Re: [math-fun] The value of PI
Henry Baker writes:
If you go through all the formulae in a large book -- e.g., Stegun, et al -- I think that "2pi" shows up more often than "pi". Sooner or later, we're bound to run up against a civilization that chose differently which to commemorate with a name.
There was a short piece (in the Mathematical Intelligencer, I think; perhaps the March 2007 issue?) that explored exactly this theme, and made the "modest proposal" that we should switch from using __ || = 3.14... to using --- ||| = 6.28... in our formulas. (An impassioned advocacy of this position would make a fun "heresy" at a college Pi Day.) Dan Asimov writes:
It is amusing to note that the sum of the V(n) for all even n is e^pi (if one reasonably takes V(0) = 1, since the 0-ball is just a point and 0-dimensional measure just counts points).
Cute, and provocative. As long as we're on this subject, permit me to bring up a curiosity I may have mentioned before on this list: The formula V(n) = (2/n) pi^(n/2) / Gamma(n/2)) for the volume of the unit n-ball gives the familiar formulas V(3) = 4 Pi / 3 V(2) = Pi V(1) = 2 (and makes V(0) blow up), but the formula also gives V(-1) = 1 / Pi V(-2) = 0 V(-3) = - 1 / 2 Pi^2 V(-4) = 0 V(-5) = 3 / 4 Pi^3 with V(-2n) = 0 and V(2n-1) V(-2n-1) = (-1)^n / Pi^2. Do the numbers V(-1), V(-2), etc. have any kind of meaning, and is there any significance to the reciprocity between V(2n-1) and V(-2n-1)? In closing, here is a counterfactual self-referential mnemonic for pi I sent Doug Hofstadter about twenty years ago (I can't remember whether he used it in his column): Had \pi only a teeny bit smaller value, this sentence would be an excellent mnemonic for it. Jim Propp
I've always been a bit puzzled that e and e^2pi were apparently unknown to Archimedes & friends. e was apparently unrecognized until the makers of log tables (c. 1600) began doing computations like 1.001 ^ 1000 . And (as Gosper points out), e^2pi arises naturally as the expansion factor of a 45-degree log spiral. Rich ________________________________________ From: math-fun-bounces@mailman.xmission.com [math-fun-bounces@mailman.xmission.com] On Behalf Of James Propp [jpropp@cs.uml.edu] Sent: Monday, May 05, 2008 9:59 PM To: math-fun@mailman.xmission.com Subject: Re: [math-fun] The value of PI Henry Baker writes:
If you go through all the formulae in a large book -- e.g., Stegun, et al -- I think that "2pi" shows up more often than "pi". Sooner or later, we're bound to run up against a civilization that chose differently which to commemorate with a name.
There was a short piece (in the Mathematical Intelligencer, I think; perhaps the March 2007 issue?) that explored exactly this theme, and made the "modest proposal" that we should switch from using __ || = 3.14... to using --- ||| = 6.28... in our formulas. (An impassioned advocacy of this position would make a fun "heresy" at a college Pi Day.) Dan Asimov writes:
It is amusing to note that the sum of the V(n) for all even n is e^pi (if one reasonably takes V(0) = 1, since the 0-ball is just a point and 0-dimensional measure just counts points).
Cute, and provocative. As long as we're on this subject, permit me to bring up a curiosity I may have mentioned before on this list: The formula V(n) = (2/n) pi^(n/2) / Gamma(n/2)) for the volume of the unit n-ball gives the familiar formulas V(3) = 4 Pi / 3 V(2) = Pi V(1) = 2 (and makes V(0) blow up), but the formula also gives V(-1) = 1 / Pi V(-2) = 0 V(-3) = - 1 / 2 Pi^2 V(-4) = 0 V(-5) = 3 / 4 Pi^3 with V(-2n) = 0 and V(2n-1) V(-2n-1) = (-1)^n / Pi^2. Do the numbers V(-1), V(-2), etc. have any kind of meaning, and is there any significance to the reciprocity between V(2n-1) and V(-2n-1)? In closing, here is a counterfactual self-referential mnemonic for pi I sent Doug Hofstadter about twenty years ago (I can't remember whether he used it in his column): Had \pi only a teeny bit smaller value, this sentence would be an excellent mnemonic for it. Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Probably this article "Pi is Wrong!" http://www.math.utah.edu/~palais/pi.pdf On Tue, May 6, 2008 at 8:54 AM, Schroeppel, Richard <rschroe@sandia.gov> wrote:
I've always been a bit puzzled that e and e^2pi were apparently unknown to Archimedes & friends. e was apparently unrecognized until the makers of log tables (c. 1600) began doing computations like 1.001 ^ 1000 . And (as Gosper points out), e^2pi arises naturally as the expansion factor of a 45-degree log spiral.
