[math-fun] Draft of my February 2017 blog post
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... 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
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
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
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
On Monday, February 13, 2017, Eugene Salamin via math-fun < math-fun@mailman.xmission.com> wrote:
While some of my suggestions have been included in the current manuscript, I see no change concerning 2π = circumference/radius being a variable.
I put in more air-quotes, and I put in the passage about the Feast of Fools, so that readers who get that far will know that talking about "different values of pi" is mathematical "street talk", not acceptable in the inner precincts of the kingdom. Eugene and Bill and others would probably prefer a more stringent approach, and if I were the first person to use the phrase "different values of pi" I'd certainly agree with them. However, this slangy way of talking is already pretty standard in parts of the math ed world and on the web, so I don't think I'd be opening any hitherto unopened floodgates of loose usage. Jim
I sort of agree with Eugene on this one; the concept of "redefining pi" is akin to "redefining 7". Yes, you can take some sort of artistic license and do that, but in doing so you risk sounding like that pot-addled hippie wondering about the universe contained in the dust speck under his fingernail. Pi is so much more than the geometric ratio you are concerned with. Pick almost any other name; be amazed when it coincides with the constant we know as pi, but please let me keep my Fourier transforms, my gamma function, and all the rest. Call it cornbread: "Pi r squared? Pi r not square. Pi r round. Cornbread r square." -tom On Mon, Feb 13, 2017 at 5:28 PM, James Propp <jamespropp@gmail.com> wrote:
On Monday, February 13, 2017, Eugene Salamin via math-fun < math-fun@mailman.xmission.com> wrote:
While some of my suggestions have been included in the current manuscript, I see no change concerning 2π = circumference/radius being a variable.
I put in more air-quotes, and I put in the passage about the Feast of Fools, so that readers who get that far will know that talking about "different values of pi" is mathematical "street talk", not acceptable in the inner precincts of the kingdom.
Eugene and Bill and others would probably prefer a more stringent approach, and if I were the first person to use the phrase "different values of pi" I'd certainly agree with them. However, this slangy way of talking is already pretty standard in parts of the math ed world and on the web, so I don't think I'd be opening any hitherto unopened floodgates of loose usage.
Jim _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- -- http://cube20.org/ -- [ <http://golly.sf.net/>Golly link suppressed; ask me why] --
This is the reason why the "punchline" to Sagan's *Contact* makes no sense. There isn't a degree of freedom, when you're designing a universe, where you get to decide the value of pi. On Mon, Feb 13, 2017 at 8:57 PM, Tomas Rokicki <rokicki@gmail.com> wrote:
I sort of agree with Eugene on this one; the concept of "redefining pi" is akin to "redefining 7". Yes, you can take some sort of artistic license and do that, but in doing so you risk sounding like that pot-addled hippie wondering about the universe contained in the dust speck under his fingernail. Pi is so much more than the geometric ratio you are concerned with.
Pick almost any other name; be amazed when it coincides with the constant we know as pi, but please let me keep my Fourier transforms, my gamma function, and all the rest.
Call it cornbread:
"Pi r squared? Pi r not square. Pi r round. Cornbread r square."
-tom
On Mon, Feb 13, 2017 at 5:28 PM, James Propp <jamespropp@gmail.com> wrote:
On Monday, February 13, 2017, Eugene Salamin via math-fun < math-fun@mailman.xmission.com> wrote:
While some of my suggestions have been included in the current manuscript, I see no change concerning 2π = circumference/radius being a variable.
I put in more air-quotes, and I put in the passage about the Feast of Fools, so that readers who get that far will know that talking about "different values of pi" is mathematical "street talk", not acceptable in the inner precincts of the kingdom.
Eugene and Bill and others would probably prefer a more stringent approach, and if I were the first person to use the phrase "different values of pi" I'd certainly agree with them. However, this slangy way of talking is already pretty standard in parts of the math ed world and on the web, so I don't think I'd be opening any hitherto unopened floodgates of loose usage.
Jim _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- -- http://cube20.org/ -- [ <http://golly.sf.net/>Golly link suppressed; ask me why] -- _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
My 2019 Pi Day essay is going to be about this very point. Jim On Monday, February 13, 2017, Allan Wechsler <acwacw@gmail.com> wrote:
This is the reason why the "punchline" to Sagan's *Contact* makes no sense. There isn't a degree of freedom, when you're designing a universe, where you get to decide the value of pi.
On Mon, Feb 13, 2017 at 8:57 PM, Tomas Rokicki <rokicki@gmail.com <javascript:;>> wrote:
I sort of agree with Eugene on this one; the concept of "redefining pi" is akin to "redefining 7". Yes, you can take some sort of artistic license and do that, but in doing so you risk sounding like that pot-addled hippie wondering about the universe contained in the dust speck under his fingernail. Pi is so much more than the geometric ratio you are concerned with.
Pick almost any other name; be amazed when it coincides with the constant we know as pi, but please let me keep my Fourier transforms, my gamma function, and all the rest.
Call it cornbread:
"Pi r squared? Pi r not square. Pi r round. Cornbread r square."
-tom
On Mon, Feb 13, 2017 at 5:28 PM, James Propp <jamespropp@gmail.com <javascript:;>> wrote:
On Monday, February 13, 2017, Eugene Salamin via math-fun < math-fun@mailman.xmission.com <javascript:;>> wrote:
While some of my suggestions have been included in the current manuscript, I see no change concerning 2π = circumference/radius being a variable.
