## My observations on the Pi Manifesto.

An enlightening discussion about pi and tau.

### My observations on the Pi Manifesto.

The author had a nice try with pi "showing up" in places, but just a few things I've seen so far (after looking for only a few minutes):

1. "Hey I have more functions than you!" isn't exactly a good argument. This author is doing himself exactly what the article is accusing "tauists" of doing --- conveniently asking questions that have pi as their answer.
2. I would argue that a lot of the functions in which pi is more convenient are much more complex and found in higher areas of mathematics, and those in which tau is found are much more elementary --- basically that pi is based off tau, rather than the other way around.
3. Some of those functions have the pi in there because it was a simplification from things cancelling out with tau ... like the formula for an ellipse. Sure, pi a b is more compact, but where does it come from? Integrating a polar form from 0 to 2pi ... or 0 to tau.
4. In the section on Euler's identity, if the author wants to say the result with pi is "stronger" he should say WHY it is stronger.
5. As for the 1/2 in the area formula for a circle ... it actually follows directly from Archimedes' original calculation of the area of a circle, which is that it has the same area as a triangle with its base equal to the circumference and its height equal to the radius --- in short, 1/2 C r.
6. The trigonometric functions having multiples of pi for their periods takes away again from the MEANING of those periods. When you say that sin(x) has a period of 2pi and tan(x) has a period of pi, big deal. You've said they're numbers. When you say the periods are tau and 1/2 tau, you realize something ... sin(x) takes 1 full circle (1 tau) to repeat itself, and tan(x) takes 1/2 a circle (1/2 tau) to do the same.
7. The argument for pi being convenient for area isn't bad, but I would argue measures of length (1-dimensional) are more fundamental than measures of area (2-dimensional). All other dimensions are extensions of 1 dimension, not 2. In grade school you learn that if you double the length in a figure, you quadruple the area, and multiply the volume by 8. Should we be teaching that if you double the area, you multiply the length by √(2) and the volume by 2√(2)?

Here's the thing ... I don't advocate COMPLETELY getting rid of pi like the author insinuates, but I do believe that tau is the more fundamental constant. As far as I'm concerned, tau should be taught as the more fundamental constant when students are learning it, and pi would be learned for convenience in calculating various other formulas.
DMAshura
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### Re: My observations on the Pi Manifesto.

DMAshura wrote:T1. "Hey I have more functions than you!" isn't exactly a good argument. This author is doing himself exactly what the article is accusing "tauists" of doing --- conveniently asking questions that have pi as their answer.

Indeed. It is a very poor argument, and both manifestos would be far better without it. However, since the point is raised by the Tau Manifesto, it seems appropriate that a Pi Manifesto should respond. I think a better answer would be "The Tau Manifesto has cherry picked equations to make tau seem like a more universal constant. We could just as easily cherry-pick equations (i.e. these...), but the argument basically devolves into ridiculousness. Call it a draw on this point."

DMAshura wrote:2. I would argue that a lot of the functions in which pi is more convenient are much more complex and found in higher areas of mathematics, and those in which tau is found are much more elementary --- basically that pi is based off tau, rather than the other way around.

I agree entirely. Which is why I generally raise the point that tau may be more useful pedagogically. Teach students to use the thing that is going to make their lives easiest in the context in which they are working, then let them figure out the generalizations later. Most students are introduced to pi/tau in the context of basic trigonometry or geometry, where the choice of tau is probably easier to understand. Hence I would prefer to teach tau, and have people learn pi as a historical artifact, or in the contexts where they might actually want to use it.

DMAshura wrote:4. In the section on Euler's identity, if the author wants to say the result with pi is "stronger" he should say WHY it is stronger.

I'm not sure that "stronger" is the word that I would choose, but I definitely prefer identity with pi over tau on purely aesthetic grounds. You get several basic constants, and only addition, multiplication, and exponentiation in an equality with a zero on the right (that being the sort that you see over and over again in basic algebra).

As to it being a "stronger" result, maybe it is because the geometric interpretation is clearer (i.e. that there is a rotation). However, if one makes that argument, maybe $e^{i\pi/2} = i$ would be an even better result, as it takes use through a one-quarter turn. Maybe $e^{i(\tau/4)} = i$ would be the best result, as the quarter turn is clear.

That being said, I repeat that I much prefer the original version with pi for purely aesthetic reasons.

DMAshura wrote:Here's the thing ... I don't advocate COMPLETELY getting rid of pi like the author insinuates, but I do believe that tau is the more fundamental constant. As far as I'm concerned, tau should be taught as the more fundamental constant when students are learning it, and pi would be learned for convenience in calculating various other formulas.

This.

I think that people on both sides of the issue are blowing things way out of proportion. Pi is not wrong, it just isnt' always the most convenient notation. That being said, I do agree that tau is better for elementary students, and that pi is the wrong thing to teach to those students. Given that Hartl and Palais seem to want to restructure the curriculum, perhaps the stridency of their position is understandable and required in order to make their point.

xander

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### Re: My observations on the Pi Manifesto.

xander wrote:I agree entirely. Which is why I generally raise the point that tau may be more useful pedagogically. Teach students to use the thing that is going to make their lives easiest in the context in which they are working, then let them figure out the generalizations later.

