## The Tau Manifesto

For the discussion of math. Duh.

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nwest
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### Re: The Tau Manifesto

Ever since I read the Tau Manifesto, it's been eating at my brain. Everywhere I see Pi, I see the "what-if" scenario of Tau having been used instead. And every single time, it's more elegant and tells me something a little bit more about what's actually going on. Somehow I missed out on "grokking" math because it was taught to me with Pi instead.

Some of the things I've been thinking about:
Pi very literally means "half a circle." At least when measured in radians. Why doesn't Pi represent a full circle? Because it was originally measured with regard to the diameter. When's the last time you saw a formula that included the diameter? We've kind of changed systems on ourselves. We derived Pi in terms of the diameter so many millennia ago, but then did all our math using the radius. So, of course there has to be an ugly 2 next to Pi every time we use it.

Quick, what's the formula for the arc length of an angle of theta radians?
Let's derive it using the circumference formula:
C = 2*Pi*r
Wait...does the 2 go with Pi to make the full circle?
Or is it Pi * (2 * r) = Pi * Diameter? Hmm... Which is it?
Since Pi was derived in terms of diameter then I would say the second one makes more sense:
C = Pi * D
Hmm...If that's the case, then why do we write it 2 * Pi * r? Again, does the 2 go with Pi or with r?

Let's try it with Tau:
C = Tau * r

Because Tau was actually derived using the radius, this statement makes sense. A full circle times its radius gives you its circumference.
So the arc length of any angle could be calculated by using some fraction of the full circle.
Arc length of 1/8 of a circle? (Theta in this case = 1/8 * Tau.)
Arc length = (1/8) * Tau * radius. That's it.
Generalizing:
Arc length = Theta * r

In this case it actually MEANS something.

The other way?
C = (1/8) * 2 * Pi * radius
C = 1/4 * Pi * radius.

Unfortunately all I see is a 1/4 where I wish I was seeing a 1/8. This is confusing when kids are learning math. How many frustrated kids gave up on math because of Pi? Granted it's not that hard to multiply by 2. Unless you're bad at math to start with. Then it's just one more obstacle.

Perhaps we should go back to the diameter approach.
C = (1/8) * Pi * (2 * r)
C = (1/8) * Pi * D

Yes, now THAT means something. 1/8 of a circle works great when you keep Pi paired with what it was derived from, the diameter. Again, how often do we use diameter? The confusion with Pi comes from pairing it with something it wasn't derived from, i.e. the radius. And how do I generalize this as neatly as I did the Tau approach?

Arc length = Theta * (D/2)
Ugly. Since I got here using Pi, I really should keep the D/2 and not change it to r.

For all of you naysayers out there, I'll just end with this. If Pi represents a half circle, then WHY do we use it for all our equations that require full circles? If Pi is HALF of something, why aren't we using the WHOLE something? It's like saying "1/2 works better for math instead of 1 because that's how we've always done it. Anytime we need a 1, we'll actually call it 'two times one half'."

"That's how we've always done it" does not make it right.

cyanyoshi
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### Re: The Tau Manifesto

nwest wrote:Ever since I read the Tau Manifesto, it's been eating at my brain. Everywhere I see Pi, I see the "what-if" scenario of Tau having been used instead. And every single time, it's more elegant and tells me something a little bit more about what's actually going on. Somehow I missed out on "grokking" math because it was taught to me with Pi instead.

That feeling never does go away, does it? I still do this all the time, too. But as with the others in a long list of conventions in mathematics and physics I don't personally agree with, I have learned to just accept them and move on, content with the deeper insight I have gained along the way. It is reassuring that this little tau idea hasn't gone away completely, though!

Xenomortis
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### Re: The Tau Manifesto

On the other hand:
n*pi are the zeroes of sin(x).
n*pi are the local maxima and minima of cos(x)

campboy
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### Re: The Tau Manifesto

IMO the tau campaign goes wrong right at the start. If you are going to write the circumference of a circle in terms of something, the diameter is the right choice. The relationship [imath]C=\pi d[/imath] generalises to Barbier's theorem, that any curve of constant diameter [imath]d[/imath] has perimeter [imath]\pi d[/imath].

nwest
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### Re: The Tau Manifesto

campboy wrote:If you are going to write the circumference of a circle in terms of something, the diameter is the right choice.

