## Tau or Pi?

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

#### SBR*

##### 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 .

#### Blizzard

##### November 10, 2012, 07:51:48 am #1 Last Edit: November 10, 2012, 08:09:21 am by Blizzard
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, 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.
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#### winkio

##### November 10, 2012, 11:27:52 am #2
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

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.

#### KK20

##### November 10, 2012, 11:37:49 am #3
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).

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#### winkio

##### November 10, 2012, 11:46:02 am #4
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.

#### Blizzard

##### November 10, 2012, 11:54:23 am #5 Last Edit: November 10, 2012, 11:57:06 am by Blizzard
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.
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#### SBR*

##### November 10, 2012, 02:19:12 pm #6
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, 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).

#### Blizzard

##### November 10, 2012, 03:40:00 pm #7 Last Edit: November 10, 2012, 03:47:08 pm by Blizzard
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
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#### winkio

##### November 10, 2012, 04:49:27 pm #8
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.

#### SBR*

##### November 10, 2012, 07:40:32 pm #9
I've always wondered: why do we measure angles between -pi and pi instead of 0 and 2pi (= 0 and tau)?

#### winkio

##### November 10, 2012, 08:13:32 pm #10
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.

#### SBR*

##### November 11, 2012, 07:48:16 am #11
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.

#### Blizzard

##### November 11, 2012, 08:38:29 am #12 Last Edit: November 11, 2012, 08:40:21 am by Blizzard
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.
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#### SBR*

##### November 11, 2012, 09:26:48 am #13
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.

#### Blizzard

##### November 11, 2012, 09:34:45 am #14
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.
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#### SBR*

##### November 11, 2012, 11:02:53 am #15
circumference/r = 2pi, not 2pi radians. What do you mean by "the radian is the more fundamental thing here"?

#### Ryex

##### November 11, 2012, 04:05:46 pm #16
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.

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#### Blizzard

##### November 11, 2012, 04:09:59 pm #17 Last Edit: November 11, 2012, 04:13:20 pm by Blizzard
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.
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#### Ryex

##### November 11, 2012, 06:21:14 pm #18 Last Edit: November 11, 2012, 06:32:06 pm by Ryex
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.
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#### SBR*

##### November 11, 2012, 06:27:04 pm #19
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.