.

Quote from: Ryex on November 11, 2012, 06:21:14 pm
But I would argue that conceptualy things have more meaning.

If they wanted to use Tau only to make it easier to understand for children, fine. But they should be aware that at one point Tau becomes the one that is harder to understand and Pi is more suitable to use. The problem here is that kids now have to know Pi as well and suddenly Tau loses its point.

Quote from: Ryex on November 11, 2012, 07:36:06 pm
In the end use which ever makes more sense to you, Calc 2 made a lot more sense however when I started thinking of 2 Pi as a whole unit (I didn't know about Tau at the time)

That's why I'm for using Tau in lower grades to make it easier to children to understand it. But funny enough, then using a definition of Tau = 2 Pi in the beginning is enough while pro-Tau people are claiming that it's not. They want to make Tau a substantial part of math.

@SBR: My bad. But you are nitpicking about units while this discussion is about the use of Tau vs. Pi.
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#### SBR*

##### November 12, 2012, 12:24:11 pm #24
Quote from: Blizzard on November 12, 2012, 03:46:08 am
Quote from: Ryex on November 11, 2012, 06:21:14 pm
so why is the fundamental circle constant described in terms of the diameter and not the radius? This is the point of using Tau

Because it's not. The fundamental formula is 2 Pi = C / r. Just because somebody derived the formula Pi = C / d from Pi = C / (2r), doesn't make it fundamental.

That's like saying: "Instead of mass, we'll use 2n = m."

#### Blizzard

##### November 12, 2012, 03:20:20 pm #25 Last Edit: November 12, 2012, 03:21:53 pm by Blizzard
You mean "2m = n".
And the very concept of Tau is using "2 Pi = Tau".
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#### SBR*

##### November 12, 2012, 05:11:27 pm #26
No. C/r is the part that makes sense. In my analogy, m is the part that makes sense. Instead of assigning a variable directly to the part that makes sense (i.e. C/r and m), you assign a variable to it so that 2 * the variable (i.e. pi or n) equals the part that makes sense.

#### Blizzard

##### November 13, 2012, 02:27:40 am #27
But m is fundamental while Tau is not.
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#### SBR*

##### November 13, 2012, 10:27:51 am #28
Let me rephrase: C/r is fundamental. Mass is fundamental. People assign the value 2pi to the fundamental value C/r. That's like assigning 2n to mass; it doesn't make sense.

#### Blizzard

##### November 13, 2012, 01:00:53 pm #29
I still don't get what you are trying to say. Yes, C/r is fundamental. And so is mass. And so is Pi. n is not fundamental, so yes, it doesn't make sense to assign 2n to mass. But Pi is fundamental so it does make sense to connect C/r and Pi. Tau on the other side is not fundamental, it is an extension of Pi.
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#### SBR*

##### November 13, 2012, 05:14:05 pm #30
Then why is pi fundamental? It's defined as C/(2r), which doesn't seem fundamental to me. Maybe I'm the one misunderstanding you, though.

#### winkio

##### November 13, 2012, 07:05:28 pm #31
Since a circle is defined in 2 dimensions, I would argue that A/r^2 is more fundamental than C/2r or C/r.

#### SBR*

##### November 14, 2012, 11:54:26 am #32 Last Edit: November 14, 2012, 03:51:46 pm by SBR*
Wow, I think you just convinced me with that one line.

EDIT: On the other hand, the area of the circle can only be calculated by deriving it from the circumference. You can prove that the relation between C and r is linear (http://www.jimloy.com/geometry/pi.htm). Because you need the circumference to calculate the area, one can argue the circumference is more fundamental than the area.

#### Blizzard

##### November 14, 2012, 04:03:20 pm #33 Last Edit: November 14, 2012, 04:31:49 pm by Blizzard
QuoteA popular way to prove the area formula is to arrange slices of the circle as shown here. As the slices get thinner, the figure gets closer and closer to a rectangle with sides of r and c/2. We can substitute 2(pi)r for c (definition of pi). Then A=(pi)r2.

To prove, not to define. Also, note how C/2 is used, not C.

EDIT: Also note Addendum #14. It's another "popular proof" which is using 2 Pi r, but has to halve it. So the first draws toward Tau and the other toward Pi. The second one isn't using circumference so the area formula is obviously fundamental if it can be proven without involving the circumference formula.

