## The Tau Manifesto

For the discussion of math. Duh.

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z4lis
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### Re: The Tau Manifesto

dissonant wrote: some minor mathematical curiousities (Fourier series, Riemann's zeta function, Euler's identity)

If Fourier series and Euler's identity are minor mathematical curiosities, what kind of math are you studying?
What they (mathematicians) define as interesting depends on their particular field of study; mathematical anaylsts find pain and extreme confusion interesting, whereas geometers are interested in beauty.

nwest
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### Re: The Tau Manifesto

I think the thing to remember is that some guy at some point in time decided to assign 3.1415... to the Pi symbol. It soon caught on as a convention. Do you really think we'd be having this discussion today (about whether he made the right choice) if instead he assigned 6.2831... to Pi? I really don't think we would. Just because it's what everyone is used to doesn't mean it's right. The area of the circle would have ALWAYS been in polynomial form. Euler's identity would have been derived in terms of a full turn. I haven't heard a single argument that has convinced me Pi is better, because they always come back to the debater not liking change, or having to throw away all the digits he's memorized. So he makes up arguments for why the status quo should just be accepted, even though it's quite obviously wrong.

Is Pi deeply rooted in our culture already? And will it be hard to change the standard? Of course. But is that really a reason to not slowly adopt the right standard? It's been done before. Perhaps in a hundred years, people will use Tau more than they use Pi.

Let me ask this. If you lived in a time when the Pi standard wasn't in use yet, and you had the opportunity to adopt a circle constant that would be used for years to come, what would you choose, and why?

gmalivuk
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### Re: The Tau Manifesto

nwest wrote:But is that really a reason to not slowly adopt the right standard?
No. The reason not to slowly change from pi to tau is that the only benefit is dropping a factor of 2 sometimes. This is such a small benefit that it's effectively outweighed by even a very small cost or inconvenience of changing it. If you want to argue for changing a standard, work on hurrying the US conversion to metric, or try to get people to switch the sign conventions for electric charge, or convince everyone to stop using QWERTY keyboard layouts. All of those changes would actually offer significant improvements in many things. Unlike saving a couple minutes over a lifetime by not writing a '2' sometimes.
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bbq
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### Re: The 2pi Manifesto

Maybe we could change it so pi=3 and then multiple every other number by (old)pi/3.
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Mavrisa
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### Re: The Tau Manifesto

gmalivuk wrote:... The reason not to slowly change from pi to tau is that the only benefit is dropping a factor of 2 sometimes. This is such a small benefit that it's effectively outweighed by even a very small cost or inconvenience of changing it. ... All of those changes would actually offer significant improvements in many things. Unlike saving a couple minutes over a lifetime by not writing a '2' sometimes.

I realize this is a bit of a dead thread, and I hope nobody gets upset with my resurrecting it, but I just have to disagree here.
The point was not solely to save people from having to write a 2 sometimes. I see it mostly as a way of making trig make sense to people who are just learning it. I took pi to be the same as 180 degrees for the longest time, and then what do I find out? Degrees are stupid and shouldn't even be taught in the first place. Why not have 1 tau = 1 turn? Everything just clicks that way. Screw degrees, as long as you understand fractions, you can understand trig, and calculus will make more sense down the line, and a few formulas look a bit prettier if you care to get that far in math, which most people don't anyway.
What I'm trying to say is that teaching tau rather than ever bothering with degrees would save LOADS of time, in everyone's life who passes through grade school.
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### Re: The Tau Manifesto

Once I see actual pedagogical research showing conclusively that students are hindered by pi and helped by tau, I will accept that a switch would be worthwhile. But so far, I've just seen people make baseless claims to that effect.
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Jyrki
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### Re: The Tau Manifesto

Mavrisa wrote: I see it mostly as a way of making trig make sense to people who are just learning it. I took pi to be the same as 180 degrees for the longest time, and then what do I find out? Degrees are stupid and shouldn't even be taught in the first place. Why not have 1 tau = 1 turn? Everything just clicks that way. Screw degrees, as long as you understand fractions, you can understand trig, and calculus will make more sense down the line, and a few formulas look a bit prettier if you care to get that far in math, which most people don't anyway.
What I'm trying to say is that teaching tau rather than ever bothering with degrees would save LOADS of time, in everyone's life who passes through grade school.

I disagree. For someone learning trig, as in learning angle measurements, degrees are the most natural unit. Children at that age are most comfortable with integers. The most common angles they will see are 90,45,30,60 degrees. I don't see them switching to radians. Can you really see them having a radian scale of whateveryoucallthoseniftytransparentpiecesofplasticwhoseenglishnameimnotawareof on a protractor (warm thanks to Dopefish), and proudly reporting that this angle is 1.57 radians plus/minus the margin of error.

Furthermore, you do realize that selecting the radian scale over degrees doesn't really say anything about pi vs 2pi. In other words, I don't see your point at all.