I put in more air-quotes, and I put in the passage about the Feast of Fools, so that readers who get that far will know that talking about "different values of pi" is mathematical "street talk", not acceptable in the inner precincts of the kingdom.
Eugene and Bill and others would probably prefer a more stringent approach, and if I were the first person to use the phrase "different values of pi" I'd certainly agree with them. However, this slangy way of talking is already pretty standard in parts of the math ed world and on the web, so I don't think I'd be opening any hitherto unopened floodgates of loose usage.
Jim _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com <javascript:;> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- -- http://cube20.org/ -- [ <http://golly.sf.net/>Golly link suppressed; ask me why] -- _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com <javascript:;> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com <javascript:;> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
It is unfortunate for someone knowledgeable in mathematics to lower himself to the level of the uninformed, instead of making an attempt to raise them up to a higher level. This reminds me of something I was told by a friend who volunteers in the Santa Cruz CA Public School. Apparently it's school doctrine that 1 is a prime number. The teacher told the class that 1 is prime, but added the qualification that mathematicians consider 1 to not be prime. Anyhow, James, it's your First Amendment right to say that π varies with the situation, and it's my First Amendment right to disagree with you. -- Gene On Mon, Feb 13, 2017 at 5:28 PM, James Propp <jamespropp@gmail.com> wrote:
On Monday, February 13, 2017, Eugene Salamin via math-fun < math-fun@mailman.xmission.com> wrote:
While some of my suggestions have been included in the current manuscript, I see no change concerning 2π = circumference/radius being a variable.
I put in more air-quotes, and I put in the passage about the Feast of Fools, so that readers who get that far will know that talking about "different values of pi" is mathematical "street talk", not acceptable in the inner precincts of the kingdom.
Eugene and Bill and others would probably prefer a more stringent approach, and if I were the first person to use the phrase "different values of pi" I'd certainly agree with them. However, this slangy way of talking is already pretty standard in parts of the math ed world and on the web, so I don't think I'd be opening any hitherto unopened floodgates of loose usage.
Jim
Getting back to the issue of whether it behooves a popularizer-who-knows-better such as myself to engage in talk about "changing the value of pi": I'm going to go ahead with the current version of the essay (which puts in lots of quotation marks but still takes a tone that some of you regard as sensationalistic boggle-mongering of the sort that sows public confusion). Nonetheless: in the spirit of First Amendment rights, I will be glad to approve of comments on my blog that make Eugene's point, whether they come from Eugene or others. Thanks, Jim On Mon, Feb 13, 2017 at 9:38 PM, Eugene Salamin via math-fun < math-fun@mailman.xmission.com> wrote:
It is unfortunate for someone knowledgeable in mathematics to lower himself to the level of the uninformed, instead of making an attempt to raise them up to a higher level. This reminds me of something I was told by a friend who volunteers in the Santa Cruz CA Public School. Apparently it's school doctrine that 1 is a prime number. The teacher told the class that 1 is prime, but added the qualification that mathematicians consider 1 to not be prime. Anyhow, James, it's your First Amendment right to say that π varies with the situation, and it's my First Amendment right to disagree with you.
-- Gene
On Mon, Feb 13, 2017 at 5:28 PM, James Propp <jamespropp@gmail.com> wrote:
On Monday, February 13, 2017, Eugene Salamin via math-fun < math-fun@mailman.xmission.com> wrote:
While some of my suggestions have been included in the current manuscript, I see no change concerning 2π = circumference/radius being a variable.
I put in more air-quotes, and I put in the passage about the Feast of Fools, so that readers who get that far will know that talking about "different values of pi" is mathematical "street talk", not acceptable in the inner precincts of the kingdom.
Eugene and Bill and others would probably prefer a more stringent approach, and if I were the first person to use the phrase "different values of pi" I'd certainly agree with them. However, this slangy way of talking is already pretty standard in parts of the math ed world and on the web, so I don't think I'd be opening any hitherto unopened floodgates of loose usage.
Jim
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Can't say I like the idea of multiple values of pi, for the same reason I don't like reading that 1 + 2 + 3 + ... = -1/12 , namely: Such claims, no matter what kind of mathematical hair-splitting can be used to justify them, just result in more confusion among the general public. —Dan
On Feb 13, 2017, at 5:28 PM, James Propp <jamespropp@gmail.com> wrote:
On Monday, February 13, 2017, Eugene Salamin via math-fun < math-fun@mailman.xmission.com> wrote:
While some of my suggestions have been included in the current manuscript, I see no change concerning 2π = circumference/radius being a variable.
I put in more air-quotes, and I put in the passage about the Feast of Fools, so that readers who get that far will know that talking about "different values of pi" is mathematical "street talk", not acceptable in the inner precincts of the kingdom.
Eugene and Bill and others would probably prefer a more stringent approach, and if I were the first person to use the phrase "different values of pi" I'd certainly agree with them. However, this slangy way of talking is already pretty standard in parts of the math ed world and on the web, so I don't think I'd be opening any hitherto unopened floodgates of loose usage.
participants (6)
-
Allan Wechsler -
Bill Gosper -
Dan Asimov -
Eugene Salamin -
James Propp -
Tomas Rokicki