I agree! I have no problem using Pi as long as the formulas use the diameter! Because then it makes sense! At least then you're using Pi paired with the thing it was derived from.

doogly
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### Re: The Tau Manifesto

nwest wrote:"That's how we've always done it" does not make it right.

It does for notation.
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gmalivuk
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### Re: The Tau Manifesto

Yeah, arguments from antiquity and from popularity are in fact completely valid when discussing communication. Where they are fallacies is when you're making a scientific argument.
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fishfry
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### Re: The Tau Manifesto

A quarter turn around the circle is tau/4. That's sensible. Sign me up for the tau revolution! Where do I get one of those T-shirts?

korona
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### Re: The Tau Manifesto

A quarter turn around the circle is pi/2. That's sensible.

I cannot believe that people who actually do math want to introduce a new convention just to get rid of a fixed multiplicative constant. I don't know a single mathematician who would be willing to introduce a new symbol for 2 pi. There isn't even an obvious benefit from doing so as pi without multiplicative factor is very common in any topic except for elementary properties of the circle.

Cleverbeans
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### Re: The Tau Manifesto

korona wrote:A quarter turn around the circle is pi/2. That's sensible.

No, no it isn't. I always use Tau now when working with angles and I teach it to all my students because it's so much more intuitive to work with. Sub in tau = 2pi right at the start and sub back at the end, voila 80% of high school radian mistakes disappear. I think the biggest increase in test scores was 40% but I generally get between 10-30% just by teaching them about tau.
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nwest
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### Re: The Tau Manifesto

Cleverbeans wrote:I always use Tau now when working with angles and I teach it to all my students because it's so much more intuitive to work with. Sub in tau = 2pi right at the start and sub back at the end, voila 80% of high school radian mistakes disappear. I think the biggest increase in test scores was 40% but I generally get between 10-30% just by teaching them about tau.

I hear stories like this all the time. How many of these stories do people need to hear before they'll stop being so stubborn about their precious Pi? No one cares that you've memorized 100+ digits. I memorized plenty myself. No one cares that books have been written on Pi. It just emphasizes the herd mentality we share as human beings...never questioning the status quo. If it's wrong, it's wrong. (If irrational behavior interests you like it interests me, read Predictably Irrational, or visit http://www.youarenotsosmart.com. Explanations of our limited brains that cause us to adopt fallacies as arrogant irrational thinkers.)

I know we're not going to change all the textbooks, but people should at least RECOGNIZE there's a better way than Pi. The test scores alone should be enough to convince anyone.

Numberphile:

The wonderful Vi Hart (every nerd's dream girl):

Of course the awesome Tau Manifesto:
http://www.tauday.com

And the article from Bob Palais that started the revolution:
http://www.math.utah.edu/~palais/pi.pdf

doogly
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### Re: The Tau Manifesto

oh my, I had almost thought that real mathematicians were involved, and then I investigated the first names
that is not the notable Palais
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nwest
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### Re: The Tau Manifesto

doogly wrote:oh my, I had almost thought that real mathematicians were involved, and then I investigated the first names
that is not the notable Palais

All 4 are mathematicians. Most of the naysayers I run across are non-mathematicians that like to troll forums. But it doesn't matter. The point is, it doesn't take a mathematician to recognize something so obvious.

doogly
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### Re: The Tau Manifesto

Sure, I was just taken aback when I thought Palais was on board, and checked.
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gmalivuk
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### Re: The Tau Manifesto

The test scores alone, if the difference is significant and replicable, show only that tau makes for a better introductory number to use when teaching plane geometry for the first time.

That's nothing to scoff at, of course, but as korona pointed out above, pi shows up in a lot of places apart from elemtary properties of Euclidean circles.
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nwest
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### Re: The Tau Manifesto

gmalivuk wrote:The test scores alone, if the difference is significant and replicable, show only that tau makes for a better introductory number to use when teaching plane geometry for the first time.