EDIT: IDK, to me it seems kinda like this. Pi is fundamental in many formulas while in some formulas 2 Pi is used because of its connection. Substituting 2 Pi for Tau in some formulas would be basically adding another constant into math and making things complicated as you now have 2 of them. But then again pro-Tau people are claiming that they don't want to do that. Formulas using 2 Pi should keep using 2 Pi. Eh, what? Why the whole fuss about it then? If Tau is only supposed to replace 2 Pi in circle and maybe sine formulas/calculations, then there is no need for Tau as it doesn't really have a purpose other than making a few formulas look nicer (and easier to understand). That's why I argue the fundamentality of Tau. To me it seems that pro-Tau people are basically trying to circumvent the argument of Pi being more fundamental than Tau by saying "yeah, but we don't want to change it everywhere so it doesn't have to be so fundamental". Either Tau is more fundamental than Pi and should replace it everywhere or it's just a substitute in some formulas to make them simpler. It can't be both, these things are mutually exclusive. If it's supposed just to make some formulas and concepts simpler, it's can't be more fundamental obviously, because that wouldn't make any sense.

So what's the point of Tau then? Making things more complicated? Adding Tau = 2 Pi on a global scale (rather than just a substitute within a formula) doesn't seem to me to be a valid arguments. It sounds more like "Gives us just this little bit! We don't want to actually make a difference. We only want this small thing to be easier." In programming this would be called an unnecessary hack (as opposed to e.g. platform specific necessary hacks), a dirty shortcut to do something or make something work just because you're too lazy to implement it in a proper way.

Let me rephrase it in a short way: Either Tau is more fundamental and should substitute Pi everywhere or it's not and shouldn't. People say it's not supposed to substitute it everywhere so obviously nobody's trying to say it's more fundamental.
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#### SBR*

##### November 17, 2012, 01:03:23 pm #34
Quote from: winkio on November 10, 2012, 04:49:27 pm
Plus, for higher dimensional spheres, you get 4pi steraidans in a sphere, 2(pi)^2 hypesteradians in a 3-sphere, etc., and there is no simple way to manage those constants.

I just found this:
Spoiler: Show

Note that, once again, 2pi is used. However, I don't know how to derive the formule for the surface/volume of a hypersphere, so my point may be invalid.

Now, I would like to note that I do not know much about steradians at all, really, but I believe the total amount of steradians in an n-sphere equals S/r^n and is therefore related to the surface, which is related to the volume which can be calculated using 2pi.

Quote from: winkio on November 13, 2012, 07:05:28 pm
Since a circle is defined in 2 dimensions, I would argue that A/r^2 is more fundamental than C/2r or C/r.

A circle can also be called a 2-sphere, in other words:

Quote from: WikipediaFor any natural number n, an n-sphere of radius r is defined as the set of points in (n + 1)-dimensional Euclidean space which are at distance r from a central point, where the radius r may be any positive real number.

You could even go as far as to say that a circle is a 1-dimensional line; although it exists in a 2-dimensional space, it is but a line. This forum post on mathforum.org supports this: http://mathforum.org/library/drmath/view/54696.html.

In other words, it's all about the line i.e. the circumference.

Quote from: Blizzard on November 14, 2012, 04:03:20 pm
QuoteA popular way to prove the area formula is to arrange slices of the circle as shown here. As the slices get thinner, the figure gets closer and closer to a rectangle with sides of r and c/2. We can substitute 2(pi)r for c (definition of pi). Then A=(pi)r2.

To prove, not to define. Also, note how C/2 is used, not C.

EDIT: Also note Addendum #14. It's another "popular proof" which is using 2 Pi r, but has to halve it. So the first draws toward Tau and the other toward Pi. The second one isn't using circumference so the area formula is obviously fundamental if it can be proven without involving the circumference formula.

Addendum #14 does use the circumference. First of all, it makes use of the fact that the formula for the circumference is a polynomial of degree 1. Secondly, to calculate the base of the triangle, you need to know the formula of the circumference.

P.S. This discussion is really interesting and I'm learning a lot, both from you guys and while researching. I would like to thank you all for this discussion!

#### Blizzard

##### November 17, 2012, 01:24:56 pm #35 Last Edit: November 17, 2012, 01:27:34 pm by Blizzard
Yeah, it's been some while since the last intellectual stimulating discussion (except for the weekly Can of Worms by Memor-X).

