Chaos Project

General => Academics => Mathematics => Topic started by: SBR* on November 10, 2012, 07:05:05 am

Title: Tau or Pi?
Post by: SBR* on November 10, 2012, 07:05:05 am
What do you guys think? Should Pi be replaced by Tau?

I am by no means an expert at Maths, but I'd like to express my opinion anyway. First of all, I think tau is a wonderful concept and much more fundamental than pi. Radians just make so much more sense when using tau and students who have a hard time understanding radians would probably be better off using tau.

However, I think pi can't be fully replaced by tau or will take a long time to be replaced. After all, everybody is already used to tau, especially the laymen. If you were to tell a random person pi is wrong and that it is to be replaced by tau, they would probably shrug their shoulders and when you ask them later what the circumference of a circle is, they'd probably answer 2 pi r. That being said, the same thing goes for radians really. Laymen and beginners always use degrees to express an angle, but those who are a bit more advanced at maths use radians too. The latter could probably learn to use tau.

Students could at first learn to use degrees and pi, but when they have to learn radians, pi gets replaced by tau.

I'd love to hear you guys' opinion on this matter :).
Title: Re: Tau or Pi?
Post by: Blizzard on November 10, 2012, 07:51:48 am
I wasn't convinced until I saw eiT = 1. Yup, 2 PI should be replaced with T, but only for understanding and teaching children about trigonometry, circles and waves. If you read up on Wiki about PI (http://en.wikipedia.org/wiki/Pi), you will notice how many representative formulas and attempts for calculation are important that PI stays PI.

What the people don't understand is that math is messy. It's not always elegant and there aren't always elegant solutions or substitutes.

I think it's really funny how the videos are biased as well. The area of a circle is r2 x PI. This means that using T would cause it to become either r2 x T / 2. And that's not confusing? There are many more formulas that use PI and not 2 PI. You're trading one demon for another, really.

The only thing that may make sense is to use both. But that makes things more complicated again so nobody's really gaining anything.
Title: Re: Tau or Pi?
Post by: winkio on November 10, 2012, 11:27:52 am
As a mechanical engineer who has setup and solved many rotational systems, tau is a laughably naive idea and is utterly and completely wrong.

The discussion here is on choosing between pi, which is a radius-defined value, and tau, which is a diameter-defined value.  Let's examine some definitions:

The word radian is used to describe an angle as a ratio of the length of the circular arc to its radius.  One radian is when the circular length is equal to the radius
Spoiler: Show

With tau, we would have to call them diametans, and would now be the ratio of the length of the circular arc to its diameter.  Now you might be saying that we could just draw the diameter in the picture above, because it is already a circle, but what about this next picture:
Spoiler: Show
(http://tanasinn.info/images/b/b6/800px-Angle_measure.png)

There is no circle.  The radius lies along the arms, with endpoints at the vertex and the intersection of the arms with the arc.  The diameter is completely irrelevant, and should not be considered.

If something is spinning, it has a nonzero angular velocity.  First, let's examine a case where both pi and tau will work: a wheel rolling on the ground without slipping.  If I want to know how fast the wheel is rotating, I simply multiply the angular velocity by the radius of the wheel.  For a wheel of radius 0.1 meters rotating at 20 rad/s, I can find that it travels at 2 m/s.  In this situation, because the wheel is radially symmetric, I can do the same thing with tau diametrans.  For a wheel of diameter 0.2 meters rotating at 10 diam/s, I can find that it travels at 2 m/s.

What happens when the object does not have a radius?  Let's say the rotating object is the arm of a baseball pitcher as the throws the baseball, and we want to find out how fast he throws it.  If we measure the length from his shoulder to the palm of his hand, then we know the radius of rotation.  Let's imagine we did this measurement and got 0.65 m, thus when we see that he rotates a quarter of a circle in 0.1 seconds, that is 5pi rad/s, so we know that he can throw the ball at 5pi * 0.65 = 10.21 m/s.

Since a diameter is not always defined in the system, it would be incorrect to define the angle based on the diameter.

Torque (Nm)
Torque is the equivalent of force for rotational systems.  The first thing to note right away is that equations already use the tau symbol to represent torque, so tau should fuck off and find a different name.  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.

tl;dr: tau is a terrible idea, pi makes sense, radians are a ratio, not an arbitrary number that you can fuck around with.
Title: Re: Tau or Pi?
Post by: KK20 on November 10, 2012, 11:37:49 am
It's just people wanting to find an easier way to represent ratios with ones. Tau is too unconventional to use. To the non-mathematical person, it makes sense. To us, it's like teaching a 5 year-old trigonometry.

Or maybe it's just because people think of PIE instead of PI, and when you say one PI, they think one PIE (damn you obesity).
Title: Re: Tau or Pi?
Post by: winkio on November 10, 2012, 11:46:02 am
I would argue that if you just want something for representing angles easily, make a circle equal to one, not pi.  For example:

Comfis: arc length over circumference.  1 comfy = 2pi radians = 360 degrees.  Half a circle is a half-comfy.  The name is a play on circumference, if you didn't get it.  It has the added bonus of 1 comfy/s = 1 Hertz for rotation speed.
Title: Re: Tau or Pi?
Post by: Blizzard on November 10, 2012, 11:54:23 am
Quote from: winkio on November 10, 2012, 11:27:52 am
tau is a laughably naive idea

<3

Also, I think the idea of making radians a bit simpler was degrees to begin with as degrees actually don't have a mathematical base either if I'm not wrong. As far as I can see, people are mostly upset that you have 2 PI instead of X whereas X could simply be 360 and it's done.
Title: Re: Tau or Pi?
Post by: SBR* on November 10, 2012, 02:19:12 pm
Quote from: Blizzard on November 10, 2012, 07:51:48 am
Spoiler: Show

I wasn't convinced until I saw eiT = 1. Yup, 2 PI should be replaced with T, but only for understanding and teaching children about trigonometry, circles and waves. If you read up on Wiki about PI (http://en.wikipedia.org/wiki/Pi), you will notice how many representative formulas and attempts for calculation are important that PI stays PI.

What the people don't understand is that math is messy. It's not always elegant and there aren't always elegant solutions or substitutes.