Some similar group of 'geniuses' once tried to change the definition of degree so that a quarter turn would be 100 'degrees'. Didn't catch on, as one third of that occurs too frequently to not have an integer value.

Counting in full turns may show a benefit in some situations, like somersaults. But (at least Finnish) snowboarders refer to 2 and half turns as back9 for short of a 900 degree twist (and similar abbreviations for 540 degree, 720 degree or 1080 degree twists).

The frequency of occurrence varies so much all depending on what you do. I would rather pick [imath]\pi/2[/imath] as the 'unit', because one of the most frequent ranges of integration on a calculus course goes from 0 to pi/2. Also, unless the Chinese world domination begins, we are unlikely to switch from our style of writing text one line at a time to one column at a time. Therefore the cost (in terms of space on a separate line, or legibility when typesetting inline) of having a fraction will be significantly larger than that of having a multiplication. This is IMHO a very pressing issue. The frequency of 2pi over pi should probably be 20-fold before that argument becomes convincing: the cost of adding several multipliers is offset by the need to have a single denominator. We can take this further!!! Yes!!!! Pi/2 it is!!!!
Last edited by Jyrki on Sun Aug 14, 2011 7:23 am UTC, edited 2 times in total.

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### Re: The Tau Manifesto

whateveryoucallthoseniftytransparentpiecesofplasticwhoseenglishnameimnotawareof

Protractor.

While degrees feel nice, I generally think that much of that arrises as a result of the fact that we were taught them from such a young age. If radians were taught early on, I would think they'd seem much more natural. The fact that pi is a scary infinite non-repeating thing can be bypassed by simply not even attaching a numerical value with pi, and letting the young un's treat the pi symbol in the same way as the degree symbol is used. While you would be dealing with fractions or terminating decimals in most measurements rather than an integer amount, I don't think that'd be more of a blockade to learning than the initial transition from degree to radians currently is.

(The above doesn't really comment on tau versus 2pi, but I figure it's far to late to change now.)

antonfire
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### Re: The Tau Manifesto

Jyrki wrote:Pi/2 it is!!!!
I think you mean pi/180.
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Jyrki
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### Re: The Tau Manifesto

antonfire wrote:
Jyrki wrote:Pi/2 it is!!!!
I think you mean pi/180.

LOL. Even a good idea can be overdone?

antonfire
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### Re: The Tau Manifesto

But surely that is the most natural unit, for someone learning trig.
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mol
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### Re: The Tau Manifesto

http://www.thepimanifesto.com/

thougts?

arbiteroftruth
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### Re: The Tau Manifesto

Most of his argument focuses on how pi works better when discussing areas. This is true, but the question is then whether area makes sense as the starting point of our conventions. I would argue that it makes more sense to define conventions starting in 1 dimension and expanding from there.

If we consider the notion of constructing a mathematical system from scratch, the first thing you need is a reference point against which all other values are compared(so, technically, conventions start with 0 dimensions), and we call this reference point "0". The next most important thing is a reference magnitude against which to compare all other numbers' distances from 0, and we call this "1". Later, when we get around to complex numbers and the need for 2 dimensions in our system, it becomes useful to define a unit of area, which we define in terms of a square with side lengths equal to our 1-dimensional reference magnitude.

Hypothetically, we could define a unit of area as an inherent convention, and work from there back down to 1-dimensional concepts and up to higher dimensions. But 1), starting with 2-dimensional conventions and getting down to 1 is less intuitive than starting with 1, and 2) expanding to higher dimensions from a 2-dimensional convention will only get you even-numbered higher dimensions, unless you first work down to 1-dimensional concepts and then expand from there.

Basically, dimensionality is a pretty inherently quantized concept, and fundamental to just about everything math is intended to work with, so starting with the lowest quantization of dimensionality makes the most sense for our conventions.

Back to pi vs. tau. If we define the size of a circle primarily by its radius, then a) if we define its geometry based on its 2-dimensional property of area, then pi makes sense, but b) if we define its geometry based on its 1-dimensional property of circumference, then tau makes sense. Given that an arbitrary area is most easily determined by integration of simple rectangular or triangular areas along the length of the arbitrary shape, and simple rectangular or triangular areas are so simple precisely because we define unit area in terms of squares of side length 1, we already work in terms of expanding out from 1-dimensional concepts, so the same should go for the circle. This is why I find much of his list of equations unconvincing: so many are simple integrals, which are essentially two-dimensional concepts, and end up dividing out the 2 from 2pi because that's what integration often does, just as illustrated in the tau manifesto's example of how the 1/2 factor is a natural result of integration in circular area. As a result, pi works better for the cleanliness of the final formula, but reveals less about the integration process that justifies the formula.

To put it another way, if we think the circle constant should be defined in terms of area, why not volume? The constant could be x=~4.1888 and the volume of a unit sphere would be x. And of course we could define similar constants for any number of dimensions. If we're to choose one number of dimensions over another, there should be something particularly special about that number of dimensions. 1 is the minimum number, and all higher dimensional geometries can be derived as expansions of that using a single repeating process, and any multipliers that result from that should help clarify that relationship. Any higher dimension used as a starting point needs at least two separate processes: 1 to expand to higher dimensions and 1 to derive the lower dimensions.