Do you have any comment about the second part of my last post? I know it's mostly subjective, but feel free to comment on it. Part of my opinion has been built as this discussion went on after all. I've been trying to picture Tau as the solution to a problem, but it always strikes me as a badly defined problem is involved or Tau isn't a good solution for a good and clearly defined problem. Kind when you try to stick two puzzle pieces together and it doesn't work, then you take another pair of puzzle pieces and try to stick them together and in the end you're wondering why you can't put the picture together.
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#### SBR*

##### November 17, 2012, 02:47:49 pm #36
Tau-ists aren't saying formulas using 2pi should continue using 2pi. 2pi should always be replaced with tau. However, some formulas may look nicer when you replace 1/2tau with pi, so you could do that if you'd wish. You know how on some calculators you've got a [SHIFT] key? I think Pi should be [SHIFT] [Tau]. However, tau is the more fundamental thing. It isn't all about whether tau or pi is more useful or makes formulas look nicer. Math should be as simple, fundamental and elegant as possible. In that aspect, to me, tau takes the cake.

Honestly, pi and tau would both work. But it isn't about which one works. That's where math and physics differ. In physics, using 9.81 as g in the formula y=gt would work, as the time isn't defined exactly. However, in Maths you often give exact answers:
Quote
x^2 = 2
x = sqrt(2)

Sure, in a practical sense, using 1.414 for sqrt(2) would work just fine. But that wouldn't be mathematics, now, would it?

Pro-Pi people often use practical examples to support pi, e.g.
Quote
A=pi*r^2 looks nicer and is easier than A=1/2*tau*r^2

However, as I just explained, that's not the point.

You said you're trying to picture tau as the solution to a problem. The thing is: there is no real problem. Tau and pi both work. It's about which one is more fundamental, beautiful and elegant.

Quote from: Blizzard on November 14, 2012, 04:03:20 pm
Pi is fundamental in many formulas while in some formulas 2 Pi is used because of its connection.

Pi is fundamental in some formulas? Pi is either fundamental or not fundamental. It's like saying: "In some formulas, the kinetic energy is fundamental." The kinetic energy can, however, be more fundamental than a function using the kinetic energy in a more complicated manner. Maybe it's not the best analogy, as I just explained physics is not like mathematics, but you get the point.

#### Blizzard

##### November 17, 2012, 03:47:17 pm #37 Last Edit: November 17, 2012, 03:50:10 pm by Blizzard
So pro-Pi people aren't allowed to use the same half-assed argument that pro-Tau are using the whole time? In that second video in the first post, the man is clearly claiming that Tau is not supposed to change all formulas. If both are used, obviously one can't be more fundamental than the other. And if both are used, then obviously you have only complicated things in an overall perspective because now there's yet another constant to keep track of. So either both sides are allowed to use that argument or none is.

The physicists vs. mathematicians is not an argument that works in favor of tau-ists. Physicists keep bending math to their favor since physics is not a science as exact as math. You have no idea how many rants I've heard from mathematicians complaining about it and how physicists simply ignore some of the proper rules of math.

If one claims to be more fundamental, beautiful and elegant, then it would make all formulas looking better, not just some of them. That argument is like saying that a car is also a truck, because both can get you to your destination, regardless of the fact that one type can clearly pack more luggage.

Quote from: SBR* on November 17, 2012, 01:03:23 pm
Quote from: Blizzard on November 14, 2012, 04:03:20 pm
QuoteA popular way to prove the area formula is to arrange slices of the circle as shown here. As the slices get thinner, the figure gets closer and closer to a rectangle with sides of r and c/2. We can substitute 2(pi)r for c (definition of pi). Then A=(pi)r2.

To prove, not to define. Also, note how C/2 is used, not C.

EDIT: Also note Addendum #14. It's another "popular proof" which is using 2 Pi r, but has to halve it. So the first draws toward Tau and the other toward Pi. The second one isn't using circumference so the area formula is obviously fundamental if it can be proven without involving the circumference formula.

Addendum #14 does use the circumference. First of all, it makes use of the fact that the formula for the circumference is a polynomial of degree 1. Secondly, to calculate the base of the triangle, you need to know the formula of the circumference.

You were using that as argument of Tau's fundamentality (as it's 2 Pi) while it's only a proof, not a definition. That's what I meant.

Quote from: SBR* on November 17, 2012, 01:03:23 pm
You could even go as far as to say that a circle is a 1-dimensional line; although it exists in a 2-dimensional space, it is but a line. This forum post on mathforum.org supports this: http://mathforum.org/library/drmath/view/54696.html.

There is no formula for calculating Pi from a 1-sphere so you can't take that as an argument as Pi "doesn't exist" in one-dimensional space. A circle is a 2D object, a sphere is a 3D object. By definition, both can be defined by only one dimension of parameters: the radius. That doesn't make them 1D.