I think it's really funny how the videos are biased as well. The area of a circle is r2 x PI. This means that using T would cause it to become either r2 x T / 2. And that's not confusing? There are many more formulas that use PI and not 2 PI. You're trading one demon for another, really.

The only thing that may make sense is to use both. But that makes things more complicated again so nobody's really gaining anything.

1/2Tr^2 shows a lot of resemblence to other formulas like K = 1/2mv^2 and U = 1/2kx^2.

A = S(circumference * dr) = S(Tr * dr) = 1/2 * T * r^2. The 1/2 is supposed to be there, but because we chose that pi = 1/2T, the 1/2 became 1. Also, I'm not saying that pi is useless. I just think pi should be used as a substitute for 1/2T, instead of the other way around. Once again, I'm no expert at math, but T just seems way more fundamental to me and should therefore be as, if not more, appreciated as pi. Also, on T day, you can eat twice as much pie.

Quote from: winkio on November 10, 2012, 11:27:52 am
Spoiler: Show

As a mechanical engineer who has setup and solved many rotational systems, tau is a laughably naive idea and is utterly and completely wrong.

The discussion here is on choosing between pi, which is a radius-defined value, and tau, which is a diameter-defined value.  Let's examine some definitions:

The word radian is used to describe an angle as a ratio of the length of the circular arc to its radius.  One radian is when the circular length is equal to the radius

With tau, we would have to call them diametans, and would now be the ratio of the length of the circular arc to its diameter.  Now you might be saying that we could just draw the diameter in the picture above, because it is already a circle, but what about this next picture:

There is no circle.  The radius lies along the arms, with endpoints at the vertex and the intersection of the arms with the arc.  The diameter is completely irrelevant, and should not be considered.

If something is spinning, it has a nonzero angular velocity.  First, let's examine a case where both pi and tau will work: a wheel rolling on the ground without slipping.  If I want to know how fast the wheel is rotating, I simply multiply the angular velocity by the radius of the wheel.  For a wheel of radius 0.1 meters rotating at 20 rad/s, I can find that it travels at 2 m/s.  In this situation, because the wheel is radially symmetric, I can do the same thing with tau diametrans.  For a wheel of diameter 0.2 meters rotating at 10 diam/s, I can find that it travels at 2 m/s.

What happens when the object does not have a radius?  Let's say the rotating object is the arm of a baseball pitcher as the throws the baseball, and we want to find out how fast he throws it.  If we measure the length from his shoulder to the palm of his hand, then we know the radius of rotation.  Let's imagine we did this measurement and got 0.65 m, thus when we see that he rotates a quarter of a circle in 0.1 seconds, that is 5pi rad/s, so we know that he can throw the ball at 5pi * 0.65 = 10.21 m/s.

Since a diameter is not always defined in the system, it would be incorrect to define the angle based on the diameter.

Torque (Nm)
Torque is the equivalent of force for rotational systems.  The first thing to note right away is that equations already use the tau symbol to represent torque, so tau should fuck off and find a different name.  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.

tl;dr: tau is a terrible idea, pi makes sense, radians are a ratio, not an arbitrary number that you can fuck around with.

I don't think tauists are replacing radians with whatever else. The problem is that 2pi radians equals one rotation instead of 1 pi. Therefore, we replace pi, not radians. 2pi radians = 1 tau radians = 360 degrees. Pi is defined as the circumference of a circle over the diameter, which is, as you pointed out, weird, because the radius is much more important. Tau, on the other hand, is defined as the circumference over the radius.

About the whole 'torque uses tau' deal: e is both the electrical value of an electron and 2.718whatever, as well as some other things. Tau is a good character, as it somewhat resembles pi. Also, they created a new character for 2pi called doublepi (http://tauday.com/tau-manifesto (http://tauday.com/tau-manifesto)).
Title: Re: Tau or Pi?
Post by: Blizzard on November 10, 2012, 03:40:00 pm
Quote from: SBR* on November 10, 2012, 02:19:12 pm
1/2Tr^2 shows a lot of resemblence to other formulas like K = 1/2mv^2 and U = 1/2kx^2.

Those formulas are like that because they are simplified forms of more complex formulas and after you integrate those formulas on the interval between 0 and a value X, you get the simpler versions (where X would be v in your first example and x in your second). Not to mention that these simplified formulas are not even complete as the infamous +C member after integration is missing. The area of a circle is not 1/2Tr2 + C. The correct formulas actually read K = x0 + 1/2mv2 and U = x0 + 1/2kx2 as there is an "initial" value (which is actually the +C constant I mentioned earlier). You have also the formula s = v0t + 1/2at2.

Quote from: SBR* on November 10, 2012, 02:19:12 pm
Also, on T day, you can eat twice as much pie.

I can eat pie every day. Still, that was a clever random argument there. xD
Title: Re: Tau or Pi?
Post by: 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.
Title: Re: Tau or Pi?
Post by: SBR* on November 10, 2012, 07:40:32 pm
I've always wondered: why do we measure angles between -pi and pi instead of 0 and 2pi (= 0 and tau)?
Title: Re: Tau or Pi?
Post by: 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.
Title: Re: Tau or Pi?
Post by: SBR* on November 11, 2012, 07:48:16 am
In general, isn't 3/4pi a different angle than -1/4pi? Most of your reasons are practical reasons, right? They don't affect whether tau or pi is a more fundamental constant. Also, tau makes more sense: 1/4tau is 1/4 of a circle.
Title: Re: Tau or Pi?
Post by: Blizzard on November 11, 2012, 08:38:29 am
Quote from: SBR* on November 11, 2012, 07:48:16 am
In general, isn't 3/4pi a different angle than -1/4pi?

Yeah, that's what winkio said.

Quote from: winkio on November 10, 2012, 08:13:32 pm
-pi/4 is a lot more intuitive than going the long way around for 7pi/4.

