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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

#31

Post by WiseWeasel »

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…

I wonder if the fourth dimension in this case might be the maximum internal surface area or something of the sort.

A fourth dimension perfect circle looks like a sphere at first glance, but has the maximum possible internal surface area upon closer inspection.

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

#32

Post by Cushman »

The last section nails it in a nutshell:

...imagine we lived in a world where we used the letter h to represent “one half” and had no separate notation for 2h. We would then observe that h is ubiquitous in mathematics. In fact, 2h is the multiplicative identity, so how can one doubt the importance of h? But this is crazy: 2h is the fundamental number, not h. Let us therefore introduce a separate symbol for 2h; call it 1. We then see that h=1/2, and there is no longer any reason to use h at all.

Color me converted as well.

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

#33

Post by blahedo »

mhartl wrote: post #18
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…

Nifty! I wasn't trolling for recognition so much as being puzzled why this was billed as a new epiphany---that makes sense though. I love the idea of recasting the presentation from "list of complex formulae for area and volume" to "sequence of constants expressing ratios".

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

#34

Post by blahedo »

cygx wrote: post #30
waterhouse wrote: post #26

Earlier quoted context omitted.

51° is not 1/7 τ. 1/7 τ is (51 3/7)°, or 51.428571...°. If you round that to 51°, then presumably you're doing something where inexact numbers are fine, in which case you might then use "0.89 (radians)". My intuition is better for degrees here, but I think that would change if I spent ten minutes doing calculations involving real radian measures.

51° is the latitude of the city I'm living in, and I chose 1/7 τ because it comes pretty close. I could have used 0.14 τ to make my point as well: If you want to avoid fractions in common cases, you'll need to multiply with an arbitrary number like, say 100 (which has obvious benefits in a base-10 system), but arguments could be made for something like 400 (in which case we end up with gon) or, say 360, which has the…

> Actual radians (ie not expressed as fractions of 2π)

I think this is the source of your problem: you view an expression like "τ/8" or "π/4" as less an "actual" radian measure than the inexact ".785". That's a function of how a lot of us were taught math, I think, and I'm quite sure that it's only made worse by the fact that the use of π camouflages the fact that this is related to fractions of a circle. But your premise is faulty. Of course radians expressed in decimal are user-unfriendly, but radians expressed in decimal are no less "actual" than radians expressed as fractions of τ or π.

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

#36

Post by mhartl »

blahedo wrote: post #33
mhartl wrote: post #18

Earlier quoted context omitted.

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…

Nifty! I wasn't trolling for recognition so much as being puzzled why this was billed as a new epiphany---that makes sense though. I love the idea of recasting the presentation from "list of complex formulae for area and volume" to "sequence of constants expressing ratios".

I know you weren't trolling, but I'm a strong believer in giving credit where credit is due. Thanks again for your comments!

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

#37

Post by mhartl »

rsanchez1 wrote: post #35

It's just semantics. Pi and tau are the same thing. You don't see physicists going around saying h is wrong and h-bar is what they need to use.

You don't see physicists going around saying h is wrong and h-bar is what they need to use.

Sure they do—or at least, they did. Physicists realized that h is "wrong", i.e., confusing and unnatural, because it's off by a factor of, um, 2 pi. They introduced h-bar precisely to rectify the problem. And the substitution worked: if you open up a standard book on quantum mechanics, the ratio of uses of h to h-bar is just about epsilon. (You'll also see lots of (2 pi)s.)

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

#38

Post by leot »

rsanchez1 wrote: post #35

It's just semantics. Pi and tau are the same thing. You don't see physicists going around saying h is wrong and h-bar is what they need to use.

"It's just semantics" is a horrible, useless phrase.

The difference between "robot" and "postlapsarian" is "just semantics", after all.

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

#39

Post by bascule »

Retric wrote: post #29
bascule wrote: post #27

Earlier quoted context omitted.

It won't amaze third graders, but it would make a lot more sense the first time you learn integral calculus

Meh, so did 2 * pi * r > 2 * pi * (r ^ 2) / 2 = pi * r ^ 2. The great thing about math is it's it works out either way. So, it really comes down to which is easier to deal with and students seem to have an easier time remembering and understanding 2 pi r and pi*r^2, but in the end it's tau day so that's what people want to argue about.

"students seem to have an easier time remembering and understanding 2 pi r and pi r^2"

Are you speaking from empirical experience as a math teacher?

What makes 2 pi r easier to understand than tau r?

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