Ultimately, the question shouldn't be simply what's more convenient, because there's nothing stopping people from using whatever notation is most convenient in a given scenario if it helps. The question should be which constant is the fundamental starting point, and which constant(s) arise from that and may deserve their own notation as a matter of convenience. Given the relationship between numbers of dimensions, and relatedly the order in which we teach students these concepts, tau works better as a starting point in my view, with pi being useful shorthand for certain situations. Like people have said, pi is already embedded in common usage, so I don't think it's going to disappear anytime soon, but that doesn't mean we should avoid using tau where it's more convenient, which just happens to be those areas that are more fundamental for learning the mathematical concepts.

Tirian
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### Re: The Tau Manifesto

arbiteroftruth wrote:Most of his argument focuses on how pi works better when discussing areas. This is true, [...]

I'm not quite willing to yield on that point. It's a common and easily-derived formula of plane geometry that the area of a regular polygon is one-half the perimeter times the apothegm, and that very naturally extends to the area of a circle being one-half the circumference times the radius. In other words, the area of a circle of radius r is one-half times tau*r times r or one-half tau r^2. Without a doubt, it's more concise to express tau/2 as pi, but to the degree that doing so hides our motivation for realizing the area of a circle there is a certain lack of elegance. Let me just say that if we had been using tau all along and this was the best argument in someone's "Pi Manifesto", it would be quite unconvincing.

arbiteroftruth
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### Re: The Tau Manifesto

Tirian wrote:
arbiteroftruth wrote:Most of his argument focuses on how pi works better when discussing areas. This is true, [...]

I'm not quite willing to yield on that point. ... Without a doubt, it's more concise to express tau/2 as pi, but to the degree that doing so hides our motivation for realizing the area of a circle there is a certain lack of elegance. Let me just say that if we had been using tau all along and this was the best argument in someone's "Pi Manifesto", it would be quite unconvincing.

That's essentially what I argued. If you define the circle constant in terms of area inherently, then pi makes sense, but it doesn't make sense to define the circle constant inherently in terms of area, because areas in general are derived from scalar dimensions, with the unit square being the reference.

In any event, I think I'll go through the details of the pi manifesto a bit more.

He then argues that just as tau/2 is half a circle of arc length, pi/2 is half a circle in area, and considers these advantages comparable. This is just a specific application of the prior argument for defining pi in terms of area, and the same counterargument applies.

Next he discusses the Gaussian integral, pointing out that it evaluates to the square root of pi. Unfortunately, this is again a case where we're discussing area, tying back to the counterarguments on that point. In this case, one of the common methods of computing the Gaussian integral involves taking the equivalent integral across 2-dimensional space and taking the square root to get the 1-dimensional equivalent. One of the ways to make the problem easier once it's in 2 dimensions is to evaluate it using polar coordinates, which means we need to integrate theta across the full circumference of a circle, which means the circle constant relating circumference to radius is the fundamentally important number that lets us compute the integral this way. Meaning again that although the use of pi may yield a cleaner final equation, the use of tau simplifies the explanation justifying that equation.

He then discusses other distributions. I'm not qualified to discuss them in depth, but I suspect that they similarly can be viewed as an application of area, tying back to the same counterargument.

His argument about the interior angles of polygons is much stronger. I'd say that's one area in which pi notation is probably preferable. However, I still don't think it is a strike against considering tau the more fundamental number. The sum of angles in polygons is a multiple of pi because any polygon can be broken into triangles, and the interior angles of triangles sum up to pi. Triangles, having the least number of sides possible for a polygon, are in that sense the fundamental polygon to consider when looking at interior angles. However, the sum of angles in a square is tau, and the square is the defining shape for Cartesian coordinates, making the square another candidate for the fundamental polygon to consider. Again, this is definitely one of the stronger argument for pi in my view, but I don't think it crushes tau or outweighs tau's advantages up to this point.

He then talks more about trigonometric functions. But all of the trigonometric functions except sin and cos are simply ratios involving sin or cos being in the denominator, and as a result his arguments in this section are ultimately a repeat of his argument regarding pi as the frequency of zero-crossings in sin and cos.

His list of other functions to consider is largely out of my league, but I do notice that many of them are integrals of functions relating to circular geometry, relating back to the argument about area. I'll leave the rest up to others to debate.

His argument about Euler's identity is a bit silly. Asking which value in Euler's formula is the first to result in an integer is literally just asking what angle is a half-circle in the complex plane. The fundamental fact expressed by Euler's formula is that complex exponentiation is equivalent to rotation in the complex plane, so the relevant constant is the constant associated with rotation around the edge of a circle, which is tau. Essentially he's arguing based on the importance of the angle of a semi-circle, and indeed that should be one of the special angles, since it's the angle that results in a straight line. But this is just a more geometric way of phrasing the zero-crossing argument about trigonometric functions. Again, that does make it a special angle, but if we have to pick one angle as the most special of all of them, it should be the one most related to what a circle fundamentally is, and this means the angle of full rotation.