Wasn't the whole point of Tau being a more practical constant than Pi? As far as I can see, the only place where Tau really makes sense is a circle. A Sine wave can also be looked at 0, Pi, 2 Pi, etc. being the zero-points of the function. Sure, Tau may be a whole period, but that's it. It's called a period and that renders Tau obsolete in Sines.
Title: Re: Tau or Pi?
Post by: SBR* on November 11, 2012, 09:26:48 am
Quote from: Blizzard on November 11, 2012, 08:38:29 am
Quote from: SBR* on November 11, 2012, 07:48:16 am
In general, isn't 3/4pi a different angle than -1/4pi?

Yeah, that's what winkio said.

Quote from: winkio on November 10, 2012, 08:13:32 pm
-pi/4 is a lot more intuitive than going the long way around for 7pi/4.

Wasn't the whole point of Tau being a more practical constant than Pi? As far as I can see, the only place where Tau really makes sense is a circle. A Sine wave can also be looked at 0, Pi, 2 Pi, etc. being the zero-points of the function. Sure, Tau may be a whole period, but that's it. It's called a period and that renders Tau obsolete in Sines.

It's not about tau being more practical. Both pi and tau have advantages. It's about tau being a more fundamental and therefore a more beautiful constant than pi, as c/r is more fundamental than c/D=c/(2r).

Also, it's debateable whether pi or tau is more practical in a circle. -pi to pi makes sense, but 1/4tau being a 1/4 of the circle is very intuitive too.
Title: Re: Tau or Pi?
Post by: Blizzard on November 11, 2012, 09:34:45 am
Quote from: SBR* on November 11, 2012, 09:26:48 am
It's about tau being a more fundamental and therefore a more beautiful constant than pi, as c/r is more fundamental than c/D=c/(2r).

Uhm, no. Pi is the one that is more fundamental. Yes, c/r equals 2 Pi radians, but the radian is the more fundamental thing here.
Title: Re: Tau or Pi?
Post by: SBR* on November 11, 2012, 11:02:53 am
circumference/r = 2pi, not 2pi radians. What do you mean by "the radian is the more fundamental thing here"?
Title: Re: Tau or Pi?
Post by: Ryex on November 11, 2012, 04:05:46 pm
Tau and Pi have nothing to do with changing radians. the size of a radian is the same if you use pi or tau. tau is just equle to 2 Pi so instead of 2pi radians you have tau radians in a circle which is that same number.

Title: Re: Tau or Pi?
Post by: Blizzard on November 11, 2012, 04:09:59 pm
Quote from: winkio on November 10, 2012, 11:27:52 am
The word radian is used to describe an angle as a ratio of the length of the circular arc to its radius.  One radian is when the circular length is equal to the radius
Spoiler: Show

And yes, it's 2 Pi radians. When angles are measure in degrees, it's 360 degrees. When they are measured in radians, it's 2 Pi radians. Saying that an angle is 2 Pi is the same as saying that a car has a velocity of 50. 50 what? Potatoes?

@Ryex: What I was trying to say is that a full circle is 2 Pi radians which means there is a connection. It's not that they depend on each other, but there are things that depend on both of them.
Title: Re: Tau or Pi?
Post by: Ryex on November 11, 2012, 06:21:14 pm
I think you completely missing the point the guy is making about tau.

A Circle is defined as the set of all points a distance of r (radius) from a central point.

Pi is the ratio of a circle's circumference to it's diameter (C / D)
Tau is the ratio of a circle's circumference to it's radius (C / r)

When you work with a circle you work with it's radius and in all but the rarest cases you don't care about the diameter. so why is the fundamental circle constant described in terms of the diameter and not the radius? This is the point of using Tau

The second link is a hour long talk the man gives on the subject and it becomes quite clear (to me at least) that the links between aspects of geometry, trig, and calculus be come a lot clearer when you knowledge that Pi introduces a factor of 2 to the task of working with a circles, and that factor of 2 often masks important concepts in simplified functions.

In short I think you and Winkio have completely misunderstood the implication of Tau, Using Tau fundamentally changes nothing, you not redefining radians, your not changing how existing functions and formulas work, your not fucking with the system. your acknowledging that the ratio of a circle's circumference to it's radius is the clearer and more fundamental way to describe a circle's size and shape and as a result the links between different parts of math become clearer.

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. But I would argue that conceptualy things have more meaning.
Title: Re: Tau or Pi?
Post by: SBR* on November 11, 2012, 06:27:04 pm
Quote from: Blizzard on November 11, 2012, 04:09:59 pm
Quote from: winkio on November 10, 2012, 11:27:52 am
The word radian is used to describe an angle as a ratio of the length of the circular arc to its radius.  One radian is when the circular length is equal to the radius
Spoiler: Show

And yes, it's 2 Pi radians. When angles are measure in degrees, it's 360 degrees. When they are measured in radians, it's 2 Pi radians. Saying that an angle is 2 Pi is the same as saying that a car has a velocity of 50. 50 what? Potatoes?

@Ryex: What I was trying to say is that a full circle is 2 Pi radians which means there is a connection. It's not that they depend on each other, but there are things that depend on both of them.

You said the circumference of a circle equals 2 pi radians.

Quote
Circumference is the linear distance around the outside of a closed curve or circular object. The circumference of a circle is of special importance to geometric and trigonometric concepts. However circumference may also describe the outside of elliptical closed curves. Circumference is a special example of perimeter.

So the circumference is to be measured in meters, kilemeters, feet, or in case of the unity circle the circumference is pretty much dimensionless.

Quote from: Ryex on November 11, 2012, 06:21:14 pm
I think you completely missing the point the guy is making about tau.

A Circle is defined as the set of all points a distance of r (radius) from a central point.

Pi is the ratio of a circle's circumference to it's diameter (C / D)
Tau is the ratio of a circle's circumference to it's radius (C / r)

When you work with a circle you work with it's radius and in all but the rarest cases you don't care about the diameter. so why is the fundamental circle constant described in terms of the diameter and not the radius? This is the point of using Tau

The second link is a hour long talk the man gives on the subject and it becomes quite clear (to me at least) that the links between aspects of geometry, trig, and calculus be come a lot clearer when you knowledge that Pi introduces a factor of 2 to the task of working with a circles, and that factor of 2 often masks important concepts in simplified functions.

