My observations on the Pi Manifesto.

[DMAshura]

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.

[xander]

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."

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.

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 [latex]e^{i\pi/2} = i[/latex] would be an even better result, as it takes use through a one-quarter turn. Maybe [latex]e^{i(\tau/4)} = i[/latex] 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.

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

[engineer]

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.