Darryl
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### Re: The Tau Manifesto

My problem with pi boils down simply to how its definition came about. Diameter is something we pretty much don't use in maths. Circles are defined by origin and radius (or origin and point on the circle, which is still going to give you radius directly). The only aspect of a circle readily defined with regards to diameter is its Circumference, which is more often done (due to having radius more commonly than diameter) as 2*pi*r. Thus, defining the circle constant in regards to something that is not used past 3rd-grade maths is a poor idea. A circle constant defined on the ratio of circumference to radius works better conceptually.

Now, pedagogically, the main benefit would be to eliminate diameter as being this "big important thing" when learning grade-school geometry and instead learning radius from the start. Thus when they switch from perimeters to areas, children are used to radius.

However, the engineering arguments for pi are very convincing. And area of the unit circle is pi, not tau.
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### Re: The Tau Manifesto

Darryl wrote:Now, pedagogically, the main benefit
Until anyone does actual research on how pi vs. tau affects math education, I'm choosing to ignore all these nice stories about how one or the other will make math easier to learn.
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skullturf
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### Re: The Tau Manifesto

gmalivuk wrote:
Darryl wrote:Now, pedagogically, the main benefit
Until anyone does actual research on how pi vs. tau affects math education, I'm choosing to ignore all these nice stories about how one or the other will make math easier to learn.

I admit you have a good point. In the absence of actual pedagogical studies, this is just speculation.

But surely it's extremely plausible speculation?

On the one hand, you have "pi = a half circle, pi/2 = a quarter circle (or a half semicircle), pi/3 = a sixth of a circle (or a third of a semicircle), etc."

On the other hand, you have "tau = a full circle, tau/2 = a half circle, tau/4 = a quarter circle, tau/6 = a sixth of a circle, etc."

Surely it's plausible that one would be more intuitive to students being introduced to measuring angles not using degrees?

However, I agree with those who say that expecting everybody to now switch from pi to tau is quixotic and impractical.

But at the very least, there's a decent argument for preferring tau over pi on intuitive or aesthetic grounds.

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### Re: The Tau Manifesto

skullturf wrote:But surely it's extremely plausible speculation?
Yes. So is "everything goes around Earth".
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### Re: The Tau Manifesto

Point taken.

But I guess part of my reason for saying what I said is:

Being a college math instructor myself, I often have to make a decision as to which way to present a topic (what notation to use, what informal verbal hand-wavy description to give, et cetera).

It's rare that we make such decisions based on actual pedagogical studies. (Mind you, we certainly should pay attention to such studies when they exist.) But a college math instructor faces a great many little decisions almost every day. (Do I use f and g, or u and v, when teaching integration by parts? What informal verbal description should I give of Green's theorem?) Not all of these specific questions have been the subject of their own pedagogical experiment.

So when teaching mathematics, it's reasonable and perhaps unavoidable to rely on our own intuitions and our own aesthetic sense when deciding how to present topics, rather than searching the literature for an actual pedagogical experiment that was done on precisely the question we're considering.

Anyway, this is perhaps all somewhat beside the point. I probably wouldn't teach tau to beginning trigonometry students, but my reasons are rather banal and prosaic: everyone uses pi.

But if we regard the tau manifesto as primarily an argument about aesthetics, I think the author makes a good case.

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### Re: The Tau Manifesto

skullturf wrote:But surely it's extremely plausible speculation?

[...]

However, I agree with those who say that expecting everybody to now switch from pi to tau is quixotic and impractical.

But at the very least, there's a decent argument for preferring tau over pi on intuitive or aesthetic grounds.

I tend to agree. On a related note, I think it's a shame that the Chaldeans & Babylonians were so hung up on 360. I reckon it'd be nicer if we divided a right angle into 120 degrees instead of 90. All the principal angles would still be integers, and you'd be able to use the trig sum and difference and half-angle formulae to find exact expressions for the trig ratios for a single degree (and binary fractions of a degree). Of course, I don't expect anyone would be interested in adopting such a convention.

Deveno
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### Re: The Tau Manifesto

personally, i think that fractions are bad. since we all agree (we do, don't we?) that we can only actually calculate with rational approximations, let's just choose a really, really large denominator, and then every number will be an integer! rocking it old-school! (besides this is just what computers do anyway, amirite? we haven't yet developed true cauchy processors, eh?).

and come on, now, this base 10 stuff is just...so..arabic! we need a truly 21-st century arithmetic, to show off our deeper understanding of numbers. i'm thinking...tally marks, pretty universal, what do you say?

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### Re: The Tau Manifesto

Tauist wrote:*Wants to get a τ Club started at my high school*- so far i have 5 τist followers. Not exactly sure what we would do tho >.<

Encourage people (preferably people you don't like) to make a π club. When they do, you can have gang wars.

