While some of my suggestions have been included in the current manuscript, I see no change concerning 2π = circumference/radius being a variable. I'm not aware of an accepted name for this ratio; perhaps call it the "circle ratio" or "circumference ratio". There are some nice formulae concerning this ratio. On the sphere of radius R, it's 2πR sin(r/R), while on the hyperbolic plane (of "radius iR"), it's 2πR sinh(r/R). The volume of the n-ball of radius R is V = (π^(n/2) / (n/2)!) R^n. -- Gene From: James Propp <jamespropp@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Sent: Monday, February 13, 2017 3:44 PM Subject: Re: [math-fun] Draft of my February 2017 blog post Re continued fractions: My 2018 pi essay will talk about the continued fraction expansion. In fact, I started out writing an essay about the c.f. expansion of pi, but by the time I started laying out all the background info on pi I wanted my readers to absorb, I realized that what I want to say is too long for one article. Re redefining pi: Eugene has convinced me to modify the way I talk about the circumference-to-radius ratio. (If you check mathenchant.wordpress.com now, you'll see the changes.) One thing that would help me out at this stage is a standard name for this ratio. (Every normed space has a "girth", but that's something else.) If the ratio doesn't have a name, would anyone care to suggest a nonce-name that doesn't already mean something else? Maybe "piety"? Jim On Mon, Feb 13, 2017 at 1:04 PM, Bill Gosper <billgosper@gmail.com> wrote:
I agree with Gene: Say more about the continued fraction. How it appears to obey Gauss's lg(1+R) distribution, with infinite expected arithmetic mean but geometric mean = Khinchin's constant. Vs the ancient Chinese value of √10 with arithmetic mean = geometric mean = 6. But if we could only prove lg(1+R), or even much less than that, we would establish the "obvious" facts that π+e, π-e, π e, etc are all irrational. For e, the arithmetic mean and geometric mean are *both* infinite, as would be a e + b for any rational a and b. --rwg The Beckmann book uses "circle ratio" to unambiguously specify π.
Date: 2017-02-11 19:23 From: Eugene Salamin via math-fun <math-fun@mailman.xmission.com> To: math-fun <math-fun@mailman.xmission.com> Reply-To: Eugene Salamin <gene_salamin@yahoo.com>, math-fun < math-fun@mailman.xmission.com>
My comments on this draft.
1. There is no such thing as a pi that varies with the choice of geometry or the size of a circle. Pi is the mathematical constant 3.14..., and that's it by definition. Sure, the ratio circumference/diameter can be different, but pi is the ratio in Euclidean geometry. You do a disservice to your readers by going against the standard usage of the Mathematical community.
2. I agree with the unnamed mathematician that calculating a gazillion decimal digits of pi is a waste of time. There's nothing special about radix 10. Far more interesting would be the continued fraction.
3. "But thanks to Einstein, we now know that the universe we live in is curved at galactic scales, and there’s no natural way to view it as a 3-dimensional curved hypersurface sitting inside an uncurved 4-dimensional space."
Actually, general relativity posits a 4-dimensional curved spacetime, and not a curved 3-dimensional space. Furthermore, the invariant metric on spacetime has a (3,1) signature rather than the Euclidean (4,0).
4. Another pi reference: https://smile.amazon.com/History-Pi-Petr-Beckmann/dp/ 0312381859/ref=sr_1_1?s=books&ie=UTF8&qid=1486861842&sr=1-1& keywords=petr+beckmann
From: James Propp <jamespropp@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Sent: Saturday, February 11, 2017 3:10 PM Subject: [math-fun] Draft of my February 2017 blog post
Hi,
I started writing a new draft titled "Three-point-one cheers for pi !" and would love to get your feedback. I plan on publishing it on the 17th. I can always use more links to content (images or videos) that are likely to interest readers of my essay. I am particularly interested in a link that would present Archimedes' method of calculating the volume of a ball in an accessible way. For that matter, if anyone has risen to Wigner's tacit challenge of explaining to a lay audience why pi is relevant to statistics, I'd love to know about it.
Keep in mind that all math-fun feedback goes into one mail-feed, so I won't know whose feedback is whose unless you sign your comment. Also, all substantive suggestions that I use will be acknowledged (unless you specifically ask me not to do this).
Please leave your feedback here: https://mathenchant.wordpress.com?p=1471&shareadraft=589f987f95cf7
Title: Three-point-one cheers for pi ! Beginning: Pi, that most celebrated of mathematical constants, leads a curiously double life. On the one hand, we define it as the ratio of the circumference of a circle to that circle's diameter, and, to the extent that we can imagine a world draped over the armature of a different geometry, we can conceive ... Read more: https://mathenchant.wordpress.com?p=1471&shareadraft=589f987f95cf7
Thanks,
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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