I don't think their discussion is really relevant for ours.

As this discussion goes further on, I'm really getting the feeling that it matters less and less. 2 Pi = Tau so the fundamentality argument can be thrown out the window: They are both equally fundamental, because they are directly linked. The question of simplicity can't be answered, because some formulas look with one simpler while others look simpler with the other. Eh. ._. I don't think we'll be getting anywhere with this discussion if we keep going in circles. (Hahaha, get it? Circles! Haha... ha... ha... yeah.)
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#### SBR*

##### November 17, 2012, 04:25:02 pm #38
Quote from: Blizzard
So pro-Pi people aren't allowed to use the same half-assed argument that pro-Tau are using the whole time?

Which argument?

Quote from: Blizzard
If both are used, obviously one can't be more fundamental than the other.

Quote from: Blizzard
2 Pi = Tau so the fundamentality argument can be thrown out the window: They are both equally fundamental, because they are directly linked.

Of course one can be more fundamental than the other. Either pi is derived from tau or the other way around. An analogy could be Planck units. For instance, one could say the Planck length is more fundamental than the regular meter.

Quote from: Blizzard
You were using that as argument of Tau's fundamentality (as it's 2 Pi) while it's only a proof, not a definition. That's what I meant.

Maybe it's but a proof, but if you can't calculate A without B (i.e. area without circumference), I'd say B (circumference) is more fundamental.

Quote from: Blizzard
There is no formula for calculating Pi from a 1-sphere so you can't take that as an argument as Pi "doesn't exist" in one-dimensional space. A circle is a 2D object, a sphere is a 3D object. By definition, both can be defined by only one dimension of parameters: the radius. That doesn't make them 1D.

Winkio argued that in a circle, the area is more important than the circumference. However, a circle is a 1-sphere and therefore a one-dimensional object in a Euclidean plane. Furthermore, to be precise, a circle doesn't have an area, only the area enclosed by the circle. A disk, however, is a 2-ball and therefore a two-dimensional object in a Euclidean plane. It's just like a sphere is a 2-sphere so a 2-dimensional object in a 3-dimensional Euclidean space and a ball is a 3-ball and therefore a 3-dimensional object in a 3-dimensional Euclidean space. You can argue whether a circle or a disk is more fundamental. However, a disk is defined as the inside of a circle, so again the A without B thing. Moreover, a circle is 1-dimensional and a disk is 2-dimensional. Therefore I'd say that a circle is more fundamental than a disk.

#### AngryPacman

##### November 17, 2012, 10:34:09 pm #39 Last Edit: November 17, 2012, 10:41:33 pm by AngryPacman
Tau. For teaching trig functions and the like, at least. I've talked to my maths teacher about it, and he agrees with me, but the thing is they can't just change the system like that. So I always have to think that, for example, pi/2 is a quarter of a circle, not half, even though that's stupid, and tau/4 is a quarter of a circle which is sensible and logical. For all other uses, I don't really care. I just think that the circle constant, when talking about circles, should be the ratio of the radius to the circumference, not the diameter to the circumference, because the diameter is less fundamental and less useful in the formulas associated with circles and trigonometry. To probably misquote Vihart, pi makes trigonometry ugly.

Circles are incredible, beautiful geometric shapes that display many of the most awesome things mathematics has to offer. Students should be taught about them in such a way that they realize this beauty. Tau is the tiara to the circle's fantasy princess, while pi is a bucket of mud. Tau makes circles and trig more beautiful, easier to understand, more appreciable, and just better, while pi makes it ugly and confusing and tedious. And no, it's not that all tedious. But the point of mathematics is to have things done in the simplest, most elegant way. Pi gets in the way of achieving this. Using pi as the circle constant is like using G/2 as the value used for gravity on Earth. Sure, it works, it gets the job done, but it could be better, there's every chance to make it better, so why wouldn't we make it better?

Quote from: winkio on November 11, 2012, 07:23:32 pm
the idea of tau is a solution in search of a problem.

For the most part, I agree with this, and I understand it. But there is a problem it solves, one I have encountered personally. Tau makes things easier to understand for people just starting to learn about the associated topics. Take it from someone who just started to learn about trig functions and circle functions just a year ago; if I was taught Tau right from the beginning, understanding it would've been f easier. But it wasn't until I stumbled upon and affixed myself to Tau that it all became clearer and more logical.
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