Exactly.
Title: Re: Tau or Pi?
Post by: Ryex on November 11, 2012, 06:37:09 pm
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?
Title: Re: Tau or Pi?
Post by: winkio on November 11, 2012, 07:23:32 pm
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.
Title: Re: Tau or Pi?
Post by: Ryex on November 11, 2012, 07:36:06 pm
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)
Title: Re: Tau or Pi?
Post by: 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.

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.
Title: Re: Tau or Pi?
Post by: SBR* on November 12, 2012, 12:24:11 pm
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."
Title: Re: Tau or Pi?
Post by: Blizzard on November 12, 2012, 03:20:20 pm
You mean "2m = n".
And the very concept of Tau is using "2 Pi = Tau".
Title: Re: Tau or Pi?
Post by: SBR* on November 12, 2012, 05:11:27 pm
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.
Title: Re: Tau or Pi?
Post by: Blizzard on November 13, 2012, 02:27:40 am
But m is fundamental while Tau is not.
Title: Re: Tau or Pi?
Post by: SBR* on November 13, 2012, 10:27:51 am
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.
Title: Re: Tau or Pi?
Post by: Blizzard on November 13, 2012, 01:00:53 pm
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.
Title: Re: Tau or Pi?
Post by: SBR* on November 13, 2012, 05:14:05 pm
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.
Title: Re: Tau or Pi?
Post by: 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.
Title: Re: Tau or Pi?
Post by: SBR* on November 14, 2012, 11:54:26 am
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 (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.
Title: Re: Tau or Pi?
Post by: 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.

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.
Title: Re: Tau or Pi?
Post by: SBR* on November 17, 2012, 01:03:23 pm
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 (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!
Title: Re: Tau or Pi?
Post by: Blizzard on November 17, 2012, 01:24:56 pm
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.
Title: Re: Tau or Pi?
Post by: SBR* on November 17, 2012, 02:47:49 pm
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.
Title: Re: Tau or Pi?
Post by: Blizzard on November 17, 2012, 03:47:17 pm
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 (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.)
Title: Re: Tau or Pi?
Post by: SBR* on November 17, 2012, 04:25:02 pm
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 (http://en.wikipedia.org/wiki/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.
Title: Re: Tau or Pi?
Post by: AngryPacman on November 17, 2012, 10:34:09 pm
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.
Title: Re: Tau or Pi?
Post by: Blizzard on November 18, 2012, 05:24:21 am
Quote from: SBR* on November 17, 2012, 04:25:02 pm
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?

The last one you mentioned in your previous post:

Quote from: SBR* on November 17, 2012, 02:47:49 pm
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.

Quote from: SBR* on November 17, 2012, 04:25:02 pm
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 (http://en.wikipedia.org/wiki/Planck_units). For instance, one could say the Planck length is more fundamental than the regular meter.

That's true, too. But I was referring to the fact that substituting all 2 Pi with Tau would make some formulas more complicated which is why I see both of the equally fundamental if anything else.

Quote from: SBR* on November 17, 2012, 04:25:02 pm
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.

Then we simply have opposing opinions. I feel that the area of an object is the base concept in 2D while I believe volume is the base concept in 3D. Circumference is 1D attribute of a circle which uses both 2D and 1D properties of a circle which is why I see it as "more complicated" and hence less fundamental.

Quote from: SBR* on November 17, 2012, 04:25:02 pm
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.

Ah, now I understand what you mean. Interesting point. But you can't call it a 1D object if it's in a 2D plane and uses both dimensions. If you go that way, you could actually call a circle just a line that is bent over the second dimension and its start and end is the same point. But then you can't argue its fundamentality in 2D space and the definition of Pi which doesn't appear anywhere in 1D space.

Quote from: AngryPacman on November 17, 2012, 10:34:09 pm
To probably misquote Vihart, pi makes trigonometry ugly.

So true. xD

I'm not arguing that Pi would make some things simpler and prettier. That's the base argument for changing 2 Pi to Tau in the first place. My argument is basically that:

1. If you substitute Pi with Tau in every existing formula, then you've done nothing, because there are many formulas that use Pi, not 2 Pi.
2. If you substitute Pi with Tau in some formulas, you are making things simpler in one place, but you have complicated math in general. People will eventually have to learn about Pi and basically you are just postponing the inevitable (and making it more complicated). Instant vs. delayed gratification if you will.

I just think that the pros are not worth the cons.
Title: Re: Tau or Pi?
Post by: SBR* on November 18, 2012, 07:09:06 am
Quote from: Blizzard
The last one you mentioned in your previous post:

Ah, I misread that. I thought you said: "Pi is fundamental in many formulas while in some formulas 2 Pi is fundamental because of its connection." My bad.

Quote from: Blizzard
That's true, too. But I was referring to the fact that substituting all 2 Pi with Tau would make some formulas more complicated which is why I see both of the equally fundamental if anything else.

Making formulas less or more complicated has nothing to do with how fundamental it is. Making formulas make more sense, however, might. We could be giving each other formulas which work in favour of either tau or pi - because of their simplicity - all day and still don't achieve anything. The tau manifesto probably uses arguments like "C=tau*r is more beautiful than C=2pi*r". I don't agree with the tau manifesto on that. On the other hand, the formula A=1/2tau*r^2 may look more complicated than A=pi*r^2, but it does make more sense (integral, area of a triangle, etc.).

Quote from: Blizzard
Ah, now I understand what you mean. Interesting point. But you can't call it a 1D object if it's in a 2D plane and uses both dimensions. If you go that way, you could actually call a circle just a line that is bent over the second dimension and its start and end is the same point. But then you can't argue its fundamentality in 2D space and the definition of Pi which doesn't appear anywhere in 1D space.

Yes, you can call a circle a line that is bent over the second dimension and its start and end are the same point. I don't argue its fundamentality in a 2D space. However, a circle doesn't have and area. A disk does. Sure enough, pi appears in neither a 0D nor a 1D object in a 1D space, for they're just a point and a line. However, pi does appear in a 1D object in a 2D space i.e. a circle.

Quote from: Blizzard
1. If you substitute Pi with Tau in every existing formula, then you've done nothing, because there are many formulas that use Pi, not 2 Pi.

Of course you've done something: you've replaced the nonsensical constant pi with the sensical constant tau and therefore made math in general more beautiful and sensical.