That's nothing to scoff at, of course, but as korona pointed out above, pi shows up in a lot of places apart from elemtary properties of Euclidean circles.

Sure. That's like saying the number 1/2 shows up a lot in math. Of course it does. But it's still half of something. Pi literally means "half a circle." Why is half a circle so much better than using a whole circle? Just about everywhere Pi shows up, there's a 2 next to it. Except in Euler's identity, which simply shows the elegance of looking at half a circle. It becomes (in my opinion) much more elegant when you actually use the whole circle.

I'm not disagreeing that Pi shows up a lot of places, but in a vast majority of cases, those formulas can be rewritten using Tau in such a way that you can more easily understand what the formula is doing. Speaking of Euler's identity, I didn't understand it until I saw it rewritten with Tau.

I personally like the elegance of the unit circle having an area of Pi. But what's really going on is that it's 1/2 Tau R^2, in other words, it's the integral of the circumference function with respect to the radius. And that actually means a lot more to me than just calling it Pi. I love how this formula matches all the other quadratic forms we're familiar with in physics, as mentioned in Bob Palais' article and in the Tau Manifesto. So it's not so special that the area is Pi; it's special that it's half of Tau, because it's an integral.

doogly
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### Re: The Tau Manifesto

On the one hand, there are some places where writing tau for 2pi would save you a little bit of ink.

On the other hand, there are a lot of integrals you do from 0 to pi, and having to put tau/2 in the limit of an integral would not only take up the ink, it would force your font size down in what is already prime real estate.
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brenok
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### Re: The Tau Manifesto

nwest wrote:Pi literally means "half a circle." Why is half a circle so much better than using a whole circle? Just about everywhere Pi shows up, there's a 2 next to it. Except in Euler's identity, which simply shows the elegance of looking at half a circle. It becomes (in my opinion) much more elegant when you actually use the whole circle.

No, Pi literally means "the sixteenth letter of the Greek Alphabet, with sound [p]". In Mathematics, it also represents the ratio between a circle's cirumference and its diameter.

I'm not disagreeing that Pi shows up a lot of places, but in a vast majority of cases, those formulas can be rewritten using Tau in such a way that you can more easily understand what the formula is doing. Speaking of Euler's identity, I didn't understand it until I saw it rewritten with Tau.

What difference does the presence of Tau or Pi makes on Euler's Identity?

fishfry
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### Re: The Tau Manifesto

doogly wrote:On the one hand, there are some places where writing tau for 2pi would save you a little bit of ink.

On the other hand, there are a lot of integrals you do from 0 to pi, and having to put tau/2 in the limit of an integral would not only take up the ink, it would force your font size down in what is already prime real estate.

You can use pi if you like! It's not going away. But why can't we give beginners a break? 1/4 of the circle is tau/4 radians. How can people object to that?

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### Re: The Tau Manifesto

fishfry wrote:1/4 of the circle is tau/4 radians. How can people object to that?

1/4 of the circle is pi/4 units of area! How can people object to that?

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### Re: The Tau Manifesto

Even as a pi proponent, I can object to that by pointing out that, at least in my experience, "circle" refers to the boundary more often than the interior of the disc.
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brenok
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### Re: The Tau Manifesto

gmalivuk wrote:Even as a pi proponent, I can object to that by pointing out that, at least in my experience, "circle" refers to the boundary more often than the interior of the disc.

Sorry, apparently in Portuguese, or at least on my school, what in English is called a disk we called "circle", and what you call a circle we called "circumference".

doogly
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### Re: The Tau Manifesto

Oh, in America they call discs circles very often. It is a more sophisticated bit of math to have rigid words to distinguish circle and disc, and then in higher dimensions, sphere and ball.
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### Re: The Tau Manifesto

Yeah, I suppose it's really a distinction I've seen made consistently only at levels of mathematics well beyond the point when anyone will still be confused by the fact that pi goes into a circle twice.
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nwest
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### Re: The Tau Manifesto

doogly wrote:On the other hand, there are a lot of integrals you do from 0 to pi, and having to put tau/2 in the limit of an integral would not only take up the ink, it would force your font size down in what is already prime real estate.