Waterice man
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### Re: The Tau Manifesto

I prefer alpha:=inf{x>0|cosx=0}

Deveno
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### Re: The Tau Manifesto

not to actively accept or reject the use of π versus τ, but i'd like to point out one thing:
how do we define π? "as the ratio of a circle's circumference to it's diameter".
what's a circumference? "the length of the circle's perimeter".
how do we know this is always the same? "well, we measure them, and compare".
how do you measure a curved length?
this. this is a complicated question, one which we were not able to give even a partial answer to, until AFTER π had been used for centuries.
it just so turns out, that the most "reasonable" definition of π, is: ∫ [-1,1] √(1 - x^2) dx.
that is: measuring the length of curves, depends on writing a curve as a function.
no fair using the trigonometric functions if we don't know what π is, first.
now, from a pedagogal point of view, you can hardly expect to teach calculus before geometry or trigonometry.
nevertheless, at the point in education where π is introduced, it is largely a matter of "teacher says so"
that π has a definite value (that it isn't say, slighty more for "really big circles") and what that definite value is (approximately).
in a similar fashion, real numbers (and irrational numbers) are often introduced in such a way as for the average
student to be totally unable to decide if something is a real number or not.
what i mean to say is: the question of π or τ, is the wrong question. why are we defining these things, and teaching facts like:
"the area of a circle is pi times the radius squared" without proof, and then in the same heartbeat require young
math students to "show their work". it's just a bit hypocritical, is it not?
the bulk of the argument for τ rests on the idea that radius is a more natural parameter than diameter.
i am willing to accept that this is true, if you can only answer me this:
parameter for what? if your answer involves "area" or "length" or "volume", i must defer to the question, then:
what are THOSE things?
Last edited by gmalivuk on Thu Oct 13, 2011 10:15 pm UTC, edited 1 time in total.
Reason: to make slightly less shitty formatting

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### Re: The Tau Manifesto

Deveno wrote:how do you measure a curved length?

this. this is a complicated question, one which we were not able to give even a partial answer to, until AFTER π had been used for centuries.

This is bullshit.

http://www.math.ubc.ca/~cass/archimedes/circle.html

Archimedes did it quite rigourously two thousand years ago.

At the very least we can talk about how much pencil it takes to draw a circle vs how much pencil it takes to draw a known length.
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### Re: The Tau Manifesto

Pretty much everything else there is also wrong, Deveno, *and* you have shitty line spacing habits to boot.

Deveno wrote:how do we know this is always the same? "well, we measure them, and compare".
Um, no. It can be proven to be the same based on the axioms of Euclidean geometry.

how do you measure a curved length?
this. this is a complicated question, one which we were not able to give even a partial answer to, until AFTER π had been used for centuries.
it just so turns out, that the most "reasonable" definition of π, is: ∫ [-1,1] √(1 - x^2) dx.
that is: measuring the length of curves, depends on writing a curve as a function.
Wrong again. We'd been approximating curves with smaller and smaller line segments for millennia before anyone invented calculus.

now, from a pedagogal point of view, you can hardly expect to teach calculus before geometry or trigonometry.
Not a problem, since pi comes out of geometry, along with trig.

nevertheless, at the point in education where π is introduced, it is largely a matter of "teacher says so"
Only if you have a shitty teacher.
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Timefly
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### Re: The Tau Manifesto

May I just say...

WHO CARES?

Seriously, maths has more to worry about than your silly notation for circle constants, could we maybe be having a look at function notation if we're going to get into notation arguments.

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### Re: The Tau Manifesto

You are welcome to start your own thread on something you find more worth discussing.
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Deveno
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### Re: The Tau Manifesto

gmalivuk wrote:Pretty much everything else there is also wrong, Deveno, *and* you have shitty line spacing habits to boot.

Deveno wrote:how do we know this is always the same? "well, we measure them, and compare".
Um, no. It can be proven to be the same based on the axioms of Euclidean geometry.

i'd like to see the proof. i have no qualms about being wrong (ever since the whole "santa claus" thing, i've kinda gotten used to it). you have shitty manners. i don't like you, as a consequence.

gmalivuk wrote:
Deveno wrote:how do you measure a curved length?
this. this is a complicated question, one which we were not able to give even a partial answer to, until AFTER π had been used for centuries.
it just so turns out, that the most "reasonable" definition of π, is: ∫ [-1,1] √(1 - x^2) dx.
that is: measuring the length of curves, depends on writing a curve as a function.
Wrong again. We'd been approximating curves with smaller and smaller line segments for millennia before anyone invented calculus.

and we had a valid number system for such "magnitudes" and a clear idea what "limit" was? gosh, someone should have told those guys who came up with limits and continuity, it was unecessary, they'd been pwned by archimedes. my point is, pi is irrational. that means there's no finite rational approximation to it. euclid's so-called proof depends on taking the limit of an n-gon, as n-->infinity. so does archimedes' "method of exhaustion". oh, but wait! we could use induction, right? because the greeks knew all about those peano axioms, didn't they?