Quote from: Blizzard
2. If you substitute Pi with Tau in some formulas, you are making things simpler in one place, but you have complicated math in general. People will eventually have to learn about Pi and basically you are just postponing the inevitable (and making it more complicated). Instant vs. delayed gratification if you will.

Try looking at it this way: at first, every formula starts out using tau. However, because people will sometimes encounter 1/2tau, they may want to learn it using pi, just because the formula looks more simple. Pi doesn't mean anything other than 1/2tau, so it doesn't really make it much more complicated. On the other hand, maybe you're right. Maybe it's better if we don't use pi at all. I'm not really sure.
Title: Re: Tau or Pi?
Post by: AngryPacman on November 18, 2012, 07:16:33 am
@Blizz - I partially agree with you. Tau should be used in circle functions and trig functions, because otherwise it's harder to understand and frankly it's stupid. In all other cases, read: where mathematics is not made confusing and tedious and ugly by it, Pi could and probably should be used.

I am not debating for the usage of Tau in all mathematics; that'd be silly, considering how much Pi is used. No, I'm simply saying that Tau should be used as the fundamental constant in trig and circular functions if anything. This is because there are only 2 things in this area of study that is associated with Pi and not 2 Pi; that being the area of a circle (Pi r^2, even though that is really (Pi D^2)/4 which isn't very pretty either) and the periodicity of the tangent graph, which I don't think should even be considered important.

The thing with the area of the circle is that it's 1/2 Tau r^2. While some may say that this isn't in it's most pristine form, terms of this form appear all the time, in fundamental equations in physics. Distance fallen = 1/2 g t^2, Spring energy = 1/2 k x^2, Kinetic energy = 1/2 m v^2, and the area of a circle is 1/2 T r^2. As for the tan graph, we see that, one, it is one equation out of dozens that pertains to T/2 (so is therefore virtually negligible), two, periodicity in T/2 is the same thing as periodicity in T anyway, and three, it still makes more sense to think of it in terms of T. "Why does the tan graph asymptote to infinite at pi over tw- OH, I SEE. IT'S THE Y CO-ORDINATE AT THAT ANGLE OVER THE X CO-ORDINATE. PI/2 IS T/4, AND I CAN SEE FROM LOOKING AT A CIRCLE THAT Y OF T/4 IS 1 AND X OF T/4 IS 0, SO TAN(T/4) OR TAN(90 degrees) IS 1/0, AND I KNOW FROM NOT BEING AN IDIOT THAT THAT'S +- INFINITY. HOW ABOUT THAT. HOW OBVIOUS AND EASY TO UNDERSTAND WITH TAU," says the loud student who yells everything that he observes.

But yeah, I don't really give a toss outside of circular and trig functions. I just think it makes more sense.
Title: Re: Tau or Pi?
Post by: Blizzard on November 18, 2012, 09:54:27 am
@AP: The formula thing is a good argument, but the context is wrong. Those are all formulas that came after integrating f(x)dx = 1/2x2 which isn't the case in a circle's area.

Yeah, as I already said, Tau does make sense and maybe should be taught in trigonometry, because it's simpler. But math isn't always elegant. In fact more often than not, it's a mess.

Quote from: SBR* on November 18, 2012, 07:09:06 am
Quote from: Blizzard
1. If you substitute Pi with Tau in every existing formula, then you've done nothing, because there are many formulas that use Pi, not 2 Pi.

Of course you've done something: you've replaced the nonsensical constant pi with the sensical constant tau and therefore made math in general more beautiful and sensical.

Quote from: SBR* on November 18, 2012, 07:09:06 am
Making formulas less or more complicated has nothing to do with how fundamental it is.

In the first you claim that the fundamentality of a constant is important in the decision while in the second one you are waiving that argument. That's why I think the fundamentalities of the constants don't help in decision making.

Quote from: SBR* on November 18, 2012, 07:09:06 am
Quote from: Blizzard
2. If you substitute Pi with Tau in some formulas, you are making things simpler in one place, but you have complicated math in general. People will eventually have to learn about Pi and basically you are just postponing the inevitable (and making it more complicated). Instant vs. delayed gratification if you will.

Try looking at it this way: at first, every formula starts out using tau. However, because people will sometimes encounter 1/2tau, they may want to learn it using pi, just because the formula looks more simple. Pi doesn't mean anything other than 1/2tau, so it doesn't really make it much more complicated. On the other hand, maybe you're right. Maybe it's better if we don't use pi at all. I'm not really sure.

Yeah, it's a real mess.
Title: Re: Tau or Pi?
Post by: SBR* on November 18, 2012, 10:30:17 am
Quote from: Blizzard
The formula thing is a good argument, but the context is wrong. Those are all formulas that came after integrating f(x)dx = 1/2x2 which isn't the case in a circle's area.

A disk's area is calculated by integrating C(x)dx = 1/2*tau*x^2: a circle can be split up in smaller circles with radius x and circumference tau*x. You call the difference in radius dx. The difference in area between two circles is C*dx = tau*x*dx. If you integrate all the differences in area, you get the total area of the circle i.e. S(C*dx) = S(tau*x*dx) = [1/2*tau*x^2] (x=0 to x=r) = 1/2*tau*r^2.

Quote from: Blizzard
In the first you claim that the fundamentality of a constant is important in the decision while in the second one you are waiving that argument. That's why I think the fundamentalities of the constants don't help in decision making.

Nope. There's a difference between how sensical a formula is and how complicated it looks. Tau makes some formulas look more complicated and some less. However, it does make a lot of formulas more sensical.
Title: Re: Tau or Pi?
Post by: Blizzard on November 18, 2012, 10:46:26 am
Quote from: SBR* on November 18, 2012, 10:30:17 am
Quote from: Blizzard
The formula thing is a good argument, but the context is wrong. Those are all formulas that came after integrating f(x)dx = 1/2x2 which isn't the case in a circle's area.

A disk's area is calculated by integrating C(x)dx = 1/2*tau*x^2: a circle can be split up in smaller circles with radius x and circumference tau*x. You call the difference in radius dx. The difference in area between two circles is C*dx = tau*x*dx. If you integrate all the differences in area, you get the total area of the circle i.e. S(C*dx) = S(tau*x*dx) = [1/2*tau*x^2] (x=0 to x=r) = 1/2*tau*r^2.