Rich
________________________________________ From: math-fun-bounces@mailman.xmission.com [math-fun-bounces@mailman.xmission.com] On Behalf Of James Propp [jpropp@cs.uml.edu] Sent: Monday, May 05, 2008 9:59 PM To: math-fun@mailman.xmission.com
Subject: Re: [math-fun] The value of PI
Henry Baker writes:
If you go through all the formulae in a large book -- e.g., Stegun, et al -- I think that "2pi" shows up more often than "pi". Sooner or later, we're bound to run up against a civilization that chose differently which to commemorate with a name.
There was a short piece (in the Mathematical Intelligencer, I think; perhaps the March 2007 issue?) that explored exactly this theme, and made the "modest proposal" that we should switch from using __ || = 3.14...
to using --- ||| = 6.28...
in our formulas. (An impassioned advocacy of this position would make a fun "heresy" at a college Pi Day.)
Dan Asimov writes:
It is amusing to note that the sum of the V(n) for all even n is e^pi (if one reasonably takes V(0) = 1, since the 0-ball is just a point and 0-dimensional measure just counts points).
Cute, and provocative.
As long as we're on this subject, permit me to bring up a curiosity I may have mentioned before on this list:
The formula V(n) = (2/n) pi^(n/2) / Gamma(n/2)) for the volume of the unit n-ball gives the familiar formulas V(3) = 4 Pi / 3 V(2) = Pi V(1) = 2 (and makes V(0) blow up), but the formula also gives V(-1) = 1 / Pi V(-2) = 0 V(-3) = - 1 / 2 Pi^2 V(-4) = 0 V(-5) = 3 / 4 Pi^3 with V(-2n) = 0 and V(2n-1) V(-2n-1) = (-1)^n / Pi^2. Do the numbers V(-1), V(-2), etc. have any kind of meaning, and is there any significance to the reciprocity between V(2n-1) and V(-2n-1)?
In closing, here is a counterfactual self-referential mnemonic for pi I sent Doug Hofstadter about twenty years ago (I can't remember whether he used it in his column):
Had \pi only a teeny bit smaller value, this sentence would be an excellent mnemonic for it.
Jim Propp
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-- Thane Plambeck tplambeck@gmail.com http://www.plambeck.org/ehome.htm
I find it interesting that there are elegant formulae involving pi seem to come with additional coefficient baggage: e.g., pi/2, pi/4, 4pi, pi^2/6. On the other hand, Euler's constant e seems to shine brightly from just one point. Are there any interesting formulae where e shows up with additional baggage? At 08:59 PM 5/5/2008, James Propp wrote:
There was a short piece (in the Mathematical Intelligencer, I think; perhaps the March 2007 issue?) that explored exactly this theme, and made the "modest proposal" that we should switch from using __ || = 3.14...
to using --- ||| = 6.28...
in our formulas. (An impassioned advocacy of this position would make a fun "heresy" at a college Pi Day.)
On Tue, May 6, 2008 at 9:04 AM, Henry Baker <hbaker1@pipeline.com> wrote:
I find it interesting that there are elegant formulae involving pi seem to come with additional coefficient baggage: e.g., pi/2, pi/4, 4pi, pi^2/6.
On the other hand, Euler's constant e seems to shine brightly from just one point. Are there any interesting formulae where e shows up with additional baggage?
Continued fractions whose terms involve arithmetic series are e + baggage. The easiest example is e-1 = [\overbar{1,1,2k}] Another example from Gosper's continued fraction entries in HAKMEM is [6; \overbar{3+6k}] = (4e^{2/3}-2) / (e^{2/3}-1) More formulae are here: http://www.numbertheory.org/php/davison.html -- Mike Stay - metaweta@gmail.com http://math.ucr.edu/~mike http://reperiendi.wordpress.com
participants (5)
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Henry Baker -
James Propp -
Mike Stay -
Schroeppel, Richard -
Thane Plambeck