I guess my problem with this argument is that we don't have an issue with writing out the number 1/2... It's more ink than writing out a symbol that means the same as 1/2. It's never been an issue, and we've never had to come up with a symbol for this number. That's because we've already assigned the symbol "1" to mean a whole. And we've derived a system of fractions for anything less than this whole value. Pi came about from someone assigning a symbol to something that means "half a circle". And this only happened because they revered the diameter more than the radius. Just as we assigned the symbol "1" to mean a single whole, we should've assigned some value to mean a whole circle. But now, because of convention, we're stuck with the whole circle symbol being "2 Pi", when it should have just been Pi so many years ago.

In addition, putting Tau/2 in your integral tells you something about how much of the range you're integrating over.

I have a feeling that if Euler had fixed this problem a long time ago (assigning Pi to be 6.28...) no one would be discussing that perhaps we should be using "half a circle" everywhere.

nwest
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### Re: The Tau Manifesto

brenok wrote:No, Pi literally means "the sixteenth letter of the Greek Alphabet, with sound [p]". In Mathematics, it also represents the ratio between a circle's cirumference and its diameter.

You're making my case for me. It represents the ratio between a circle's circumference and its diameter. (Circumference has 3 C's by the way.) When was the last time you used a diameter in your equations? If Pi was derived using the diameter, then rather than using the radius everywhere, we should use D, and re-derive all our formulas with respect to the diameter. THEN a quarter circle would actually be 1/4 Pi diametrians (or some other name for this new made up unit.) Because then you would actually have Pi diametrians in a full circle.

Tau, on the other hand, is derived from the ratio of a circle's circumference to its radius. Therefore you have Tau radians in your circle.

So, if you're going to combine Pi with radians (something it was NOT derived from), then Pi LITERALLY means half a circle in radians. But it means a full circle in diametrians.

I suggest we change over to this new unit, the diametrian, so that Pi will actually make sense. And we should re-derive all our formulas to be in terms of the diameter instead.

Who's with me!??!

OK, obviously that's nonsense. A circle is defined by its radius more naturally than its diameter. So the radian is the more natural unit. And Tau is derived from the radius, which is why it makes so much more sense when kids are learning this stuff.

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### Re: The Tau Manifesto

Using tau/2 instead of pi in an integral only gives you more information about the range in those specific cases that come from something that goes around an actual whole circle. Which isn't always the case in the integrals I think doogly is talking about.
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nwest
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### Re: The Tau Manifesto

gmalivuk wrote:Using tau/2 instead of pi in an integral only gives you more information about the range in those specific cases that come from something that goes around an actual whole circle. Which isn't always the case in the integrals I think doogly is talking about.

That's a fair argument. But the point still stands that it's like integrating from 0 to 1/2. Pi is just the circular version of the number 1/2. That number will show up a lot in math, but no one's developed a symbol for it to make our lives easier. Maybe someone should write a manifesto adopting the symbol & to mean 1/2. Then, anytime we want to talk about a whole, we call it 2 &.

If that sounds ridiculous, this is what the Pi argument looks like to me (and legions of others.)

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### Re: The Tau Manifesto

It's not the same sort of argument, though, because pi advocates are suggesting keeping a convention while hypothetical & advocates would be recommending a whole new one.

Personally, my resistance to tau is analogous to my resistance to proposed English spelling reforms and the like. Sure, reformers may have the best of intentions and may have some underlying good points about the shortcomings of the current system, but their proposed change(s) and justifications thereof often leave a whole lot to be desired.
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nwest
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### Re: The Tau Manifesto

gmalivuk wrote:It's not the same sort of argument, though, because pi advocates are suggesting keeping a convention while hypothetical & advocates would be recommending a whole new one.

Personally, my resistance to tau is analogous to my resistance to proposed English spelling reforms and the like. Sure, reformers may have the best of intentions and may have some underlying good points about the shortcomings of the current system, but their proposed change(s) and justifications thereof often leave a whole lot to be desired.

So, if I understand correctly, you're not opposed to the idea of Tau. You're opposed to bucking established traditions.