Can you reference that calculation? While it does make sense to some extent, it seems weird to me (kinda like I mentioned math vs. physics earlier where things are used wrongly in physics because they "fit").

Quote from: SBR* on November 18, 2012, 10:30:17 am
Quote from: Blizzard
In the first you claim that the fundamentality of a constant is important in the decision while in the second one you are waiving that argument. That's why I think the fundamentalities of the constants don't help in decision making.

Nope. There's a difference between how sensical a formula is and how complicated it looks. Tau makes some formulas look more complicated and some less. However, it does make a lot of formulas more sensical.

I wasn't arguing the simplicity or complexity of the formulas, I'm arguing the fact that you were saying in the first "let's make things simpler and nicer because it's fundamental" and in the second one "making things nicer has nothing to do with them being fundamental" which is contradictory.
Title: Re: Tau or Pi?
Post by: SBR* on November 18, 2012, 11:17:45 am
Quote from: Blizzard
Can you reference that calculation? While it does make sense to some extent, it seems weird to me (kinda like I mentioned math vs. physics earlier where things are used wrongly in physics because they "fit").

http://en.wikipedia.org/wiki/Area_of_circle
It's the 'onion proof'.
Also, do you remember Archimedes' triangle method? The one with the triangle with base=C and height=r? That's basically the same calculation - if you were to draw the graph of the formula y=tau*x, you would end up with a triangle.

Quote from: Blizzard
I wasn't arguing the simplicity or complexity of the formulas, I'm arguing the fact that you were saying in the first "let's make things simpler and nicer because it's fundamental" and in the second one "making things nicer has nothing to do with them being fundamental" which is contradictory.

I think you misunderstood me. In the first one, I said: "Of course you've done something: you've replaced the nonsensical constant pi with the sensical constant tau and therefore made math in general more beautiful and sensical." I said it makes more sense using tau. In the second one, I said: "Making formulas less or more complicated has nothing to do with how fundamental it is." With this, I mean the way it looks: 1/2*tau*r^2 doesn't look as nice and is a bit more difficult to remember, but it makes more sense.
Title: Re: Tau or Pi?
Post by: Blizzard on November 18, 2012, 12:05:44 pm
Quote from: SBR* on November 18, 2012, 11:17:45 am
http://en.wikipedia.org/wiki/Area_of_circle
It's the 'onion proof'.
Also, do you remember Archimedes' triangle method? The one with the triangle with base=C and height=r? That's basically the same calculation - if you were to draw the graph of the formula y=tau*x, you would end up with a triangle.

Ah, I see now. Thanks.

Quote from: SBR* on November 18, 2012, 11:17:45 am
I think you misunderstood me. In the first one, I said: "Of course you've done something: you've replaced the nonsensical constant pi with the sensical constant tau and therefore made math in general more beautiful and sensical." I said it makes more sense using tau. In the second one, I said: "Making formulas less or more complicated has nothing to do with how fundamental it is." With this, I mean the way it looks: 1/2*tau*r^2 doesn't look as nice and is a bit more difficult to remember, but it makes more sense.

Yeah, I misunderstood that.
Title: Re: Tau or Pi?
Post by: AngryPacman on November 19, 2012, 04:04:13 am
Quote from: Blizzard on November 18, 2012, 09:54:27 am
But math isn't always elegant. In fact more often than not, it's a mess.

If you could have your house painted by either Michelangelo or a colony of wasps dipped in food dye rolling around on your walls, which would you choose? I'd say you'd probably choose the renowned Renaissance painter whose work is remembered five and a half centuries after his death over a swarm of angry arthropods each trying to have their way with your wall.

And that's what this boils down to. You're right in saying that when people say that maths is beautiful and elegant they're stretching the truth and forgetting all about the ugly, hairier side of things. But we are given a choice here. A choice to have a beautiful set of equations that are all logical, sensible and (for some people, myself included) pleasing to think about - a house painted by the orange Teenage Mutant Ninja Turtle - or to have a set of equations that all make sense, all work, but are completely missing the point and have a glaring mistake that just gets in the way - a facade constructed by a raging horde of vespines drenched in pigment. And even though there have been wannabee hornets rolling all over it for several millenia, it's never too late to get your Italian in there to make your garage look like the Sistine Chapel - while Pi may have been the convention since the time of Archimedes, it's still not too late to rectify his mistake.

In short, Pi = pissed off hornets drunk on acrylic, and Tau = a god-damn teenage mutant turtle painter who dual-wields nunchaku. It may seem that I've warped this argument in favour of my side (mainly because, well, I have), but my point remains objective and valid; the only reason Tau shouldn't be used over Pi is because everyone's already used to Pi. Nobody sees the fault with the abstractly-thrown-together wasp wall because they've been looking at it for two and a half thousand years - they're used to it. But show them your beautiful Renaissance art and they'll have a whole new appreciation for your living room. Or mathematics. Whatever.

Yeah?
Title: Re: Tau or Pi?
Post by: winkio on November 19, 2012, 04:21:22 am
Those metaphors make no sense, I can't tell if you are being sarcastic or if you actually think that such an over-the-top argument is effective.  We had some good points being made earlier though.
Title: Re: Tau or Pi?
Post by: Blizzard on November 19, 2012, 04:42:44 am
Beauty != simplicity. The aren't mutually exclusive, but they aren't synonyms either. Also, math has no artistic value, it's an exact science, not something you bring creativity into. You could say it's not about beauty or simplicity, it's about usability and then the whole Michelangelo argument really makes no sense.
Title: Re: Tau or Pi?
Post by: AngryPacman on November 19, 2012, 07:46:05 am
Perhaps the analogy was too drawn-out. I did stray from my point quite a bit. The analogy overall was... ridiculous to begin with, and perhaps a little inspired by lack out thinking statements through :)

Quote from: Blizzard on November 19, 2012, 04:42:44 am
math has no artistic value, it's an exact science, not something you bring creativity into.

