## Tau or Pi?

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

#### Blizzard

##### November 18, 2012, 05:24:21 am #40 Last Edit: November 18, 2012, 05:25:22 am by Blizzard
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. 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.
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#### SBR*

##### November 18, 2012, 07:09:06 am #41
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.

#### AngryPacman

##### November 18, 2012, 07:16:33 am #42
@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.
G_G's a silly boy.

#### Blizzard

##### November 18, 2012, 09:54:27 am #43 Last Edit: November 18, 2012, 09:55:33 am by Blizzard
@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.
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#### SBR*

##### November 18, 2012, 10:30:17 am #44
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.

#### Blizzard

##### November 18, 2012, 10:46:26 am #45
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.
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#### SBR*

##### November 18, 2012, 11:17:45 am #46
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.

#### Blizzard

##### November 18, 2012, 12:05:44 pm #47
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.
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#### AngryPacman

##### November 19, 2012, 04:04:13 am #48
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?
G_G's a silly boy.

#### winkio

##### November 19, 2012, 04:21:22 am #49
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.

#### Blizzard

##### November 19, 2012, 04:42:44 am #50 Last Edit: November 19, 2012, 04:43:51 am by Blizzard
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.
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#### AngryPacman

##### November 19, 2012, 07:46:05 am #51
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.)
G_G's a silly boy.

#### Blizzard

##### November 19, 2012, 09:30:57 am #52
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.
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#### SBR*

##### November 19, 2012, 10:45:42 am #53
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.

#### winkio

##### November 19, 2012, 10:58:26 am #54
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.

#### SBR*

##### November 19, 2012, 11:47:51 am #55
I figured they didn't program calculators to use completing the square to solve quadratic equations.

#### winkio

##### November 19, 2012, 11:55:12 am #56
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.

#### SBR*

##### November 19, 2012, 12:19:28 pm #57
Hmmm, interesting. I always thought calculators just filled in a bunch of numbers and gave you the one closest to the answer. My bad.

#### Blizzard

##### November 19, 2012, 02:53:16 pm #58
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.
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#### AngryPacman

##### November 19, 2012, 03:18:08 pm #59
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
G_G's a silly boy.