I'm not trying to kill Pi. Just trying to bring awareness to the fact that we've been doing things kind of backward all this time, and that there are better ways. And it seems like you don't necessarily disagree with what I've been saying.... You just don't like getting rid of the traditional way of doing things. Do I understand correctly?

There really are 2 different questions in the debate:
1) Have we been doing it wrong all this time, and is there a better way?
2) Should we adopt Tau wholesale, change all our textbooks, and try to change the momentum of Pi?

All of my posts are really about the first question. Unfortunately, I believe the ship has sailed for number 2, but I certainly like to see how many people agree that the answer to number 1 is a resounding YES. And I like to see if I can use reason and logic to bring them around to agreeing with that assessment. (And I get the feeling you kind of agree with that answer since you seem to be more adamant about answering the second question.) I don't think the answer to number 2 is a YES, but I'm kind of disappointed it's probably a NO. It's hard to change the course of a ship that massive.

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### Re: The Tau Manifesto

When you integrate from 0 to pi, you are usually not doing "half" of something. Maybe an entire azimuth, for example.

If anything, the more you talk about tau, the more resistant I get, because it seems like you are oblivious to the big picture. Instead of "it shows up all over," you've really just got the volume of S^1 in mind, and are riding this hobby horse way past its expiration.
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### Re: The Tau Manifesto

doogly wrote:If anything, the more you talk about tau, the more resistant I get, because it seems like you are oblivious to the big picture.

See also: spelling reformers, Lojban enthusiasts, Esperanto evangelists, and anyone else who thinks things could be suddenly and drastically improved if only everyone would buy into this one weird trick they have.
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nwest
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### Re: The Tau Manifesto

doogly wrote:When you integrate from 0 to pi, you are usually not doing "half" of something. Maybe an entire azimuth, for example.

If anything, the more you talk about tau, the more resistant I get, because it seems like you are oblivious to the big picture. Instead of "it shows up all over," you've really just got the volume of S^1 in mind, and are riding this hobby horse way past its expiration.

If this has taught me anything, it's that humans are closed-minded about new ideas. You know what it's called when people get more entrenched in their ideas in internet forums? The Backfire Effect. Regardless of logical new information, people that have already made up their mind will never see the good in anything else. You are a prime example of this.

I didn't want to give up Pi. I memorized more digits than I can recall now. It was difficult for me at first, but I had to have an open mind when it came to Tau, and it has changed the way I see math. It has helped kids learn math more quickly. It simplifies a lot of things. I did my own research. I re-derived a lot of things. That's the great thing about math. You can re-derive it from the ground up. All by yourself. You don't have to believe a thing that you were taught in your math class, because you can re-discover it all on your own. I re-discovered all sorts of topics in math, on my own, including geometry, trigonometry, calculus and beyond. And when I was done, Pi wasn't there.

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### Re: The Tau Manifesto

I'm confused now. What is there to "derive" ? Do you think something deeper than absorbing a factor of two is happening?

I can't imagine how it could change the way you "see math," unless you are looking at multiplication by the number two as something way, way more significant than it is.
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### Re: The Tau Manifesto

nwest wrote:
doogly wrote:When you integrate from 0 to pi, you are usually not doing "half" of something. Maybe an entire azimuth, for example.

If anything, the more you talk about tau, the more resistant I get, because it seems like you are oblivious to the big picture. Instead of "it shows up all over," you've really just got the volume of S^1 in mind, and are riding this hobby horse way past its expiration.

If this has taught me anything, it's that humans are closed-minded about new ideas. You know what it's called when people get more entrenched in their ideas in internet forums? The Backfire Effect. Regardless of logical new information, people that have already made up their mind will never see the good in anything else. You are a prime example of this.
Be all that as it may, you'll note that doogly actually gave a logical explanation for his resistance, and it didn't include or imply closed-mindedness or stubbornness anywhere.

I, for example, am opposed to proposed spelling reforms of English not because I think there are absolutely no problems with existing spellings, but rather because the proposals I've seen are themselves often quite shortsighted and oblivious to a lot of the nuances of English as it exists. I argue against the likes of Lojban and Esperanto not because I think those languages are stupid in themselves, but because they're usually most vocally promoted by idealists who are ignorant of how natural language works and develops, and who claim miraculous properties of their pet auxlangs that almost surely never existed.