gmalivuk wrote:
now, from a pedagogal point of view, you can hardly expect to teach calculus before geometry or trigonometry.
Not a problem, since pi comes out of geometry, along with trig.

i disagree. apparently that doesn't sit well with you. i think "the question of pi" points to a certain deficiency (a certain inability to define things we would like to define) of geometry. to show how subtle the problem is, look at this discussion http: //www.physicsforums.com/archive/index.php/t-98476.html (i put extra spaces between the http: and the //, i am a bit unsure as to whether or not this breaks the forum rules).
gmalivuk wrote:
nevertheless, at the point in education where π is introduced, it is largely a matter of "teacher says so"
Only if you have a shitty teacher.
[/quote]

you seem fond of that word.

antonfire
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### Re: The Tau Manifesto

It sounds like you are suggesting that we try to teach people calculus before we teach them about pi. Your reasoning, as far as I can tell, also suggests that we try to teach them the peano axioms before we teach them how to add. Both of these are Bad Ideas™.

Yes, math is complicated, and there's more subtleties lurking under the hood than we let on in primary school. There is nothing particularly special about pi in that respect.
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Deveno
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### Re: The Tau Manifesto

antonfire wrote:It sounds like you are suggesting that we try to teach people calculus before we teach them about pi. Your reasoning, as far as I can tell, also suggests that we try to teach them the peano axioms before we teach them how to add. Both of these are Bad Ideas™.

Yes, math is complicated, and there's more subtleties lurking under the hood than we let on in primary school. There is nothing particularly special about pi in that respect.

not at all. i am merely suggesting that one of the soundest mathematical definitions of pi is that of a certain integral, which just so happens to be the area of a half-circle. why the area, and not the arc-length? mostly because arc-lengths are more complicated to compute. nor am i suggesting that we teach zermelo fraenkel set theory before arithmetic (although the results might be, erm, amusing).

tau-radians make more sense to me than pi-radians. yes, i read the manifesto. on a scale of 1 to 10 as important mathematical choices for teaching, i give this issue a 4. i see some conceptual benefit, not a lot. the entire thrust of what i was saying about the integral was, if that (the definite integral of √(1-x^2) from -1 to 1) is (and i believe so, though some may prefer to dispute this) the "shortest" educational route to a "mathematically sound" definition of pi, that it might not be so un-motivated to start with pi, rather than tau, even though this motivation won't be understood for years to come.

rather than argue that on its own terms, however, the replies so far, seem to consist of people attempting to make themselves appear smart at my expense.

i'll repeat some of what i said before, for emphasis: we teach young people to accept a lot of mathematics on faith. commutativity of addition, for example. i was 17 years old before i saw a satisfactory proof of that, although i "learned" it in 3rd grade. that's 9 years of faith, baby. for some, it may be a shorter gap, some schools put a lot of good stuff in "pre-cal" courses these days, and induction proofs are filtering down to middle-school levels. it's unavoidable, of course; arithmetic is a skill people need rather early in life, and as a practical matter, the theory can wait.

but we (presumably) are not children, here, but adults, thinking about a (albeit somewhat minor) way to change how we think about a number. and (as such) may contemplate how a "provisional" definition of pi (or tau, if you lean that way) will give way to a more sophisticated definition. the process should be "transparent" so that when the circle is complete (that's a pun, btw), both the provisional definition, and the newer one are seen at once to be equivalent. and that's NOT "Bad Math".

i have seen by the way (gmalivuk, i'm looking at YOU) the geometrical arguments that the polygonal approximations of a circle do lead one to the (intuitive) notion that "in the limit" circles are all similar, which then does imply that the circuference/diameter is a circle-invariant. but the limiting process is not defined (explicitly), and does not constitute a "proof" (just a plausibility argument). for a better, more thorough discussion, i defer to:

http: //math.stackexchange.com/questions/3198/proof-that-pi-is-constant-the-same-for-all-circles-without-using-limits

so, in response to your assertion that "most of what i said is wrong" i reply:

don't think so.

Talith
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### Re: The Tau Manifesto

You seem to have a problem with a geometric definition which relies on taking a limit of a sequence as the definition of pi, and yet your definition relies on a much more difficult concept of integration which has limits tied up right in the definition (take your favourite Riemann or Lebesgue integral and you'll soon find limits). Good definitions of objects in mathematics should strike a balance between 'intuitively obvious' in some sense but also 'easy to work with' from a practical point of view. This is why we sometimes take the definition of the trigonometric functions to be their Taylor expansion instead of their original geometric interpretation as angles of triangles inside a circle, because it's often nicer to work with.