I disagree. Science != !Creativity (forgive me if there's a less stupid way of writing that expression, I haven't done any programming in a rather long time XD)
I understand why you say that. However, I do not agree.

Quote from: Blizzard on November 19, 2012, 04:42:44 am
You could say it's not about beauty or simplicity, it's about usability

Yes, you could say that. But beauty and simplicity should be made the best of, because they make things easier to understand, especially in a situation where there is nothing to lose. Pi works, Pi is usable, yes. But Tau also works just as well, Tau is just as usable, and it makes things more elegant and nice and all of those words I've already said. Besides, isn't simplicity a large role in usability anyway? The simpler formula is the more usable one. And we know that mathematical beauty is basically simplicity and elegance of concepts that seem complex or confusing at first glance.
Mathematical beauty is comparable to music. When something works (usability), that's expected, it's what music is supposed to do. When something sounds nice (beauty), that's good. When something is interesting, something achieved by veiling complexity with simplicity, that's great. But when a piece works, sounds good, and is interesting musically, that's the epitome of musical perfection. It's better. Why would we settle for Mozart's 40th (which works, is simple and sounds nice) when we can have Beethoven's 3rd (which works, is interesting and sounds awesome. Sorry, bit of opinion going on here. But I think my point's been made) ? The same idea applies to this argument of Tau vs Pi. Again, I'll say that I only care about this argument in the domain of trigonometry and circular functions. I'm neutral about this in any other domain of mathematics. I just think that students should be given the opportunity to understand trig and circles in the best possible way, and to me, that's with Tau.

I think this has gotten to the point where we're arguing about arguing instead of arguing about Tau and Pi and maths and stuff.

(Also, was that analogy any better or am I just digging my grave here?)
(Also also, I shouldn't have chosen Mozart's 40th as my go-to average piece. It's just the first piece that came to mind. Don't get me wrong, it's a fantastic piece. It's just that Beethoven's 3rd is my favourite piece of classical music.)
(Also also also, I am spending way too long on these posts and I think I'm just repeating myself in progressively stupider ways. So I'm gonna go to sleep.)
Title: Re: Tau or Pi?
Post by: Blizzard on November 19, 2012, 09:30:57 am
Quote from: AngryPacman on November 19, 2012, 07:46:05 am
Quote from: Blizzard on November 19, 2012, 04:42:44 am
math has no artistic value, it's an exact science, not something you bring creativity into.

I disagree. Science != !Creativity (forgive me if there's a less stupid way of writing that expression, I haven't done any programming in a rather long time XD)
I understand why you say that. However, I do not agree.

You didn't understand what I was trying to say. Science != exact science. An exact science has no room for creativity. You can use creativity to maybe find new ways to solve problems, but 2 + 2 will always be 4. It will never be 4.2 just because you feel a bit creative that day.
Title: Re: Tau or Pi?
Post by: SBR* on November 19, 2012, 10:45:42 am
Quote from: Blizzard on November 19, 2012, 04:42:44 am
You could say it's not about beauty or simplicity, it's about usability and then the whole Michelangelo argument really makes no sense.

Sure, it's about usability too. But personally, I think math is not only about that. I'm sure there are tons of things discussed in mathematics that have no practical use - most equations can be solved by calculaters nowadays anyway. I like to think math is a perfect world of numbers, variables, dimensions, equations, etc. In other words, there's no direct creativity, but it shows resemblance to creative arts: making it as beautiful as possible. I think you could call math art.
Title: Re: Tau or Pi?
Post by: winkio on November 19, 2012, 10:58:26 am
Quote from: SBR* on November 19, 2012, 10:45:42 am
But personally, I think math is not only about that. I'm sure there are tons of things discussed in mathematics that have no practical use - most equations can be solved by calculators nowadays anyway.

How do you think they programmed the calculators?  Magic?  What about solving complicated mathematical systems, like the stresses on a car undergoing a collision?  Computer math is so big it has its own department at my university (Computational and Applied Mathematics).

Math has a resemblance to language much more than art.
Title: Re: Tau or Pi?
Post by: SBR* on November 19, 2012, 11:47:51 am
I figured they didn't program calculators to use completing the square to solve quadratic equations.
Title: Re: Tau or Pi?
Post by: winkio on November 19, 2012, 11:55:12 am
No, they use the quadratic formula for that.  But for partial fraction expansion, computers do complete the square.  You should realize that algebra and trigonometry only scratch the surface of mathematics.
Title: Re: Tau or Pi?
Post by: SBR* on November 19, 2012, 12:19:28 pm
Hmmm, interesting. I always thought calculators just filled in a bunch of numbers and gave you the one closest to the answer. My bad.
Title: Re: Tau or Pi?
Post by: Blizzard on November 19, 2012, 02:53:16 pm
I noticed something in this discussion. All the people who have learned math in-depth (at least to some extent or a specific area) are actually against Tau while the rest is pro-Tau.
Title: Re: Tau or Pi?
Post by: AngryPacman on November 19, 2012, 03:18:08 pm
Quote from: Blizzard on November 19, 2012, 09:30:57 am
You didn't understand what I was trying to say. Science != exact science. An exact science has no room for creativity. You can use creativity to maybe find new ways to solve problems, but 2 + 2 will always be 4. It will never be 4.2 just because you feel a bit creative that day.

Oh, yep. K. We're not trying to change numbers though. The numbers are fine. We're just trying to change the way they're represented. We're not changing the value of 2 Pi, we're just calling it something less confusing for people learning about the subject. I'm committed to my opinion because I had a lot of trouble with trig and circles when I had to think in Pi. I know that, for me at least, Tau works better, so why shouldn't it be used? You wouldn't define the number 1 as 2 h where h is 1/2, because it's just 1 and there's no need to complicate it. So why would you define a full revolution as 2 Pi when it can just be Tau?

Quote from: Blizzard on November 19, 2012, 02:53:16 pm
I noticed something in this discussion. All the people who have learned math in-depth (at least to some extent or a specific area) are actually against Tau while the rest is pro-Tau.

I'm an enthusiast >:(
Title: Re: Tau or Pi?
Post by: Ryex on November 19, 2012, 03:46:05 pm
I'm not against or for Tau. they are both logical constants that fit and their benefits to using either one as well as cons. IMHO neither one has an advantage as every equation that use Pi could be re written to use Tau but it's simplified form would look completely different than it's Pi counterpart.