For now, while π is ubiquitous, just teach the students that τ is more fundamental, that's π is more or less a mistake, and that they'll encounter situations using π in which things don't make intuitive sense. Do some equations mixing and matching the two, then keep teaching them the π-based equations they'll actually use. "You'll often experience equations involving 2π. Keep it in the back of your head why it's there." Make sure to show the cases of 2π/2 explicitly before cancelling them out.
engineer
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### Re: My observations on the Pi Manifesto.

DMAshura wrote:1. "Hey I have more functions than you!" isn't exactly a good argument. This author is doing himself exactly what the article is accusing "tauists" of doing --- conveniently asking questions that have pi as their answer.

Definitely true, especially considering that the Tau Manifesto's list came in the introduction and was obviously just meant to show a few examples, not to be an exhaustive list. I have started a topic about this point here: viewtopic.php?f=30&t=403
DMAshura wrote:2. I would argue that a lot of the functions in which pi is more convenient are much more complex and found in higher areas of mathematics, and those in which tau is found are much more elementary --- basically that pi is based off tau, rather than the other way around.

I think the main reason most of the formulas are from higher areas of mathematics is that there are more formulas in those areas, making them easier to cherry-pick.
DMAshura wrote:3. Some of those functions have the pi in there because it was a simplification from things cancelling out with tau ... like the formula for an ellipse. Sure, pi a b is more compact, but where does it come from? Integrating a polar form from 0 to 2pi ... or 0 to tau.

Yeah, this is basically the basis of why tau is more natural, even when pi is simpler. Even in formulas like the area of a circle, pi is only there because of cancellation with a factor of 1/2 that comes via triangles or the power rule. The same thing goes for the area of an ellipse since it is just a circle of radius a stretched in one direction by a factor of b/a, meaning the area will be multiplied by b/a.
In addition, there is actually another version of the formula that is simpler with tau. For an ellipse defined by the equation ax^2+bxy+cy^2=1 is τ/√(4ac-b^2)
DMAshura wrote:4. In the section on Euler's identity, if the author wants to say the result with pi is "stronger" he should say WHY it is stronger.

Exactly what I was thinking. I assume he meant because the tau version comes "trivially" by squaring the pi version; the problem with this is that it's a lie--I could just as easily say the pi version comes "trivially" by squaring the eta version, and in general the n version comes "trivially" by squaring the n/2 version. Both versions really come from Euler's formula, and neither is more trivial than the other.
DMAshura wrote:5. As for the 1/2 in the area formula for a circle ... it actually follows directly from Archimedes' original calculation of the area of a circle, which is that it has the same area as a triangle with its base equal to the circumference and its height equal to the radius --- in short, 1/2 C r.
6. The trigonometric functions having multiples of pi for their periods takes away again from the MEANING of those periods. When you say that sin(x) has a period of 2pi and tan(x) has a period of pi, big deal. You've said they're numbers. When you say the periods are tau and 1/2 tau, you realize something ... sin(x) takes 1 full circle (1 tau) to repeat itself, and tan(x) takes 1/2 a circle (1/2 tau) to do the same.

Yet another reason why tau is just so much more elegant. It's actually pretty funny to look at the Wikipedia article for pi where it talks about how pi is important because the periods of certain functions are multiples of pi, such as 2π. How about tau is important because the period of many functions is equal to tau? In addition, the period of radians themselves is tau.
Also, all of the "tau is just a multiple of pi" arguments don't work anyway, because we can just as easily say "pi is just a fraction of tau."
DMAshura wrote:7. The argument for pi being convenient for area isn't bad, but I would argue measures of length (1-dimensional) are more fundamental than measures of area (2-dimensional). All other dimensions are extensions of 1 dimension, not 2. In grade school you learn that if you double the length in a figure, you quadruple the area, and multiply the volume by 8. Should we be teaching that if you double the area, you multiply the length by √(2) and the volume by 2√(2)?

Agreed. Pi-ists will often try to think of crazy convoluted reasons to say that area is more fundamental, but really it all boils down to this. There is no way to seriously say that two dimensions are more fundamental than one.
DMAshura wrote:Here's the thing ... I don't advocate COMPLETELY getting rid of pi like the author insinuates, but I do believe that tau is the more fundamental constant. As far as I'm concerned, tau should be taught as the more fundamental constant when students are learning it, and pi would be learned for convenience in calculating various other formulas.

I would agree with this as well. I think as long as we already have pi, we might as well keep it for convenience once we switch over and just define it as τ/2. That way we still know tau is more fundamental, but in the rare case where pi is simpler we have a convenient shorthand--like using semiperimeter instead of 1/2*perimeter.
τ>π
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### Re: My observations on the Pi Manifesto.

xander wrote:As to it being a "stronger" result, maybe it is because the geometric interpretation is clearer (i.e. that there is a rotation). However, if one makes that argument, maybe $e^{i\pi/2} = i$ would be an even better result, as it takes use through a one-quarter turn. Maybe $e^{i(\tau/4)} = i$ would be the best result, as the quarter turn is clear.

How is the turn clearer in the pi version? One of the reasons I prefer the tau version is the e^(iτ)=1 represents a full turn very clearly by getting you back where you started (at 1 or e^0), while the pi version is only a half turn. The pi version, e^(iπ)=-1 is still pretty clearly a half-turn, but the commonly used e^(iπ)+1=0 version completely destroys the geometric interpretation. It's not even just a turn anymore; it's a turn followed by a random shift to the right. In order for the rotation to be obvious from the equation, the result needs to be on the unit circle, but this version moves the result off the unit circle, to an inelegant zero.
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