Though that is the case however, we don't teach this definition of the trig functions at an early age for the obvious reasons and we can still get a 'mathematically sound' definition of the trig functions without having to know differential calculus or the theory of infinite polynomials and series (there's just no need). In the same way, the geometric definition of pi is a perfectly good, 'mathematically sound' definition in any way you could want (be it from axioms or just 'a good enough model to work with') and there doesn't seem to be any advantage at an early stage in education to change to your integral definition. In fact, there doesn't seem to be any reason to use your definition even at a later stage in education or research.

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### Re: The Tau Manifesto

Deveno wrote:and we had a valid number system for such "magnitudes" and a clear idea what "limit" was? gosh, someone should have told those guys who came up with limits and continuity, it was unecessary, they'd been pwned by archimedes. my point is, pi is irrational. that means there's no finite rational approximation to it. euclid's so-called proof depends on taking the limit of an n-gon, as n-->infinity. so does archimedes' "method of exhaustion". oh, but wait! we could use induction, right? because the greeks knew all about those peano axioms, didn't they?
Are you seriously suggesting that no one knew how to properly use anything in math until people came along centuries later and gave it rigorous logical foundations?
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Deveno
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### Re: The Tau Manifesto

Talith wrote:You seem to have a problem with a geometric definition which relies on taking a limit of a sequence as the definition of pi, and yet your definition relies on a much more difficult concept of integration which has limits tied up right in the definition (take your favourite Riemann or Lebesgue integral and you'll soon find limits). Good definitions of objects in mathematics should strike a balance between 'intuitively obvious' in some sense but also 'easy to work with' from a practical point of view. This is why we sometimes take the definition of the trigonometric functions to be their Taylor expansion instead of their original geometric interpretation as angles of triangles inside a circle, because it's often nicer to work with.

Though that is the case however, we don't teach this definition of the trig functions at an early age for the obvious reasons and we can still get a 'mathematically sound' definition of the trig functions without having to know differential calculus or the theory of infinite polynomials and series (there's just no need). In the same way, the geometric definition of pi is a perfectly good, 'mathematically sound' definition in any way you could want (be it from axioms or just 'a good enough model to work with') and there doesn't seem to be any advantage at an early stage in education to change to your integral definition. In fact, there doesn't seem to be any reason to use your definition even at a later stage in education or research.

I think you miss my point (don't worry, i'm used to this by now). i am NOT suggesting we force this "integral" definition down the throats of kids in elementary geometry classes. they don't have the tools to comprehend it (a lot of ideas need to be fleshed out before-hand). i am saying that the "elementary definiton" of circumference/diameter relies on notions that cannot be established with the tools they have on hand. which is ok, i am not trying to re-vamp the educational system, here.

what i AM saying, is that when (if?) you reach the point (amd some people never do) where you are finally in a position to prove that pi is a real number, the integral definiton is then a clear one to use (one could, i suppose, go back to the geometrical proofs, armed with the new notion of "limit" and prove those rigorously. that ok, too). and that the integral definiton does provide some motivation for using pi, instead of tau.

your analogy with taylor series is apt. if one has complex analysis at one's disposal, then it's far more satisfactory to DEFINE the complex exponential as a certain complex power series, and obtain the trigonometric definitons as corollaries. the motivation behind the use of "e" and the transcendantal nature of the trigonometric functions becomes clearer, and proof of their properties are analytic, rather than geometrical. but that's not going to help someone taking trig for the first time, and i see no reason to abandon SOHCAHTOA on that account.

of course an integral definiton of pi isn't "elementary", and of course, some simpler definiton has to be given if we want students to be familiar with pi (or tau) at all. for those people will never use pi/tau all that much, any definition that is reasonably correct will serve. for those who will one day learn more math, one hopes that they will be trained in the habit of not using unproven facts. this entire thread is about the "reasonableness" or pi versus tau, and i was merely giving one rationale for pi (ironic, too, because tau makes sense to me).

let's back up a bit. assume we know "somehow" (in some unspecified way) that circumference/diameter is a constant. well, if it's rational, we know it's a number. but what if it's not? how do you even begin to say what a real, irrational number is (well, ok, algebraic numbers are roots of polynomials. too bad pi isn't algebraic) without some notion of what "convergence" is? perhaps dedekind cuts might be the most basic, although showing they have all the field properties we desire (so that we can, in particular, embed the rationals in them) is a bit messy. it's not a "given" that "pi is a number" until we a) say what "numbers are" and b)show pi is one of them. and sure, we gloss over that point in teaching number systems to begin with, because the explanation and proof would just be too hard.

gmalivuk wrote:
Deveno wrote:and we had a valid number system for such "magnitudes" and a clear idea what "limit" was? gosh, someone should have told those guys who came up with limits and continuity, it was unecessary, they'd been pwned by archimedes. my point is, pi is irrational. that means there's no finite rational approximation to it. euclid's so-called proof depends on taking the limit of an n-gon, as n-->infinity. so does archimedes' "method of exhaustion". oh, but wait! we could use induction, right? because the greeks knew all about those peano axioms, didn't they?
Are you seriously suggesting that no one knew how to properly use anything in math until people came along centuries later and gave it rigorous logical foundations?

