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A new section of The Tau Manifesto: Getting to the bottom of pi

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Re: A new section of The Tau Manifesto: Getting to the bottom of pi

#11

Post by Retric »

blahedo wrote: post #7
WiseWeasel wrote: post #4

It seems like these guys are fighting an uphill battle, not that I'd want to discourage it. It's simply much easier to visualize the path traced when you relate the radius to the circumference of a circle rather than its diameter, since there's only one point to focus on at a time as you rotate around the axis. Pi is insidious as it's only much later that this foundation begins to appear problematic. [My analogy was…

No, the tau manifesto agrees: radius is the more fundamental concept. That's why tau is better! Pi is an expression relating the diameter to the circumference, which as you point out is a little counterintuitive.

The problem with tau for introductory math is area = pi * r^2 = (tau/2) * r^2. Given the choice between having equations with 2x or other with 1/2 y most people feel more comfortable with 2x.

EX: e^(i * tau / 2) = -1 wait what?

Re: A new section of The Tau Manifesto: Getting to the bottom of pi

#12

Post by bascule »

Retric wrote: post #11
blahedo wrote: post #7

Earlier quoted context omitted.

No, the tau manifesto agrees: radius is the more fundamental concept. That's why tau is better! Pi is an expression relating the diameter to the circumference, which as you point out is a little counterintuitive.

The problem with tau for introductory math is area = pi * r^2 = (tau/2) * r^2. Given the choice between having equations with 2x or other with 1/2 y most people feel more comfortable with 2x. EX: e^(i * tau / 2) = -1 wait what?

If you read the Tau Manifesto you'll see why tau * r^2 / 2 is actually AWESOME (he actually refers to it as pi's coup de grace)

Just think of it like an integral (that's what it is, after all!), raise the power and divide by the power. Suddenly the origin of this formula is no longer obscured.

Regarding Euler's identity: e^(i * tau) = 1

Re: A new section of The Tau Manifesto: Getting to the bottom of pi

#13

Post by [deleted] »

bascule wrote: post #12
Retric wrote: post #11

Earlier quoted context omitted.

The problem with tau for introductory math is area = pi * r^2 = (tau/2) * r^2. Given the choice between having equations with 2x or other with 1/2 y most people feel more comfortable with 2x. EX: e^(i * tau / 2) = -1 wait what?

If you read the Tau Manifesto you'll see why tau * r^2 / 2 is actually AWESOME (he actually refers to it as pi's coup de grace) Just think of it like an integral (that's what it is, after all!), raise the power and divide by the power. Suddenly the origin of this formula is no longer obscured. Regarding Euler's identity: e^(i * tau) = 1

[deleted]

Re: A new section of The Tau Manifesto: Getting to the bottom of pi

#14

Post by Retric »

bascule wrote: post #12
Retric wrote: post #11

Earlier quoted context omitted.

The problem with tau for introductory math is area = pi * r^2 = (tau/2) * r^2. Given the choice between having equations with 2x or other with 1/2 y most people feel more comfortable with 2x. EX: e^(i * tau / 2) = -1 wait what?

If you read the Tau Manifesto you'll see why tau * r^2 / 2 is actually AWESOME (he actually refers to it as pi's coup de grace) Just think of it like an integral (that's what it is, after all!), raise the power and divide by the power. Suddenly the origin of this formula is no longer obscured. Regarding Euler's identity: e^(i * tau) = 1

Telling a 3rd grader to think of it like an integral seems to be putting the cart WAAAAAAAY before the horse.

Edit: Also that's is hardly a more fundamental equation because you could also say e^(1024 pi * i) = e^(4096 * tau * i) = 1.

Re: A new section of The Tau Manifesto: Getting to the bottom of pi

#15

Post by cygx »

I live with 60 minutes per hour, 24 hours per day, 7 days per week, 360° in a full circle, a speed of light of 299,792,458 m/s, to name only a few cases of historical 'accidents'.

2π is arguably more fundamental than π, but given the things above, I don't care much.

If you want to tackle something worthwhile, update theoretical physics lectures to 'modern' notation (where modern means 1960s).

Re: A new section of The Tau Manifesto: Getting to the bottom of pi

#16

Post by sallowswine »

"For example, consider integrals over all space in polar coordinates:

\[ \int_0^{2\pi}\int_0^\infty f(r, \theta) r dr d\theta \]

The upper limit of the $\theta$ integration is always 2$\pi$."

This statement is false. If $r = r(\theta)$, then it can be the case (e.g., for a "looped" limacon $r = \cos(\theta)$) that $\theta$ will not range from $[0,2\pi]$. The upper limit is not necessarily "always" $2\pi$.

Re: A new section of The Tau Manifesto: Getting to the bottom of pi

#17

Post by mhartl »

tungstentim wrote: post #8

"pi is a lie" and "pi is wrong" are incredibly disingenuous slogans. I would be really excited about this site if it weren't for these sensationalist remarks at the top. It's sad, because their main content is all about mathematical reasoning, which overall they do a fine job presenting. pi is not a lie. It's not part of some conspiracy theory among math elites to mislead the masses. It's not about math bureaucrats r…

"pi is a lie" and "pi is wrong" are incredibly disingenuous slogans.

For the former, you just need to know your meme:

http://www.urbandictionary.com/define.php?term=the+cake+is+a...

For the latter, you need to know that it comes from Bob Palais' original article.

For both, you might need to lighten up.

Re: A new section of The Tau Manifesto: Getting to the bottom of pi

#18

Post by mhartl »

blahedo wrote: post #6

Huh. Section 5.1 (one of the new ones) is presented as this new epiphany about hyperspheres and tau, but I made a post back in 2010 about this ( http://www.blahedo.org/blog/archives/001083.html ), emailed Hartl at the time, and he said he'd already thought of that. Still, cool to see that my observation was worth (eventually) including in the manifesto itself. (He did elaborate on the idea considerably, of course.) ;…

When I said I'd already thought of it, I was referring to the general observation that n-sphere volumes support tau, not pi. You can find my original comments, dating from July 2010, at http://forums.xkcd.com/viewtopic.php?f=17&t=61958.

The "epiphany" refers to the specific realization that there are three families of constants, and in particular that pi is not a member of the family of volume constants but rather is part of a useless family of its own.

In any case, you certainly deserve mention, so I've added you to the acknowledgments. Thanks!

Re: A new section of The Tau Manifesto: Getting to the bottom of pi

#19

Post by aidenn0 »

Retric wrote: post #11
blahedo wrote: post #7

Earlier quoted context omitted.

No, the tau manifesto agrees: radius is the more fundamental concept. That's why tau is better! Pi is an expression relating the diameter to the circumference, which as you point out is a little counterintuitive.

The problem with tau for introductory math is area = pi * r^2 = (tau/2) * r^2. Given the choice between having equations with 2x or other with 1/2 y most people feel more comfortable with 2x. EX: e^(i * tau / 2) = -1 wait what?

We should thus not teach anyone the formula for area of a triangle then

Re: A new section of The Tau Manifesto: Getting to the bottom of pi

#20

Post by mhartl »

cygx wrote: post #15

I live with 60 minutes per hour, 24 hours per day, 7 days per week, 360° in a full circle, a speed of light of 299,792,458 m/s, to name only a few cases of historical 'accidents'. 2π is arguably more fundamental than π, but given the things above, I don't care much. If you want to tackle something worthwhile, update theoretical physics lectures to 'modern' notation (where modern means 1960s).

360° in a full circle

That's one of the things tau can help solve. I suspect the main reason that people still use degrees so much is because using pi undermines the beauty and intuitiveness of radians.

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