Teaching tau, like many proposed educational reforms, may very well improve some limited areas of student understanding, while causing problems elsewhere and failing to produce positive results on par with the cost and effort of the reform in the first place.

And when the advocates so rarely venture beyond the single fact of the volume of S^1, you don't inspire a great deal of confidence in any positive results outside of that fairly narrow realm of mathematics.
Unless stated otherwise, I do not care whether a statement, by itself, constitutes a persuasive political argument. I care whether it's true.
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### Re: The Tau Manifesto

All these posts supporting tau seem to have this underlying (false) assumption that circle arclength is the be-all, end-all of what pi is.

So I guess I'd sum up my resistance to tau in the following way: What arguments for tau exist that don't involve arclength on a circle?
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fishfry
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### Re: The Tau Manifesto

MartianInvader wrote:All these posts supporting tau seem to have this underlying (false) assumption that circle arclength is the be-all, end-all of what pi is.

That's a very unfair characterization of the tau-ist position. Of course pi shows up everywhere. Math, physics, probability, and so on. It needn't go away. My advocacy is primarily based on beginner pedagogy. I believe that tau is a much more natural unit of arc length on the circle for people hearing about radians for the first time. One-quarter is one-quarter, I've already mentioned that.

MartianInvader wrote:So I guess I'd sum up my resistance to tau in the following way: What arguments for tau exist that don't involve arclength on a circle?

For me arclength of the circle is the primary motivator. Everyone's free to use tau or pi as they like. But the basic unit taught in trigonometry class would be tau, because it's one less thing to confuse students.

But if you really want another application, a good characterization of pi is that it's half the period of the cosine and sine functions. Tau would win there too ... tau is the period of the cosine and the sine functions.

Nobody's suggesting that pi should ripped out of the number line, you know! It'll still be there. But in high school trig, we define the arclength of the unit circle to be tau radians and go from there.

I don't expect this to ever happen. But math curricula are always changing. You never know.

nwest
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### Re: The Tau Manifesto

I guess my point is that if Pi had been properly defined in the beginning to be 6.28..., then, today, 3.141592... wouldn't have any other significance besides being half of the full circle. You can make all the excuses for Pi that you want, but all I see is you making excuses for something that was wrongly defined in the first place. Obviously history did not turn out this way. But in the alternate history where Pi was properly defined, we wouldn't be having these discussions. The only reason you're using Pi today is because it's been culturally pounded into you. I truly believe that if Euler had fixed this, we wouldn't even be talking about this today.

Like Vi Hart says, "Stop making excuses for Pi."

I think I'm going to take a break from this.... So have fun continuing the discussion on your own.

doogly
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### Re: The Tau Manifesto

We might be having these discussions, but we'd probably be pretty dismissive of the folks who thought tau should equal pi/2... or maybe of the fermion fans who want tau = 2pi (even with your new pi.) Or maybe someone would want tau = sqrt(3/2)pi (again, with your pi) so that pi^2 = 1 + 1/4 + 1/9 + 1/16 + 1/25 + ... Doesn't that seem neat? The right way for things to be? We should just tidy it up. Everyone knows that plane geometry is a special case, but the Reimann zeta function is deep, important mathematics. But yes, we'd dismiss such things.

This is because it is just convention. There is no mathematical content inherent in the choice to put a 2 in or out. So of course if it had gone the other way, nobody would really care. It's a factor of 2.
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korona
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### Re: The Tau Manifesto

nwest wrote:Like Vi Hart says, "Stop making excuses for Pi."

I would not care if pi was pi/2 or pi/4 or 2pi, there is no "right choice" and elementary geometry is the least convincing argument to change a convention.
If we could really rewrite history you could as well consider setting pi = pi / 2 so that all maxima and minima of sin and cos are integral. But then again, that is not a convincing argument because pi shows up in every topic, from complex analysis, topology, algebra to the representation theory of Lie groups and it is hard to say which value of pi would involve writing the least fractions.

I don't see any person, not working in elementary geometry (i.e. teaching in high school), who is annoyed by a multiplicative constant.