no, i'm suggesting that no one knew how to properly evaluate limits and define real numbers until the right concepts were created. yes, newton sometimes did calculus wrong. people had all sorts of erroneous ideas about infinity, limits, series and "magnitudes". before a proper inductive definiton of the natural numbers was put forth, the use of proof by induction had no logical justification. the best you could say about it was: "it always seems to work". that doesn't mean that no one came up with genuinely good results. euclid's elements is a masterpiece of deductive reasoning, euler, gauss, too many others to name all made great contributions to our mathematical understanding, even if the foundations of their investigations were uncertain. even now, we still don't really "know what a number is" (well, we might, if we actually knew what a set was). what we do know, however (and this is a great stride, and fairly recent), is how numbers have to behave. we know what numbers aren't, and we leverage that knowledge. when set theory was finally given some sort of shape, and arithmetic axiomatized, and topological ideas given form, we looked back, and saw that most of what had come before could be salvaged. this gives the hope that "we're on the right track".

again, i'm not arguing that young school-children should be versed in the logical foundational underpinnings of math. in many ways, they are recapitulating the historical development of math, working on faith that, ultimately, it does "make sense".

antonfire
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### Re: The Tau Manifesto

Deveno wrote:no, i'm suggesting that no one knew how to properly evaluate limits and define real numbers until the right concepts were created.
Define the real numbers, perhaps not. Turns out the greeks were not that bad at "evaluating limits" despite this. They did not have a formal definition for a number system which included pi, no. As you have pointed out, they didn't have a formal definition for a number system which included 7 either. They still knew that 7 was smaller than 12, and that pi was smaller than 22/7.

The idea that they did not know how to measure lengths of curves (or areas bounded by curves) is preposterous. They just didn't do those things in a formal framework.

Jerry Bona wrote:The Axiom of Choice is obviously true; the Well Ordering Principle is obviously false; and who can tell about Zorn's Lemma?

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### Re: The Tau Manifesto

Deveno wrote:i have seen by the way (gmalivuk, i'm looking at YOU) the geometrical arguments that the polygonal approximations of a circle do lead one to the (intuitive) notion that "in the limit" circles are all similar, which then does imply that the circuference/diameter is a circle-invariant. but the limiting process is not defined (explicitly), and does not constitute a "proof" (just a plausibility argument). for a better, more thorough discussion, i defer to:

http://math.stackexchange.com/questions ... ing-limits
You mean that thorough discussion where the very first answer explains why pi, even if it's just identified with the sequence rather than its limit, must be the same for all Euclidean circles? And where said answer does not appeal to limits at any point?
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Deveno
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### Re: The Tau Manifesto

gmalivuk wrote:
Deveno wrote:i have seen by the way (gmalivuk, i'm looking at YOU) the geometrical arguments that the polygonal approximations of a circle do lead one to the (intuitive) notion that "in the limit" circles are all similar, which then does imply that the circuference/diameter is a circle-invariant. but the limiting process is not defined (explicitly), and does not constitute a "proof" (just a plausibility argument). for a better, more thorough discussion, i defer to:

http://math.stackexchange.com/questions ... ing-limits
You mean that thorough discussion where the very first answer explains why pi, even if it's just identified with the sequence rather than its limit, must be the same for all Euclidean circles? And where said answer does not appeal to limits at any point?

said answer, if one reads further down, is shown to not actually define pi, but approximants π_k(C). which is a fairly sophisitcated answer, which foreshadows a later possible definiton of real numbers. which implies, for example, that if euclid would have seen a modern exposition of real numbers, he would see the similarity. i don't think the greeks were stupid, i think they did great with the tools they had. they didn't have a notion corresponding to rational number, but they did have a notion of geometrical congruence, and damned if ol' eudoxus didn't find a way to make that work by comparing ratios.

if one really wants to say that pi is a "sequence" (which is technically accurate, if one uses the cauchy construction of the reals), then i'm ok with that, but equating that with a "constant" is again, probably a bit more sophisticated than you are going to find in an elementary geometry class. this seems to be a feature of all transcendental numbers, that "you can't get ahold of them" without involving some non-finitary process. one wonders (ok, me, i wonder) what plato or aristotle would have made of cantor's 2nd diagonal argument.

i don't like capital letters (it's one of the reasons i double-space, to keep it all from running together). it's not that i don't know how to do it.

gmalivuk
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### Re: The Tau Manifesto

Deveno wrote:i don't like capital letters (it's one of the reasons i double-space, to keep it all from running together). it's not that i don't know how to do it.
There is a happy medium, you know. Like, you can still refuse to use capital letters while also making simple paragraph breaks where paragraph breaks should rightly be. Instead of, for instance, every other line.
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Talith
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### Re: The Tau Manifesto

Deveno wrote:i don't like capital letters (it's one of the reasons i double-space, to keep it all from running together). it's not that i don't know how to do it.

Ok, I'm pretty sure this guy is just trolling us now.