Pro Tau argues that the new forms make more sense as they are closer in form to similarly derived equation that are not meant for circles

The question comes down to "do we want slightly more complex looking equation that bare more resemblance to non circle equations but ave to re write everything and relearn it? or do we want to say where we are as everything works anyway"

Personalty I think we have a better chance of getting the entirety of the american general public to change to the metric system than we do getting the math and science community to change to Tau
Title: Re: Tau or Pi?
Post by: SBR* on November 19, 2012, 04:03:47 pm
Quote from: Blizzard on November 19, 2012, 02:53:16 pm
I noticed something in this discussion. All the people who have learned math in-depth (at least to some extent or a specific area) are actually against Tau while the rest is pro-Tau.

Once you learn more about a subject, you tend to become more conformistic. I'm not saying you guys are conformistic, but I'd just like to point out that knowing more about a topic not necessarily results in wiser choices.

I think the question comes down to this: Do you think mathematics is all about beauty and elegance or about practicality. If it's the former, changing 2pi for tau seems good to me. If it's the latter, I'd say changing to tau would be too much of a hassle.
Title: Re: Tau or Pi?
Post by: Blizzard on November 19, 2012, 04:44:00 pm
Quote from: SBR* on November 19, 2012, 04:03:47 pm
Once you learn more about a subject, you tend to become more conformistic. I'm not saying you guys are conformistic, but I'd just like to point out that knowing more about a topic not necessarily results in wiser choices.

I beg to differ. Without experience, knowledge and wisdom, you can't make a good choice. If you are lacking a lot knowledge and experience compared to me and we are both equally wise, you simply can't make a good decision based on your knowledge and experience alone, especially because you don't see the whole picture. (Not that I do, but I surely can see more than you in this case.)

You wouldn't go vote for somebody either without first looking into what they are actually standing for.

Quote from: SBR* on November 19, 2012, 04:03:47 pm
I think the question comes down to this: Do you think mathematics is all about beauty and elegance or about practicality. If it's the former, changing 2pi for tau seems good to me. If it's the latter, I'd say changing to tau would be too much of a hassle.

Look around you. Look at your keyboard. Look at your screen. There's math appliance everywhere. It's a highly theoretical science, but it's being applied basically everywhere. I believe that practicality is far more important because of this.
Title: Re: Tau or Pi?
Post by: SBR* on November 19, 2012, 05:23:39 pm
Quote from: Blizzard
I beg to differ. Without experience, knowledge and wisdom, you can't make a good choice. If you are lacking a lot knowledge and experience compared to me and we are both equally wise, you simply can't make a good decision based on your knowledge and experience alone, especially because you don't see the whole picture. (Not that I do, but I surely can see more than you in this case.)

You wouldn't go vote for somebody either without first looking into what they are actually standing for.

I'm not saying experience makes you choose unwisely. I just wanted to show that inexperience can also give birth to new visions. For example, I often ask my Physics teacher questions about certain problems, to which he responds with: "I never looked at it that way before."

Quote from: Blizzard
Look around you. Look at your keyboard. Look at your screen. There's math appliance everywhere. It's a highly theoretical science, but it's being applied basically everywhere. I believe that practicality is far more important because of this.

That's not exactly what I meant. I, for one, am more interested in the theoretical part of mathematics and not so in the practical part. Those are two different sides of math. Maybe the theoretical side would care for tau, but the practical not so. That being said, this is but a hypothesis.
Title: Re: Tau or Pi?
Post by: AngryPacman on November 20, 2012, 05:44:37 am
I think Blizz should be banned from arguing because he clearly has the unfair advantage of always being right in a sensible way. Totally unfair.
Title: Re: Tau or Pi?
Post by: Blizzard on November 20, 2012, 06:08:46 am
Uhm, what? So far everybody has been acting civilized.
Title: Re: Tau or Pi?
Post by: G_G on November 20, 2012, 08:15:36 am
He was just making a joke Blizzard. xD
Title: Re: Tau or Pi?
Post by: Blizzard on November 20, 2012, 08:51:56 am
I hope so, because I can't see any smilies indicating that. SUCH AS :V:
Title: Re: Tau or Pi?
Post by: AngryPacman on November 21, 2012, 03:00:10 am
It was a joke but I was playing the straight man. The  :V: would've ruined it for everyone.
Title: Re: Tau or Pi?
Post by: Ryex on December 03, 2012, 05:01:40 pm

as anyone who uses math on a regular basis knows using Tau isn't hard it, changes nothing. we simply define it and move on with our derivation ect using Tau. the fact are that all the leg work has been done with 2pi is inconsequential if we really wanted to we could go through an replace with Tau and perhaps simplify a few equation see what we can get. hell in a computer system if you pre define 2pi as Tau and use you might cut out 8 simple multiplication operations and perhaps get a 3% speedup across that operation (provided it's a interpreted language as any compiler worth it's salt would see 2pi and compile a constant). but that's not the point.

the fact is that it would of been a hell of a lot easier to wrap our minds around it in the beginning if we had used Tau and not Pi. as far as education goes we should make an effort to at the very least present the concept of Tau and let student use it to ease the learning process.

those of us who have already learned this may have the concept down but that doesn't me we shouldn't strive to make the process easier for future learners so that we may advance together.
Title: Re: Tau or Pi?
Post by: Apidcloud on December 04, 2012, 05:50:04 am
After watching that video, I can say that I agree with "numberphile" as well. It's not about making a specific formula easier, it's about teaching.
Some formulas get easier using Tau, others don't, but that's really not the matter as everyone tries to prove. Tau may be a better concept when teaching. I remember of having some problems when trying to learn 2pi 'n stuff  :facepalm:

Quote from: Riex's
the fact is that it would of been a hell of a lot easier to wrap our minds around it in the beginning if we had used Tau and not Pi. as far as education goes we should make an effort to at the very least present the concept of Tau and let student use it to ease the learning process.

those of us who have already learned this may have the concept down but that doesn't me we shouldn't strive to make the process easier for future learners so that we may advance together.

:haha: