Pi day
Martin Ward
Pi day
I saw a posting in one of the humour groups to the effect that 14 March
is Pi Day because 3/14 was a good approximation to pi. Someone had
replied that 3/14 was closer to 0.2142857...
Any Briton and most other Europeans would consider 14 March as 14/3 any
way, and 4.666666... is still not very close to pi.
At this point any mathematically literate reader will think as I did:
*22 July* should be Pi Day. 22/7 = 3.142857... is the good
approximation to pi known since the days of the ancient Greeks.
That made me think, what about e Day? This should be 19 July, since
19/7 = 2.714285 is close to e = 2.718281... The problem is to find the
best rational approximation to e with denominator no more than 12 (and
some constraints on the numerator that create no difficulty).
Then there is Phi Day (phi is the so called golden ratio defined by
1/phi = phi - 1), this turns out to be 13 August.
A calendar of interesting number days can be constructed, and may be of
interest to maths students and of use to maths teachers. It is
presented in the table below. The numbers marked * have an entry in
"The Penguin Dictionary of Curious and Interesting Numbers" by David
Wells - look there if you wonder why pi/sqrt(18) is worth celebrating on
3 April and you do not happen to know about sphere packing.
You may want to add more. What other interesting numbers are there in a
suitable range? As an exercise sqrt(14), sqrt(15) etc can be found a
day each...
Some are not unique. I have presented all the possibilities but I think
we should accept the canceled form as the day, so i^i Day is 1 May
rather than 2 October. The latter could be (i^i Day)'.
Martin Ward
(5 Day)'' 1997
1/pi 0.3183098862 01-Mar *
2^sqrt(2) 2.6651441427 08-Mar *
root 11 3.3166247904 10-Mar
lg2(10) 3.3219280949 10-Mar * log of 10 to the base 2
pi/sqrt(18) 0.7404804897 03-Apr *
2^(1/3) 1.2599210499 05-Apr * cube root of two
2*pi 6.2831853072 25-Apr *
i to the i 0.2078795764 01-May * i^i = exp(-pi/2)
6/pi^2 0.6079271019 03-May *
zeta(3) 1.2020568753 06-May * =1/1^3+1/2^3+1/3^3+...
root 13 3.6055512755 18-May
1/pi 0.3183098862 02-Jun *
1/zeta(3) 0.8319073919 05-Jun * =1/(1/1^3+1/2^3+1/3^3+...)
2^sqrt(2) 2.6651441427 16-Jun *
root 8 2.8284271247 17-Jun
root 10 3.1622776602 19-Jun
root 11 3.3166247904 20-Jun
lg2(10) 3.3219280949 20-Jun * log of 10 to the base 2
log(e) 0.4342944819 03-Jul *
gamma 0.5772156649 04-Jul *
e^(1/e) 1.4446678610 10-Jul * Steiner's number
e 2.7182818285 19-Jul *
pi 3.1415926536 22-Jul *
phi-1 0.6180339887 05-Aug * =1/phi
pi/sqrt(18) 0.7404804897 06-Aug *
2^(1/3) 1.2599210499 10-Aug * cube root of two phi
1.6180339887 13-Aug *
phi^2 2.6180339887 21-Aug *
Liouville's 0.1100010000 01-Sep *
1/pi 0.3183098862 03-Sep *
root pi 1.7724538509 16-Sep * =gamma(1/2)
root 5 2.2360679775 20-Sep *
root 6 2.4494897428 22-Sep
2^sqrt(2) 2.6651441427 24-Sep *
root 11 3.3166247904 30-Sep
lg2(10) 3.3219280949 30-Sep * log of 10 to the base 2
i to the i 0.2078795764 02-Oct * i^i = exp(-pi/2)
log(2) 0.3010299957 03-Oct *
6/pi^2 0.6079271019 06-Oct *
ln(2) 0.6931471806 07-Oct
1/sqrt(2) 0.7071067812 07-Oct * =sqrt(2)/2
zeta(3) 1.2020568753 12-Oct * =1/1^3+1/2^3+1/3^3+...
Brun's 1.9019500000 19-Oct * Brun's constant
ln(10) 2.3025850930 23-Oct *
1/e 0.3678794412 04-Nov *
pi/2sqrt(3) 0.9068996821 10-Nov *
pi^2/6 1.6449340668 18-Nov * =zeta(2)
root 3 1.7320508076 19-Nov *
root 7 2.6457513111 29-Nov
1/pi 0.3183098862 04-Dec *
pi/sqrt(18) 0.7404804897 09-Dec *
1/zeta(3) 2.2020568753 10-Dec * =1/(1/1^3+1/2^3+1/3^3+...)
pi^4/90 1.0823232337 13-Dec * =1/1^4+1/2^4+1/3^4+...
2^(1/3) 1.2599210499 15-Dec * cube root of two root 2
1.4142135624 17-Dec *
--
Martin Ward
_________________________________________________________________________
"When *I* use a word", Humpty Dumpty said in rather a scornful tone, "it
means just what I choose it to mean - neither more nor less".
"The question is", said Alice, "whether you *can* make words mean so many
different things".
"The question is", said Humpty-Dumpty, "which is to be master - that's all".
"Through the Looking-Glass" - Lewis Carroll.
_________________________________________________________________________
Turnpike evaluation. For information, see http://www.turnpike.com/
Scott Phung
Martin Ward <mar...@ward-hull.demon.co.uk> wrote:
MW> Pi day
MW>
MW> I saw a posting in one of the humour groups to the effect that 14 March
MW> is Pi Day because 3/14 was a good approximation to pi. Someone had
MW> replied that 3/14 was closer to 0.2142857...
They probably meant 3.14, not 3/14.
MW> Any Briton and most other Europeans would consider 14 March as 14/3 any
MW> way, and 4.666666... is still not very close to pi.
MW> At this point any mathematically literate reader will think as I did:
MW> *22 July* should be Pi Day. 22/7 = 3.142857... is the good
MW> approximation to pi known since the days of the ancient Greeks.
It's the best approximation with a denomiator <= 12.
MW> That made me think, what about e Day? This should be 19 July, since
MW> 19/7 = 2.714285 is close to e = 2.718281... The problem is to find the
MW> best rational approximation to e with denominator no more than 12 (and
MW> some constraints on the numerator that create no difficulty).
Uhm, the best thing you can do is to do continued fractions. I forgot
how to do continued fractions off my head, but it'll give you
sucessive *BEST* approximations with increasing denimators. Look
in a number theory book.
MW> Then there is Phi Day (phi is the so called golden ratio defined by
MW> 1/phi = phi - 1), this turns out to be 13 August.
MW>
MW> Some are not unique. I have presented all the possibilities but I think
MW> we should accept the canceled form as the day, so i^i Day is 1 May
MW> rather than 2 October. The latter could be (i^i Day)'.
Isn't it ironic how the imaginary number raised to the imaginary
power equals a real number? (Involving the important constants pi,
and e) :-)
MW> Martin Ward
MW> (5 Day)'' 1997
MW>
<snip>
(Impressive list, by the way)
MW> Martin Ward
Scott Phung
Hagbard Celine
In article <8WRuzAAX...@ward-hull.demon.co.uk>,
Martin Ward <mar...@ward-hull.demon.co.uk> wrote:
>Pi day
>
>I saw a posting in one of the humour groups to the effect that 14 March
>is Pi Day because 3/14 was a good approximation to pi. Someone had
>replied that 3/14 was closer to 0.2142857...
Well, maybe they were saying that 3.14, not 3/14 was a good approximation to
pi. Just think it over.>
--
ab jc ik bc ai kj
R R R =R R R
ij kf de ij dk ef
Hagbard Celine
Jill Russell
dates might be commemorated by maths teachers
>1/pi 0.3183098862 01-Mar *
>2^sqrt(2) 2.6651441427 08-Mar *
>root 11 3.3166247904 10-Mar
What a good idea! We could make them into Bank Holidays, or at least
school holidays. Children could study at home that day - the calendar.
--
Jill
Dik T. Winter
In article <5gk8lm$q...@opus.vcn.bc.ca> land...@vcn.bc.ca (Scott Phung) writes:
> MW> At this point any mathematically literate reader will think as I did:
> MW> *22 July* should be Pi Day. 22/7 = 3.142857... is the good
> MW> approximation to pi known since the days of the ancient Greeks.
>
> It's the best approximation with a denomiator <= 12.
Actually it is the best approximation with a denominator <= 56.
...
> Uhm, the best thing you can do is to do continued fractions. I forgot
> how to do continued fractions off my head, but it'll give you
> sucessive *BEST* approximations with increasing denimators. Look
> in a number theory book.
Best in some sense. The most informative book in English is (as far as
I know) by Olds, with the unimaginative title "Continued Fractions".
The most informative book at all is in German by Perron, and a bit
old (1913).
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
Laurent PICOULEAU
Martin Ward <mar...@ward-hull.demon.co.uk> wrote:
>Pi day
>I saw a posting in one of the humour groups to the effect that 14 March
>is Pi Day because 3/14 was a good approximation to pi. Someone had
>replied that 3/14 was closer to 0.2142857...
>Any Briton and most other Europeans would consider 14 March as 14/3 any
>way, and 4.666666... is still not very close to pi.
>At this point any mathematically literate reader will think as I did:
>*22 July* should be Pi Day. 22/7 = 3.142857... is the good
>approximation to pi known since the days of the ancient Greeks.
>That made me think, what about e Day? This should be 19 July, since
>19/7 = 2.714285 is close to e = 2.718281... The problem is to find the
>best rational approximation to e with denominator no more than 12 (and
>some constraints on the numerator that create no difficulty).
[...]
>--
>Martin Ward
[...]
And for the number i, I suggest 32 january as the first imaginary date for the
first of imaginary numbers ;-)
Laurent PICOULEAU
Dik T. Winter
In article <E77u5...@cwi.nl> I wrote:
> In article <5gk8lm$q...@opus.vcn.bc.ca> land...@vcn.bc.ca (Scott Phung) writes:
> > MW> At this point any mathematically literate reader will think as I did:
> > MW> *22 July* should be Pi Day. 22/7 = 3.142857... is the good
> > MW> approximation to pi known since the days of the ancient Greeks.
> >
> > It's the best approximation with a denomiator <= 12.
>
> Actually it is the best approximation with a denominator <= 56.
Ok, now how to find things like this. First start with subsequent
approximations from continued fractions. For pi they are: 3/1, 22/7,
333/106, 355/113 and so on (%). These give best approximations in some
sense (*). However if you have successive approximants pn/qn and pm/qm
(m = n + 1) you also have to look through (pn+k.pm)/(qn+k.qm). With
pn=3, qn=1, pm=22, qm=7 and k=8 this gives you 179/57, which is better
than 22/7. Increasing k will give you better approximations until
you hit k=16, when 333/106 (the next approximant) is already very much
better.
--
* I disremember quite a bit (it is more than 10 years ago that I did
have a look at it). What I particularly disremember is the meaning of
"some sense". What I *do* remember is that the method outlined above
gives subsequent better approximations such that there are *no* better
approximations between them.
--
% How to get a continued fraction expansion. Start with the number,
subtract the integral part. Get the reciprocal of the remainder,
subtract the integral part. Continue. Note the subsequent integer
parts. For pi these are 3, 7, 15, 1, 292, 1, etc. To get an approximant
you truncate this series and reverse the process on the remainder.
Stephen Brierley
>And for the number i, I suggest 32 january as the first imaginary date for
>the first of imaginary numbers ;-)
Nice one, Laurent.
But in what sense is i the first of the imaginary numbers?
Stephen.
ste...@brierley.demon.co.uk
Christian, Maths teacher, pianist/organist, cricket lover, general nutcase
uk.religion.christian who's who: http://www.brierley.demon.co.uk/ukrc.html
View the second most basic website ever at http://www.brierley.demon.co.uk
Brian Raiter
> And for the number i, I suggest 32 january as the first imaginary date for
> the first of imaginary numbers ;-)
Wouldn't the first imaginary date be January 0?
b
Jeffrey G. Montgomery
In article <5gkt94$gp4$4...@news.internetsat.com>,
Laurent PICOULEAU <lcr...@a2points.com> wrote:
>[...]
>
>And for the number i, I suggest 32 january as the first imaginary date for the
>first of imaginary numbers ;-)
Actually; Feb. 29th would be even better -
y = year
if (y-1)/4 is an integer, i
if y/2 is an integer but y/4 is not, -1
if (y+1)/4 is an integer, -i
and finally
if y/4 is an integer, 1
Not valid if y/100 is an integer except if y/400 is an integer.
:)
Jeff
Bertel Lund Hansen
Laurent PICOULEAU wrote:
>And for the number i, I suggest 32 january as the first imaginary date for the
>first of imaginary numbers ;-)
How about february 30th? It is lesse obvious.
--
Venlig hilsen, Bertel
http://home3.inet.tele.dk/bertellh
Ulrich Berger
Martin Ward wrote:
> You may want to add more. What other interesting numbers are there in a
> suitable range?
But any number in any range is interesting, don't you know that?
For, if there were any uninteresting numbers, then there would be such a
thing like the SMALLEST uninteresting number. Obviously this would make
this number very interesting. This is a contradiction, and therefore
there is no uninteresting number!
Uli
Ulrich Berger
> >Pi day
> >
> >I saw a posting in one of the humour groups to the effect that 14 March
> >is Pi Day because 3/14 was a good approximation to pi. Someone had
> >replied that 3/14 was closer to 0.2142857...
>
> pi. Just think it over.>
Well, maybe Martin Ward IS aware of that. Already heard of irony???
Uli
Dave Wilson
In article <332EB5...@pap.univie.ac.at>,
Ulrich Berger <ber...@pap.univie.ac.at> wrote:
>Martin Ward wrote:
>
>> You may want to add more. What other interesting numbers are there in a
>> suitable range?
>
>But any number in any range is interesting, don't you know that?
>For, if there were any uninteresting numbers, then there would be such a
>thing like the SMALLEST uninteresting number. Obviously this would make
>this number very interesting. This is a contradiction, and therefore
>there is no uninteresting number!
>
A nice definition is given in David Wells' Book of Interesting Numbers is
given when discussing 39, interesting because it is the first uninteresting
number. He then continues somewhat in this vein, ' from 39 we will not
define as interesting any number which is uninteresting'
Dave A letter always arrives at
Da...@zizek.demon.co.uk its destination
D.Wi...@mmu.ac.uk J Lacan
http://s13a.math.aca.mmu.ac.uk
Mark Gibson
Martin Ward (mar...@ward-hull.demon.co.uk) wrote:
: Pi day
:
:
: *22 July* should be Pi Day. 22/7 = 3.142857... is the good
: approximation to pi known since the days of the ancient Greeks.
A very clever and entertaining post. I agree that July 22 should be Pi
Day. Incidentally it is also the Deast of St. Mary Magdalene, patron saint
of prostitutes and "fallen" women in the Calendar of the Catholic Chu7rch.
But as for 22/7 being "close" to pi, "close" is a pretty bague term.
Compared with the spiral nebula in Andromeda, Betelgeuse is pretty close
to Earth but don't hold your breath waiting for some airline to announce
weekend excursions there.
3.14159265379 is a closer approximation to pi that I usually use. But that
isn't very close compared with the approximation developed by the guy who
used a computer to work it out to millions of sig figs.
The most fascinating thing about all irrational numbers is that no matter
how hard we try we can never really get "close" to them. The number itself
always recedes over the horizon away from our attempts at precision.
That's why IMHO there really can't be any Pi day, e day, sqrt(2) day, etc.
Incidentelly I'm sure it must have ocurred to someone that there are
DEGREES of irrationality among the irrational numbers. For axample some
irrational numbers when subjected to a simple operation can be mapped to a
rational number. e.g. sqrt(2)*sqrt(2) = 2. Whilst as far as I know e*e and
pi*pi, e*pi, etc are still irrational. Since this ocurred to a
mathematical dunce like me it can't be that important but I though it was
interesting, so there.
MSG
---
THEY'RE COMING TO TAKE ME AWAY, HA-HA
THEY'RE COMING TO TAKE ME AWAY...
Tony Buckland
In article <5gmrkk$d00$1...@news-s01.ny.us.ibm.net>, <cha...@ibm.net> wrote:
>In <AF53F4E8...@brierley.demon.co.uk>, ste...@brierley.demon.co.uk (Stephen Brierley) writes:
>Another idea for an imaginary date--
>How about any date in the year 0 (zero)?
If only there really had been a year zero.
Then we wouldn't have to be concerned about whether the new
century and millenium begin Jan 1 2000 or Jan 1 2001.
cha...@ibm.net
>>the first of imaginary numbers ;-)
>
>
>But in what sense is i the first of the imaginary numbers?
>
Another idea for an imaginary date--
How about any date in the year 0 (zero)?
Robert R. Chapman, Jr.
cha...@ibm.net
Lynnwood, WA 98036 USA
**********************************************************
It's interesting how superstition thrives in unsettled times,
and how many are prepared to listen to it, at least with
half an ear.
--Dietrich Bonhoeffer
Letters and Papers from Prison, The Enlarged Edition
(New York: Macmillan Paperbacks Edition 1972), page 153
**********************************************************
Miro Jurisic
In article <5gndo3$5...@flounder.trans.ge.com>, bau...@erie.ge.com wrote:
> Anybody got a proof for that? i^i = exp(-pi/2)?
>
> Thanks,
>
> Ron Bauerle
i = e^(i*pi/2)
i^i = (e^(i*pi/2))^i = e^((i*pi/2)*i) = e^(-pi/2)
meeroh
--
Miroslav Jurisic | mee...@mit.edu | http://www.mit.edu/people/meeroh/
MIT Information Systems (Mac development) | Athena On-Line Consulting
Lt. Commander Meeroh Ambellus, ATS TrekMUSH (ats.trekmush.org:1701)
"It's not hatred that's important; it's the desire to anihilate." [101
Dalamatians]
G.W. Owen
Brian Raiter (b...@drizzle.com) wrote:
: > And for the number i, I suggest 32 january as the first imaginary date for
: > the first of imaginary numbers ;-)
: Wouldn't the first imaginary date be January 0?
:
How about the 12th of Never?
--
Gareth Owen <G.W....@keele.ac.uk>
Drugs affect children in the opposite way they affect adults.
Jaime Pina
On Tue, 18 Mar 1997, Ulrich Berger wrote:
{>Martin Ward wrote:
{>
{>> You may want to add more. What other interesting numbers are there in a
{>> suitable range?
{>
{>But any number in any range is interesting, don't you know that?
{>For, if there were any uninteresting numbers, then there would be such a
{>thing like the SMALLEST uninteresting number. Obviously this would make
{>this number very interesting. This is a contradiction, and therefore
{>there is no uninteresting number!
{>
{>Uli
{>
{>
WRONG! If there are uninteresting numbers, there does not have to
be a SMALLEST uninteresting number, just like there is no smallest number
in the real numbers.
--------------------------------------------------------------------
"Once you have flown you will walk the Earth with your eyes turned
skyward, for there you have been, there you long to return."
---Leonardo da Vinci
Keith Hearn
>>>the first of imaginary numbers ;-)
>>
>>
>>But in what sense is i the first of the imaginary numbers?
>
>Another idea for an imaginary date--
>
>How about any date in the year 0 (zero)?
How about any date in the year i?
Keith Hearn
khe...@pyramid.com
Perry Winkel
In article <01bc3440$d8a87040$LocalHost@default>, "Belinda Berry"
<BJB...@onaustralia.com.au> wrote:
> How do you actually calculate pi anyway. (without loooking it up in a
> book.) I suppose if you drew up a really accurate circle and did some
> measuring it might work, but nothing I draw could possibly get more than
> about 2 decimal places.
> --
> Belinda Berry
> <BJB...@onaustralia.com.au>
>
> >
...as I recall from long ago in high school, there is in fact a very easy
way to determine pi. You need a stick (a pencil works fine also) and a
floor with a pattern of square tiles. What you do is this:
1-Throw the pensil in the air randomly.
2-Count the number of lines (dividing the tiles) that the pencil crosses
after it has landed.
3-Repeat step 1 and two and add up the results...also remember the number
of times that you have thrown up the pencil...
and finally use the formula that I learned at highschool, but have
forgotten since then...
Repeating this process for a large number of times will come up with a
fairly accurate value of pi.
Perry Winkel
Ron Bauerle
Anybody got a proof for that? i^i = exp(-pi/2)?
Thanks,
Ron Bauerle
K. Edgcombe
In article <Pine.GSO.3.95.970318...@shrike.depaul.edu>,
>{>For, if there were any uninteresting numbers, then there would be such a
>{>thing like the SMALLEST uninteresting number. Obviously this would make
>{>this number very interesting. This is a contradiction, and therefore
>{>there is no uninteresting number!
>
>be a SMALLEST uninteresting number, just like there is no smallest number
>in the real numbers.
I think the original discussion was about the natural numbers (the non-negative
integers), and it is clear that the poster of the above proof was thinking of
the natural numbers. By the well-ordering axiom, therefore, the proof holds.
By the way, even if you think someone is wrong, there is no need to SHOUT.
Katy Edgcombe
Scott Phung
d...@cwi.nl (Dik T. Winter) wrote:
DW> In article <E77u5...@cwi.nl> I wrote:
DW> Ok, now how to find things like this. First start with subsequent
DW> approximations from continued fractions. For pi they are: 3/1, 22/7,
DW> 333/106, 355/113 and so on (%). These give best approximations in some
DW> sense (*). However if you have successive approximants pn/qn and pm/qm
DW> (m = n + 1) you also have to look through (pn+k.pm)/(qn+k.qm). With
DW> pn=3, qn=1, pm=22, qm=7 and k=8 this gives you 179/57, which is better
DW> than 22/7. Increasing k will give you better approximations until
DW> you hit k=16, when 333/106 (the next approximant) is already very much
DW> better.
Thanks for the information. I hadn't learned about this, and it
can be useful (like say, what is the best approx. to pi with a
denominator less than 100, or the like).
Do you know if there is a way to calculate the value or approximate
value of k such that with this value of k, it will produce a better
approx. than pn, but not than pm? (Ie-in this case, k = 8).
Or does a such a calculation not exist? I imagine in the gap
between 333/106 and 355/113, there does not exist any rational
number more accurate than 333/106 but less accurate than 355/113.
Does this method also work for other irrational numbers like
square roots, nth roots, quadratic irrationals, e, etc?
DW> % How to get a continued fraction expansion. Start with the number,
DW> subtract the integral part. Get the reciprocal of the remainder,
DW> subtract the integral part. Continue. Note the subsequent integer
DW> parts. For pi these are 3, 7, 15, 1, 292, 1, etc. To get an approximant
DW> you truncate this series and reverse the process on the remainder.
The above method is quick and short (I like it), and hopefully easy
to remember.
It's also quite hard to beleive (for me) how continued fractions
don't pick up these 'holes'.
Scott Phung
David Madore
Jaime Pina wrote:
Jaime Pina wrote:
> WRONG! If there are uninteresting numbers, there does not have to
> be a SMALLEST uninteresting number, just like there is no smallest number
> in the real numbers.
Here is a proof that all real numbers are interesting:
Let I be the set of interesting real numbers.
* I is not empty because 0, 1, e and pi are certainly in I.
* Any number sufficiently close to a given interesting number is
interesting.
(For example, a number which has the same first billion decimals as pi
is
clearly interesting.) Therefore I is open.
* A number which is the limit of a convergent sequence of interesting
numbers
is certainly interesting. Therefore I is closed.
But R is connected. Therefore I=R, and all real numbers are interesting.
QED.
David A. Madore
(david....@ens.fr,
http://www.eleves.ens.fr:8080/home/madore/index.html.en)
Belinda Berry
Ulrich Lange
David Madore (mad...@clipper.ens.fr) wrote:
: Here is a proof that all real numbers are interesting:
: Let I be the set of interesting real numbers.
: * I is not empty because 0, 1, e and pi are certainly in I.
: * Any number sufficiently close to a given interesting number is
: interesting.
: (For example, a number which has the same first billion decimals as pi
: is
: clearly interesting.) Therefore I is open.
: * A number which is the limit of a convergent sequence of interesting
: numbers
: is certainly interesting. Therefore I is closed.
: But R is connected. Therefore I=R, and all real numbers are interesting.
ROTFL :-)
Since your argument for I being open might be questioned by some people,
an alternative proof would be:
1) All natural numbers are interesting because a smallest uninteresting
natural number cannot exist.
2) Sums and products, as well as additive and multiplicative inverses
of interesting numbers are clearly interesting, hence all rational
numbers are interesting.
3) The limit of a sequence of convergent sequence of interesting numbers
is interesting, hence all real numbers are interesting.
However, I guess both of us used the axiom that algebra and topology are
interesting, which might also be questioned by some people :-)
--
Ulrich Lange Dept. of Chemical Engineering
University of Alberta
la...@gpu.srv.ualberta.ca Edmonton, Alberta, T6G 2G6, Canada
cha...@ibm.net
In <5gmt4q$a40$1...@nntp.ucs.ubc.ca>, buck...@unixg.ubc.ca (Tony Buckland) writes:
>In article <5gmrkk$d00$1...@news-s01.ny.us.ibm.net>, <cha...@ibm.net> wrote:
>>In <AF53F4E8...@brierley.demon.co.uk>, ste...@brierley.demon.co.uk (Stephen Brierley) writes:
>>Another idea for an imaginary date--
>>How about any date in the year 0 (zero)?
>
> Then we wouldn't have to be concerned about whether the new
> century and millenium begin Jan 1 2000 or Jan 1 2001.
>
Personally, I am not losing much sleep over when the new millenium begins.
On the other hand, the people I contract with are real worried what will
happen to their computers on 1 Jan 2000. And we thought we only
needed to worry about Michelangelo's birthday.
Brian Raiter
> How do you actually calculate pi anyway. (without loooking it up in a
> book.) I suppose if you drew up a really accurate circle and did some
> measuring it might work, but nothing I draw could possibly get more than
> about 2 decimal places.
There are numerous web pages devoted entirely to pi. Most of them list
numerous methods to calculate its value. Check yahoo.
The one I remember off the top of my head:
pi/4 = 1/1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + ...
This one, however, takes a very very long time to converge. There are
much better ones.
b
G.W. Owen
Miro Jurisic (mee...@mit.edu) wrote:
: In article <5gndo3$5...@flounder.trans.ge.com>, bau...@erie.ge.com wrote:
:
: > Anybody got a proof for that? i^i = exp(-pi/2)?
: >
: > Thanks,
: >
: > Ron Bauerle
:
: i = e^(i*pi/2)
:
: i^i = (e^(i*pi/2))^i = e^((i*pi/2)*i) = e^(-pi/2)
:
: meeroh
Of course i^i can also take an infinite number of other solutions, in
a analogous way with the fact that 3^(1/2) can take 2 values.
Riemann surfaces and all that.
--
Gareth Owen <G.W....@keele.ac.uk>
Looking up is as scary as looking down.
G.W. Owen
Belinda Berry (BJB...@onaustralia.com.au) wrote:
: How do you actually calculate pi anyway. (without loooking it up in a
: book.) I suppose if you drew up a really accurate circle and did some
: measuring it might work, but nothing I draw could possibly get more than
: about 2 decimal places.
:
There are a number of ways.
There are some infinite series that can be shown to add up to pi, or
pi/2 or somesuch and taking as many terms as you like can give you as
many decimal places as you want/need.
Failing that, you can find a formula for the area of a circle, using
integration, and then approximate it using Simpson's rule, taking
sections as thin as necessary for the accuracy you want.
--
Gareth Owen <G.W....@keele.ac.uk>
Pleidiol Wyf I'm Gwlad
Tom Waters
Tony Buckland (buck...@unixg.ubc.ca) wrote:
: If only there really had been a year zero.
: Then we wouldn't have to be concerned about whether the new
: century and millenium begin Jan 1 2000 or Jan 1 2001.
Does anyone remember whether the French revolutionary calendar had a year
zero?
Tom
--
Thomas Waters
twa...@use.usit.net
1021 East Oak Hill Avenue, Knoxville TN 37917
Dig And Be Dug In Return
David Madore
Tom Waters wrote:
> Does anyone remember whether the French revolutionary calendar had a year
> zero?
They did not. The dates were written in roman numerals, so they did not
make provision for a zero. The calendar wasn't supposed to be used to
refer to pre-revolutionary days, either, so there would be no negative
years (they would be written in the old gregorian calendar). Neither did
they have a month zero or a day zero in each month.
Dik T. Winter
In article <5goas8$s...@opus.vcn.bc.ca> land...@vcn.bc.ca (Scott Phung) writes:
> d...@cwi.nl (Dik T. Winter) wrote:
> DW> (m = n + 1) you also have to look through (pn+k.pm)/(qn+k.qm). With
> DW> pn=3, qn=1, pm=22, qm=7 and k=8 this gives you 179/57, which is better
> DW> than 22/7. Increasing k will give you better approximations until
> DW> you hit k=16, when 333/106 (the next approximant) is already very much
> DW> better.
>
> Thanks for the information. I hadn't learned about this, and it
> can be useful (like say, what is the best approx. to pi with a
> denominator less than 100, or the like).
>
> Do you know if there is a way to calculate the value or approximate
> value of k such that with this value of k, it will produce a better
> approx. than pn, but not than pm? (Ie-in this case, k = 8).
[It has been over 10 years ago that I looked at this stuff, so some
of the minor details may be incorrect.]
Lets look at pi and use some notation. The continued fraction expansion of
pi is [3; 7, 15, 1, 292, 1, ...], which does not terminate. The first 3
is set off as the initial integer part (and that is the only element that
can be 0). Now 3/1 = [3], 22/7 = [3; 7] and 333/106 = [3; 7, 15]
(terminating because they are rationals). If we let k increase to 15 we
get the next approximant, 333/106 (I made an error here in my original
post); and it is not by coincidence that the last number in the expansion
of 333/106 is also 15. Better approximations start for k one half of that
number (I disremember whether it is also so for even numbers or that you
have to start one further).
> Or does a such a calculation not exist? I imagine in the gap
> between 333/106 and 355/113, there does not exist any rational
> number more accurate than 333/106 but less accurate than 355/113.
That is right if you modify your statement to "any rational number
with denominator between 106 and 113". Because the first number omitted
from the continued fraction is 1, only k=1 would work but that gives you
the next approximant: (22+333)/(7+106) = 355/113. So when the first
number omitted is 1, there is no intermediate to look at. Subsequent
approximants alternate between being too small and too large. Intermediates
are all on the same side as the next approximant.
>
> Does this method also work for other irrational numbers like
> square roots, nth roots, quadratic irrationals, e, etc?
Yes, even for rationals themselves. For instance, the common rational
constant 2.54 has continued fraction expansion [2; 1, 1, 5, 1, 3].
Subsequent approximants from this are 2, 3, 5/2, 28/11, 33/13, 127/50.
And we can get some intermediate approximants. A large number in the
approximation indicates that at that place the denominator will increase
by a large part. Also the numbers in the continued fraction expansion
of algebraic numbers can not get "too large" because rationals can not
approximate them "too close". (For some notion of "too large" and
"too close" which can be quantified.)
> It's also quite hard to beleive (for me) how continued fractions
> don't pick up these 'holes'.
The reason you can also look at the intermediate forms is because
the continued fraction approximants only approximate closest in some
sense (it has to do with the difference compared to 1/q, where q
is the denominator of the approximant). When you also look at the
intermediates you get closest approximation in another sense.
Given two successive intermediates (including approximants) pn/qn
and pm/qm than there is *no* rational number other than pn/qn closer
to the number approximated with a denominator smaller than pm. (A
bit garbled, I hope you get the meaning.)
On computers continued fraction approximations are very useful in
fixed point arithmetic.
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
Martin Ward
In article <01bc3440$d8a87040$LocalHost@default>, Belinda Berry
>How do you actually calculate pi anyway. (without loooking it up in a
>book.) I suppose if you drew up a really accurate circle and did some
>measuring it might work, but nothing I draw could possibly get more than
>about 2 decimal places.
Quite right, Belinda, not even the most accurate drawing and measuring
could produce results good for many more places than that.
Series are the traditional way of calculating numbers like pi to lots of
places. For example:
(pi^2)/6 = 1+(1/2^2)+(1/3^2)+(1/4^2)+...
The series goes on for ever, but the terms get smaller and smaller and
the sum is said to "converge", in this case "converge to pi squared over
six". For a given degree of accuracy we can stop after a while. This
particular series is easy to state but converges very slowly, so "after
a while" means after a *long* while. For example, to get four decimal
places of accuracy we have to add up 6918 terms. Fortunately there are
other series which converge more rapidly.
--
Best wishes
Martin Ward
__________________________________________________________________
"No-one is ever hurt by the truth - he is injured who continues in
self-deception and ignorance." Marcus Aurelius
__________________________________________________________________
Turnpike evaluation. For information, see http://www.turnpike.com/
Martin Ward
In article <5gndo3$5...@flounder.trans.ge.com>, Ron Bauerle
<bau...@erie.ge.com> writes
>In article <5gk8lm$q...@opus.vcn.bc.ca>, land...@vcn.bc.ca (Scott Phung) writes:
>>Martin Ward <mar...@ward-hull.demon.co.uk> wrote:
>
>>MW> Some are not unique. I have presented all the possibilities but I think
>>MW> we should accept the canceled form as the day, so i^i Day is 1 May
>>MW> rather than 2 October. The latter could be (i^i Day)'.
>>
>>Isn't it ironic how the imaginary number raised to the imaginary
>>power equals a real number? (Involving the important constants pi,
>>and e) :-)
>
>
>Thanks,
>
>Ron Bauerle
>
= 0 + i*1
= i
so i^i = (e^(i*pi/2))^i
= e^(i*i*pi/2)
= e^(-pi/2)
--
Martin Ward
_________________________________________________________________________
"When *I* use a word", Humpty Dumpty said in rather a scornful tone, "it
means just what I choose it to mean - neither more nor less".
"The question is", said Alice, "whether you *can* make words mean so many
different things".
"The question is", said Humpty-Dumpty, "which is to be master - that's all".
"Through the Looking-Glass" - Lewis Carroll.
_________________________________________________________________________
Turnpike evaluation. For Turnpike information, mailto:in...@turnpike.com
Martin Ward
In article <5gmqsn$b...@milo.vcn.bc.ca>, Mark Gibson <mgi...@vcn.bc.ca>
writes
>Martin Ward (mar...@ward-hull.demon.co.uk) wrote:
>: Pi day
>:
>:
>: *22 July* should be Pi Day. 22/7 = 3.142857... is the good
>: approximation to pi known since the days of the ancient Greeks.
>:
>A very clever and entertaining post. I agree that July 22 should be Pi
>Day. Incidentally it is also the Deast of St. Mary Magdalene, patron saint
>of prostitutes and "fallen" women in the Calendar of the Catholic Chu7rch.
>
>But as for 22/7 being "close" to pi, "close" is a pretty bague term.
Yes, but the point is that it is as close a rational approximation as we
can get with a small denominator. In this case with denominator <= 12.
>Compared with the spiral nebula in Andromeda, Betelgeuse is pretty close
>to Earth but don't hold your breath waiting for some airline to announce
>weekend excursions there.
>
>3.14159265379 is a closer approximation to pi that I usually use. But that
>isn't very close compared with the approximation developed by the guy who
>used a computer to work it out to millions of sig figs.
>
>The most fascinating thing about all irrational numbers is that no matter
>how hard we try we can never really get "close" to them. The number itself
>always recedes over the horizon away from our attempts at precision.
>That's why IMHO there really can't be any Pi day, e day, sqrt(2) day, etc.
We can get as close as we like to an irrational in the sense that if you
say get within a millionth of pi, I can produce a rational number that
close; and if you say a billionth I can get that close too; and so on to
any degree of accuracy you want.
What we cannot do is state an irrational *exactly* as a fraction or
decimal.
>
>Incidentelly I'm sure it must have ocurred to someone that there are
>DEGREES of irrationality among the irrational numbers. For axample some
>irrational numbers when subjected to a simple operation can be mapped to a
>rational number. e.g. sqrt(2)*sqrt(2) = 2. Whilst as far as I know e*e and
>pi*pi, e*pi, etc are still irrational. Since this ocurred to a
>mathematical dunce like me it can't be that important but I though it was
>interesting, so there.
>
>MSG
>---
>THEY'RE COMING TO TAKE ME AWAY, HA-HA
>THEY'RE COMING TO TAKE ME AWAY...
You are quite right about this, and it is an important insight.
Irrational numbers like sqrt(2) are said to be algebraic, because they
are the roots of an algebraic equation (with rational coefficients). In
this case:
x^2 - 2 = 0
Irrational numbers like e and pi which are not the root of any such
equation are said to be transcendental. (It is rather hard to *prove*
that pi and e are transcendental).
Martin Ward
In article <5gp93t$o...@nyx.cs.du.edu>, Charles Carroll
<ccar...@nyx.com> writes
>"Belinda Berry" <BJB...@onaustralia.com.au> writes:
>
>>How do you actually calculate pi anyway. (without loooking it up in a
>>book.) I suppose if you drew up a really accurate circle and did some
>>measuring it might work, but nothing I draw could possibly get more than
>>about 2 decimal places.
>>Belinda Berry
>><BJB...@onaustralia.com.au>
>
>One way to do it is with this series. This series doesn't converge
>as quickly as some other series,
And how! This takes over 5000 terms to reach even one decimal place of
accuracy, even worse (in this respect) than the one I suggested!
> but it's probably the simplest
>one there is. Try it and see.
>
>pi = 4/1 - 4/3 + 4/5 - 4/7 + 4/9 - 4/11 + 4/13 - 4/15 ...
>
>
--
Martin Ward
_________________________________________________________________________
"When *I* use a word", Humpty Dumpty said in rather a scornful tone, "it
means just what I choose it to mean - neither more nor less".
"The question is", said Alice, "whether you *can* make words mean so many
different things".
"The question is", said Humpty-Dumpty, "which is to be master - that's all".
"Through the Looking-Glass" - Lewis Carroll.
_________________________________________________________________________
Turnpike evaluation. For information, see http://www.turnpike.com/
Martin Ward
In article <5gk8lm$q...@opus.vcn.bc.ca>, Scott Phung <land...@vcn.bc.ca>
writes
<snip>
>MW> Then there is Phi Day (phi is the so called golden ratio defined by
>MW> 1/phi = phi - 1), this turns out to be 13 August.
>MW>
>MW> Some are not unique. I have presented all the possibilities but I think
>MW> we should accept the canceled form as the day, so i^i Day is 1 May
>MW> rather than 2 October. The latter could be (i^i Day)'.
>
>Isn't it ironic how the imaginary number raised to the imaginary
>power equals a real number? (Involving the important constants pi,
>and e) :-)
>
>Scott Phung
Yes, wonderful isn't it.
I am not sure if I prefer that result, or
e^(i*pi) +1 = 0
I even managed to buy a t-shirt with this printed on it.
--
Martin Ward
_________________________________________________________________________
"When *I* use a word", Humpty Dumpty said in rather a scornful tone, "it
means just what I choose it to mean - neither more nor less".
"The question is", said Alice, "whether you *can* make words mean so many
different things".
"The question is", said Humpty-Dumpty, "which is to be master - that's all".
"Through the Looking-Glass" - Lewis Carroll.
_________________________________________________________________________
Turnpike evaluation. For Turnpike information, mailto:in...@turnpike.com
Kuddel
Martin Ward <mar...@ward-hull.demon.co.uk> wrote:
> Pi day
>
> I saw a posting in one of the humour groups to the effect that 14 March
> is Pi Day because 3/14 was a good approximation to pi. Someone had
> replied that 3/14 was closer to 0.2142857...
Pi can be approximated by 22/7 (that's what the anciet Greeks did, when
they build wheels) So we 22nd of July should be the Pi day, shouldn't
it?
Yours sincerly Kuddel
Kathrin Paschen
Mark Gibson (mgi...@vcn.bc.ca) wrote:
: Incidentelly I'm sure it must have ocurred to someone that there are
: DEGREES of irrationality among the irrational numbers. For axample some
: irrational numbers when subjected to a simple operation can be mapped to a
: rational number. e.g. sqrt(2)*sqrt(2) = 2. Whilst as far as I know e*e and
: pi*pi, e*pi, etc are still irrational. Since this ocurred to a
: mathematical dunce like me it can't be that important but I though it was
: interesting, so there.
That's because pi and e are transcendental, while sqrt(2) is algebraic.
Transcendental numbers cannot solve a rational polynomial (they cannot
be 'mapped to a rational number' the way you did with sqrt(2)).
All rational numbers are algebraic, but not all irrational ones are.
It's a very fascinating subject :).
Mark Meiss
>Incidentelly I'm sure it must have ocurred to someone that there are
>DEGREES of irrationality among the irrational numbers. For axample some
>irrational numbers when subjected to a simple operation can be mapped to a
>rational number. e.g. sqrt(2)*sqrt(2) = 2. Whilst as far as I know e*e and
>pi*pi, e*pi, etc are still irrational. Since this ocurred to a
>mathematical dunce like me it can't be that important but I though it was
>interesting, so there.
It sure has--e, pi, etc. are "transcendental" numbers, a subset of
irrational numbers. sqrt(2), sqrt(3), etc. are "algebraic" numbers.
An algebraic number can be a root of a polynomial equation. A
transcendental number can't. And actually it wasn't too long ago
(relatively speaking) that pi and e were proven to be transcendental
numbers. So it's very interesting, actually.
--Mark
--
Mark Meiss (mme...@indiana.edu) | No eternal reward will
Indiana University CS Dept. | forgive us now for - Jim Morrison
Bloomington, Indiana, USA | wasting the dawn.
David
d...@cwi.nl (Dik T. Winter) wrote:
Just interjecting:
Continued Fractions are discussed at some length in Hall and Knight's
"Higher Algebra." The text (and presumably the topic) are 1800's 1st
publication. The latest publication I have is 1932, but there were
likely later ones.
Hope that this may be of some assistance.
Regards,
David.
John Ross
Tom Waters wrote <5grnjb$i44$1...@news.usit.net>:
> Tony Buckland (buck...@unixg.ubc.ca) wrote:
> Does anyone remember whether the French revolutionary calendar had a year
> zero?
Remember? Good grief! No one's that old, it just feels that way!
--
John Ross
Southampton
Martin Ward
In article <5gob11$6...@lyra.csx.cam.ac.uk>, "K. Edgcombe"
<ke...@cus.cam.ac.uk> writes
>In article <Pine.GSO.3.95.970318...@shrike.depaul.edu>,
>Jaime Pina <jp...@shrike.depaul.edu> wrote:
>>On Tue, 18 Mar 1997, Ulrich Berger wrote:
>>
>>{>
>>{>But any number in any range is interesting, don't you know that?
>>{>For, if there were any uninteresting numbers, then there would be such a
>>{>thing like the SMALLEST uninteresting number. Obviously this would make
>>{>this number very interesting. This is a contradiction, and therefore
>>{>there is no uninteresting number!
>>
>>in the real numbers.
>
>I think the original discussion was about the natural numbers (the non-negative
>integers), and it is clear that the poster of the above proof was thinking of
>the natural numbers. By the well-ordering axiom, therefore, the proof holds.
>
>By the way, even if you think someone is wrong, there is no need to SHOUT.
>
>Katy Edgcombe
I was referring to real numbers (like pi - see the subject line) so
Jaime's crticism is valid. I presume Ulrich was thinking of the natural
numbers, or possibly the rationals, or possibly rationals of the form
a/b where a,b are natural numbers with a<=31 and b<=12. Any of these
sets can be well-ordered, and then Ulrich's argument "works" as Katy
says.
I like this argument, but of course it has a more subtle flaw...
--
Martin Ward
_________________________________________________________________________
"When *I* use a word", Humpty Dumpty said in rather a scornful tone, "it
means just what I choose it to mean - neither more nor less".
"The question is", said Alice, "whether you *can* make words mean so many
different things".
"The question is", said Humpty-Dumpty, "which is to be master - that's all".
"Through the Looking-Glass" - Lewis Carroll.
_________________________________________________________________________
Turnpike evaluation. For information, see http://www.turnpike.com/
Martin Ward
In article <332FC191...@clipper.ens.fr>, David Madore
<mad...@clipper.ens.fr> writes
>Jaime Pina wrote:
>Jaime Pina wrote:
>> be a SMALLEST uninteresting number, just like there is no smallest number
>> in the real numbers.
>
>Let I be the set of interesting real numbers.
> * I is not empty because 0, 1, e and pi are certainly in I.
> * Any number sufficiently close to a given interesting number is
>interesting.
>(For example, a number which has the same first billion decimals as pi
>is clearly interesting.)
uncountably infinite, with the same cardinal number as the reals - this
is therefore a rather large claim. I would find it hard to work up much
interest in more than one or two of them!
>Therefore I is open.
> * A number which is the limit of a convergent sequence of interesting
>numbers
>is certainly interesting. Therefore I is closed.
>
>But R is connected. Therefore I=R, and all real numbers are interesting.
--
Martin Ward
In article <199703211...@isdn78.hrz.uni-siegen.de>, Kuddel
<holg...@studm.hrz.uni-siegen.de> writes
Yes, that's what I said! You must have missed the origin posting (or
much of it) Kuddel. I will email a copy to you.
--
Best wishes
Martin Ward
_________________________________________________________________________
"When *I* use a word", Humpty Dumpty said in rather a scornful tone, "it
means just what I choose it to mean - neither more nor less".
"The question is", said Alice, "whether you *can* make words mean so many
different things".
"The question is", said Humpty-Dumpty, "which is to be master - that's all".
"Through the Looking-Glass" - Lewis Carroll.
_________________________________________________________________________
Turnpike evaluation. For Turnpike information, mailto:in...@turnpike.com
Sitaram Iyer
On 19 Mar 97 09:50:35 GMT, Belinda Berry wrote:
: How do you actually calculate pi anyway. (without loooking it up in a
: book.) I suppose if you drew up a really accurate circle and did some
: measuring it might work, but nothing I draw could possibly get more than
: about 2 decimal places.
: --
: Belinda Berry
: <BJB...@onaustralia.com.au>
There was this very fast converging iteration:
Start with a(0) = sqrt(2)-1 and b(0) = 6 - 4*sqrt(2)
t = (1-a[i-1]^4)^(0.25)
a[i] = (1-t)/(1+t)
b[i] = (1+a[i])^4 * b[i-1] - 2^(2*i+1) * a[i] * (1+a[i]+a[i]^2)
1/b[i] is supposed to converge to PI VERY fast, within 3 iterations it
fills up double precision and within a (don't remember) dozen or so
iterations it gives millions of digits. Can someone elaborate ?
- Sitaram.
--
Thanx.
+-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=+
|All that is gold does not glitter, From the ashes a fire shall be woken, |
|Not all those who wander are lost; A light from the shadows shall spring;|
|The old that is strong does not wither, Renewed shall be blade that was broken|
|Deep roots are not reached by the frost The crownless again shall be king. |
| - J. R. R. Tolkien, The Lord of the Rings |
+-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=+
| Sitaram Iyer <sitaram@[144.16.111.2]> http://www.cse.iitb.ernet.in/~sitaram |
+-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=+
Robert James Aguirre
Martin Ward wrote:
> I was referring to real numbers (like pi - see the subject line) so
> Jaime's crticism is valid. I presume Ulrich was thinking of the natural
> numbers, or possibly the rationals, or possibly rationals of the form
> a/b where a,b are natural numbers with a<=31 and b<=12. Any of these
> sets can be well-ordered, and then Ulrich's argument "works" as Katy
> says.
>
> I like this argument, but of course it has a more subtle flaw...
> --
Hmm I suppose the reals "can" be well ordered as well (or must I leave
choice axioms alone...
Jonathan Cunningham
In article <5gmqsn$b...@milo.vcn.bc.ca>,
mgi...@vcn.bc.ca (Mark Gibson) wrote:
>Martin Ward (mar...@ward-hull.demon.co.uk) wrote:
>: Pi day
>:
>:
(snip)
>
>3.14159265379 is a closer approximation to pi that I usually use. But that
Since I haven't seen anyone else say it yet, if you are going to use a 12
digit approximation, how about using 3.14159265359 which is an even better
approximation than the one you use :-).
I find "3" is accurate enough for many purposes, though.
Be careful if you are determining pi by measuring the ratio of the
circumference of a circle to its diameter: a one tonne weight in a 10 metre
diameter circle will affect the value of pi in about the 24th or 25th
decimal place (general relativity and curvature of space-time etc.). If
you wish to try this, it's worth knowing that the correct value of
pi should be around 3.14159265358979323846264338 in flat space, although
the last two or three digits will be different with the one tonne weight
in place. :-).
Jonathan
Aaron J. Dinkin
In article <199703211...@isdn78.hrz.uni-siegen.de>,
holg...@studm.hrz.uni-siegen.de (Kuddel) wrote:
>Martin Ward <mar...@ward-hull.demon.co.uk> wrote:
>
>> Pi day
>>
>> I saw a posting in one of the humour groups to the effect that 14 March
>> is Pi Day because 3/14 was a good approximation to pi. Someone had
>> replied that 3/14 was closer to 0.2142857...
>
>Pi can be approximated by 22/7 (that's what the anciet Greeks did, when
>they build wheels) So we 22nd of July should be the Pi day, shouldn't
>it?
This isn't just a humor group thing - Pi Day was actually observed in
numerous elemantary schools.
And it's 3.14, people. Not 3/14. And the 22d of July is 7/22.
-Aaron J. Dinkin
Dr. Whom
Steve Carter
Aaron J. Dinkin (adi...@commschool.org) wrote:
~ In article <199703211...@isdn78.hrz.uni-siegen.de>,
~ holg...@studm.hrz.uni-siegen.de (Kuddel) wrote:
~ >Martin Ward <mar...@ward-hull.demon.co.uk> wrote:
~ >
~ >> Pi day
~ >>
~ >> I saw a posting in one of the humour groups to the effect that 14 March
~ >> is Pi Day because 3/14 was a good approximation to pi. Someone had
~ >> replied that 3/14 was closer to 0.2142857...
~ >
~ >Pi can be approximated by 22/7 (that's what the anciet Greeks did, when
~ >they build wheels) So we 22nd of July should be the Pi day, shouldn't
~ >it?
~ This isn't just a humor group thing - Pi Day was actually observed in
~ numerous elemantary schools.
~ And it's 3.14, people. Not 3/14. And the 22d of July is 7/22.
Not in the UK it's not. That's the 7th of next October. :)
You can get away with 7-22-97 in UK but / is strictly heirarchic!
-------------------------------------
Steve Carter, shc...@york.ac.uk, http://www.york.ac.uk/~shc103
"I'm an ex-citizen of nowhere, and sometimes I get mighty homesick."
- Ben Rumson in Paint Your Wagon
Spam mailings to this address form an invitation to Megabytes of junk.
John Cartmell
In article <AF5DA45C...@sofluc.demon.co.uk>, Jonathan Cunningham
<URL:mailto:j...@sofluc.demon.co.uk> wrote:
>
> In article <5gmqsn$b...@milo.vcn.bc.ca>,
> mgi...@vcn.bc.ca (Mark Gibson) wrote:
>
> >Martin Ward (mar...@ward-hull.demon.co.uk) wrote:
> >: Pi day
> >:
> >:
> (snip)
> >
> >3.14159265379 is a closer approximation to pi that I usually use. But that
>
> Since I haven't seen anyone else say it yet, if you are going to use a 12
> digit approximation, how about using 3.14159265359 which is an even better
> approximation than the one you use :-).
>
> I find "3" is accurate enough for many purposes, though.
Quite true. Making sausage rolls the pastry needs to be three widths
of sausage and a bit. The hardest thing to teach is when an
approximation is appropriate.
--
_/_/_/ _/ John Cartmell
_/ __/_/ _/_/ _/_/ using Acorn Risc PCs - and StrongARMed
_/ _/ _/ _/ _/ _/ _/ _/ UK designed and made - British software
_/_/ _/__/ _/ _/ _/ _/ supporting our own; even if it is the best ;-)
Mike Hall
In article <ant26224...@cartmell.demon.co.uk>, John Cartmell
<jo...@cartmell.demon.co.uk> wrote:
> In article <AF5DA45C...@sofluc.demon.co.uk>, Jonathan Cunningham
> <URL:mailto:j...@sofluc.demon.co.uk> wrote:
> >
> > In article <5gmqsn$b...@milo.vcn.bc.ca>,
> > mgi...@vcn.bc.ca (Mark Gibson) wrote:
> >
> > >Martin Ward (mar...@ward-hull.demon.co.uk) wrote:
> > >: Pi day
> > Since I haven't seen anyone else say it yet, if you are going to use a 12
> > digit approximation, how about using 3.14159265359 which is an even better
> > approximation than the one you use :-).
> >
> > I find "3" is accurate enough for many purposes, though.
>
> Quite true. Making sausage rolls the pastry needs to be three widths
> of sausage and a bit. The hardest thing to teach is when an
> approximation is appropriate.
Don't I recall that the Temple of Solomon had a golden "bath" which was 10
cubits in diameter, and a circle of 30 cubits "encompassed it round
about"??? It comes as a shock to many to find that the Bible says pi is
equal to 3 ...
Mike
==============================================
To be good is noble, but to teach others how to be good is nobler--
and less trouble! (Mark Twain)
==============================================
I'll look at yours if you'll look at mine!! Visit me at:
http://www.elite.net/~thehalls
==============================================
Dick Smith
In message <adinkin-ya0231800...@news.usa1.com> Aaron J. Dinkin wrote:
] This isn't just a humor group thing - Pi Day was actually observed in
] numerous elemantary schools.
]
] And it's 3.14, people. Not 3/14. And the 22d of July is 7/22.
Not in this country; the 22nd of July is 22/7/yy, although I'm aware that in
some countries that order is reversed.
Dick
--
=============================================================================
Dick Smith Acorn Risc PC di...@risctex.demon.co.uk
=============================================================================
Greffindel
John Cartmell (jo...@cartmell.demon.co.uk) wrote:
: In article <AF5DA45C...@sofluc.demon.co.uk>, Jonathan Cunningham
: <URL:mailto:j...@sofluc.demon.co.uk> wrote:
: >
: > In article <5gmqsn$b...@milo.vcn.bc.ca>,
: > mgi...@vcn.bc.ca (Mark Gibson) wrote:
: >
: > >Martin Ward (mar...@ward-hull.demon.co.uk) wrote:
: > >: Pi day
: >
: > Since I haven't seen anyone else say it yet, if you are going to use a 12
: > digit approximation, how about using 3.14159265359 which is an even better
: > approximation than the one you use :-).
: >
: > I find "3" is accurate enough for many purposes, though.
: Quite true. Making sausage rolls the pastry needs to be three widths
: of sausage and a bit. The hardest thing to teach is when an
: approximation is appropriate.
(Feel free to shoot me down if I've got a detail slightly bobbled, but...)
I read once (and subsequently verified) that with 3.1415927 as an
approximation for pi, you can compute the circumference of the earth
(or rather, a perfect sphere having the same diameter as the earth)
and be off by only 6 or 7 inches... So it really begs the question--
when, if ever, are we likely to need more than 10 digits of pi?
For the sake of argument, consider space travel. Here's a rather
stupid story problem on the astronomical scale: The obscure star
Smurfulus Eta is 1000 light-years from the centre of the galaxy. As
the galaxy makes one full rotation about its own axis, how far does it
travel? (Circular orbit, by the way.)
So, it's pi x 1000 light-years, right?
Speed of light: 3x10E8 m/s
Seconds in a year: 60 x 60 x 24 x 365.25 = 31557600
sec/min min/hour hour/day day/year
m in a light-year: 9467280000000000 =9.46728 x 10E15 (I'm doing this
all w/pencil and paper, so hit me if I biff the arithmetic).
m in 1000 light-years: 9.46728 X 10E18
Now, running a little c program on my home computer,
3.141592653589793239 is as much of pi as I can get without inventing
my own quadruple=precision storage class, so let's see what that gives
us:
(much later)
2.852008848434121673571912 x 10E19, which in real numbers is
28520088484341216735.1912 meters.
18 decimal digits of pi gives you the distance correct down to the
millimeter. How much more pi do you need?
: --
: _/_/_/ _/ John Cartmell
: _/ __/_/ _/_/ _/_/ using Acorn Risc PCs - and StrongARMed
: _/ _/ _/ _/ _/ _/ _/ _/ UK designed and made - British software
: _/_/ _/__/ _/ _/ _/ _/ supporting our own; even if it is the best ;-)
--
--Greffindel the Plaid
/\/ \/ \/\
/ ( O O ) \ Art consists of drawing the line somewhere.
/ */\_/\_/\__\_ --G. K. Chesterton
/ *( o \
|*\ --v-v-v-v | Logic is like a one-legged duck,
\*\==\ Able to paddle only in circles.
\*\==\ --Ancient Chinese Secret
.sig |*|==|
V2.0 |*|==| I always think I'm going to break
|*|==| But in the end I only bend.
http://eagle.cc.ukans.edu/~ipsifend/dragon.html *NO SOLICITING ON PREMISES*
John Ross
Aaron J. Dinkin wrote <adinkin-ya0231800...@news.usa1.com>:
> This isn't just a humor group thing - Pi Day was actually observed in
> numerous elemantary schools.
>
> And it's 3.14, people. Not 3/14. And the 22d of July is 7/22.
Not (I believe) where this thread started.
We're not all back to front. :-)
--
John Ross
Southampton
John Ross
Mike Hall wrote <thehalls-260...@news.elite.net>:
> <jo...@cartmell.demon.co.uk> wrote:
> > Quite true. Making sausage rolls the pastry needs to be three widths
> > of sausage and a bit. The hardest thing to teach is when an
> > approximation is appropriate.
>
> Don't I recall that the Temple of Solomon had a golden "bath" which was 10
> cubits in diameter, and a circle of 30 cubits "encompassed it round
> about"??? It comes as a shock to many to find that the Bible says pi is
> equal to 3 ...
Perhaps He created the universe more curved than we know it. Now He's
loosening or losing his grip. Either way space is becoming less
curved. Discuss. :-)
--
John Ross
Southampton
Martin Ward
In article <slrn5jd935....@mohin.cse.iitb.ernet.in>, Sitaram
Iyer <sit...@mohin.cse.iitb.ernet.in> writes
When I tried this iteration 1/b[i] did not converge on pi at all.
Perhaps I misunderstood. Can anyone make the statement of the iteration
clearer?
--
Martin Ward
In article <adinkin-ya0231800...@news.usa1.com>, "Aaron J.
Dinkin" <adi...@commschool.org> writes
>In article <199703211...@isdn78.hrz.uni-siegen.de>,
>holg...@studm.hrz.uni-siegen.de (Kuddel) wrote:
>
>>Martin Ward <mar...@ward-hull.demon.co.uk> wrote:
>>
>>> Pi day
>>>
>>
>
>This isn't just a humor group thing - Pi Day was actually observed in
>numerous elemantary schools.
>
>And it's 3.14, people. Not 3/14. And the 22d of July is 7/22.
>
>Dr. Whom
You missed the beginning of the thread. The "someone" knew very well
that pi is close to 3.14, and why 14th March was proposed for "Pi Day",
it was a quip.
And 22nd July may be 7/22 if you are an American, but us Europeans
mostly write it the other way round. I won't clog up the bandwidth by
re-posting my origin article so soon, but I will e-mail it to you.
--
Best wishes
Martin Ward
Martin Ward
In article <33375C6B...@owlnet.rice.edu>, Robert James Aguirre
<ro...@owlnet.rice.edu> writes
Yes, you are right, given the Axiom of Choice any set can be well-
ordered. However, we cannot assume that the "smallest" element in the
reals under an arbitrarily chosen well-ordering will be interesting,
since this is tantamount to assuming that every real is interesting.
(Given a well ordering of the reals we can easily construct another with
any desired smallest element). Petitio principii.
This also applies to the rationals, of course - I was sloppy there, it
is too long since I really thought about this sort of thing. However,
the natural numbers are well-ordered by the usual order <, and 1 is
interesting - this is what Ulrich's argument refers to - and the
rationals can be well-ordered by a bijection with the natural numbers
which maps interesting rationals on to 1,2,... and it is that that I was
thinking about above. The reals cannot be well-ordered in any such
simple way.
--
Martin Ward
_________________________________________________________________________
"When *I* use a word", Humpty Dumpty said in rather a scornful tone, "it
means just what I choose it to mean - neither more nor less".
"The question is", said Alice, "whether you *can* make words mean so many
different things".
"The question is", said Humpty-Dumpty, "which is to be master - that's all".
"Through the Looking-Glass" - Lewis Carroll.
_________________________________________________________________________
Turnpike evaluation. For information, see http://www.turnpike.com/
Dik T. Winter
In article <5heh6s$o...@raven.cc.ukans.edu> ipsi...@eagle.cc.ukans.edu (Greffindel) writes:
> So, it's pi x 1000 light-years, right?
> Speed of light: 3x10E8 m/s
> Seconds in a year: 60 x 60 x 24 x 365.25 = 31557600
> sec/min min/hour hour/day day/year
> m in a light-year: 9467280000000000 =9.46728 x 10E15 (I'm doing this
> all w/pencil and paper, so hit me if I biff the arithmetic).
> m in 1000 light-years: 9.46728 X 10E18
> Now, running a little c program on my home computer,
> 3.141592653589793239 is as much of pi as I can get without inventing
> my own quadruple=precision storage class, so let's see what that gives
> us:
> 2.852008848434121673571912 x 10E19, which in real numbers is
> 28520088484341216735.1912 meters.
> 18 decimal digits of pi gives you the distance correct down to the
> millimeter. How much more pi do you need?
Are you sure? I think the error in the speed of light and in the number
of days in a year is much too large to warrant this assertion.
It is true, in real life 22/7 is mostly good enough, and if not you
go to 355/133 which probably is excellent (it would be good enough
in the calculation above). On the other hand, there are indeed
situations where you need an extremely good approximation of pi, or
your calculations can go wrong. There are highly unstable calculations
(like some of the higher zeros of the Riemann zeta function) that
require this. (In the latter case to verify that all non-trivial zeros
are on a particular line, you need to separate them. The problem is
that they can get arbitrarily close to each other. And the calculations
involve cosines.)
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
Reg Braddock
On Tue, 18 Mar 1997 23:56:55 +0100, Perry Winkel <win...@worldonline.nl> wrote:
>In article <01bc3440$d8a87040$LocalHost@default>, "Belinda Berry"
><BJB...@onaustralia.com.au> wrote:
>
>> How do you actually calculate pi anyway. (without loooking it up in a
>> book.) I suppose if you drew up a really accurate circle and did some
>> measuring it might work, but nothing I draw could possibly get more than
>> about 2 decimal places.
>...as I recall from long ago in high school, there is in fact a very easy
>way to determine pi. You need a stick (a pencil works fine also) and a
>floor with a pattern of square tiles. What you do is this:
>
Or borrow a trick from monte carlo integration. On a computer,
generate two sets of random variables, X and Y which are uniformly
distributed [0,1]. Taking each x and y and calculate the distance
between the origin (0,0) and the point defined by x and y. If it's
less than 1, increment a counter, say INSIDE otherwise one called
OUTSIDE. After a suitably large number of (x,y), PI can be determined
by 4* (INSIDE/(INSIDE+OUTSIDE)). In effect, it would be the
integration of the area inside a unit circle which we know to have an
area of PI*r*r.
Reg.
--
----------------------------------------------------------
Friends help you move. Real friends help you move bodies.
Greffindel
John Ross (jo...@jross.demon.co.uk) wrote:
: Mike Hall wrote <thehalls-260...@news.elite.net>:
: > <jo...@cartmell.demon.co.uk> wrote:
Hey, why not? I've always liked these theories where everything in
the universe is slowly but steadily decaying... we know space is
expanding-- maybe that extra expansion is affecting curvature; and for
that matter, how about the speed of light? Sure, it's basically
constant, but what if over the long term that's decaying too?
(no joke: I heard one of the oxymorons known as Creation Scientists
claim once that he had proved that the speed of light was undergoing a
logarithmic decay, and that tracing the curve backwards it rises to an
asymptote at right around 4004 B. C.... a good laugh was had by all)
Anyway, hey, the Bible's not a math text; apparently 3 was a good
enough approximation for Solomon. Or maybe it was about 10 cubits
across, give or take a cubit, and about 30 cubits around, give or take
a cubit... It would be extensively silly to take this passage (as
some have done) to mean The Bible Says Pi = 3.000 ...
: --
: John Ross
: Southampton
Greffindel
R.W.Dirksen (R.W.D...@caiw.nl) wrote:
: Dik T. Winter wrote:
: >
: > > So, it's pi x 1000 light-years, right?
: > > Speed of light: 3x10E8 m/s
: > > 18 decimal digits of pi gives you the distance correct down to the
: > > millimeter. How much more pi do you need?
: >
: > Are you sure? I think the error in the speed of light and in the number
: > of days in a year is much too large to warrant this assertion.
Well, presumably anyone who was doing the calculation for real would want
as many significant figures for the speed of light and the number of
seconds in a year; I just picked easy numbers to facilitate doing the
multiplication by hand... (in fact, odds are there's at least one
error in my multiplication as big as any caused by c being
2.9something x 10E8 rather than 3.
: >
: > It is true, in real life 22/7 is mostly good enough, and if not you
: > go to 355/133 which probably is excellent (it would be good enough
: > in the calculation above). On the other hand, there are indeed
: > situations where you need an extremely good approximation of pi, or
: > your calculations can go wrong. There are highly unstable calculations
: > (like some of the higher zeros of the Riemann zeta function) that
: > require this. (In the latter case to verify that all non-trivial zeros
: > are on a particular line, you need to separate them. The problem is
: > that they can get arbitrarily close to each other. And the calculations
: > involve cosines.)
Cool. I can buy that.
: > --
: > dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
: > home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
: The speed of light is known very exact because it is a constant value.
: They actually made it a definition:
: 1 meter = the distance that light travels in 1/299792458 seconds
: Therefore, the light speed c is defined as 299,792.458 km/s.
Ah, there's the number. :)
: p.s. why celebrate pi-day anyway. I personally like the number
: 0.3157894736842105263157894736842105263157894736842105263157894736...
: or 6/19. That's why I celebrate that day instead!
: Gerben Dirksen
Happy 6/19 day, then! :)
Greffindel
Jonathan Cunningham (j...@sofluc.demon.co.uk) wrote:
: In article <5gmqsn$b...@milo.vcn.bc.ca>,
: mgi...@vcn.bc.ca (Mark Gibson) wrote:
: >Martin Ward (mar...@ward-hull.demon.co.uk) wrote:
: >: Pi day
: >:
: >:
: (snip)
: >
: >3.14159265379 is a closer approximation to pi that I usually use. But that
: Since I haven't seen anyone else say it yet, if you are going to use a 12
: digit approximation, how about using 3.14159265359 which is an even better
: approximation than the one you use :-).
: I find "3" is accurate enough for many purposes, though.
: Be careful if you are determining pi by measuring the ratio of the
: circumference of a circle to its diameter: a one tonne weight in a 10 metre
: diameter circle will affect the value of pi in about the 24th or 25th
: decimal place (general relativity and curvature of space-time etc.). If
: you wish to try this, it's worth knowing that the correct value of
: pi should be around 3.14159265358979323846264338 in flat space, although
: the last two or three digits will be different with the one tonne weight
: in place. :-).
: Jonathan
(thinks) (thinks) (thinks) You know, that makes sense. What could
you say about the set of weights such that under their influence, the
"effective pi" would become rational? Algebraic? How big a weight
would you need to get a "pi" of 4?
R.W.Dirksen
Dik T. Winter wrote:
>
> In article <5heh6s$o...@raven.cc.ukans.edu> ipsi...@eagle.cc.ukans.edu (Greffindel) writes:
> > So, it's pi x 1000 light-years, right?
> > Speed of light: 3x10E8 m/s
> > sec/min min/hour hour/day day/year
> > m in a light-year: 9467280000000000 =9.46728 x 10E15 (I'm doing this
> > all w/pencil and paper, so hit me if I biff the arithmetic).
> > m in 1000 light-years: 9.46728 X 10E18
> > Now, running a little c program on my home computer,
> > 3.141592653589793239 is as much of pi as I can get without inventing
> > my own quadruple=precision storage class, so let's see what that gives
> > us:
> ...
> > 2.852008848434121673571912 x 10E19, which in real numbers is
> > 28520088484341216735.1912 meters.
> > millimeter. How much more pi do you need?
>
> Are you sure? I think the error in the speed of light and in the number
> of days in a year is much too large to warrant this assertion.
>
> go to 355/133 which probably is excellent (it would be good enough
> in the calculation above). On the other hand, there are indeed
> situations where you need an extremely good approximation of pi, or
> your calculations can go wrong. There are highly unstable calculations
> (like some of the higher zeros of the Riemann zeta function) that
> require this. (In the latter case to verify that all non-trivial zeros
> are on a particular line, you need to separate them. The problem is
> that they can get arbitrarily close to each other. And the calculations
> involve cosines.)
> dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
> home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
The speed of light is known very exact because it is a constant value.
They actually made it a definition:
1 meter = the distance that light travels in 1/299792458 seconds
Therefore, the light speed c is defined as 299,792.458 km/s.
p.s. why celebrate pi-day anyway. I personally like the number
Dik T. Winter
In article <333C26...@caiw.nl> "R.W.Dirksen" <R.W.D...@caiw.nl> writes:
> > > Speed of light: 3x10E8 m/s
> > Are you sure? I think the error in the speed of light and in the number
> > of days in a year is much too large to warrant this assertion.
> The speed of light is known very exact because it is a constant value.
> They actually made it a definition:
>
> 1 meter = the distance that light travels in 1/299792458 seconds
> Therefore, the light speed c is defined as 299,792.458 km/s.
I know. So the assumed speed of light had an error exceeding that for
pi. And what is that to the discussion?
Eric Roy White
>In article <5gmqsn$b...@milo.vcn.bc.ca>,
>mgi...@vcn.bc.ca (Mark Gibson) wrote:
>
>>Martin Ward (mar...@ward-hull.demon.co.uk) wrote:
>>
>>3.14159265379 is a closer approximation to pi that I usually use. But that
>
>Since I haven't seen anyone else say it yet, if you are going to use a 12
>digit approximation, how about using 3.14159265359 which is an even better
>approximation than the one you use :-).
>
>I find "3" is accurate enough for many purposes, though.
>
>Be careful if you are determining pi by measuring the ratio of the
>circumference of a circle to its diameter: a one tonne weight in a 10 metre
>diameter circle will affect the value of pi in about the 24th or 25th
>decimal place (general relativity and curvature of space-time etc.). If
>you wish to try this, it's worth knowing that the correct value of
>pi should be around 3.14159265358979323846264338 in flat space, although
>the last two or three digits will be different with the one tonne weight
>in place. :-).
>
>Jonathan
>
¿ not sure of the method you are describing for measuring the
Circumference of the Sphere?
¿ how accurate have real tools for measuring Circumference and
Diameter?
the best ruler i have seen is only 1 decimal places with the 2 decimal
place being approx.
__________________________________________________________________
thanks for your time & patience
¨¨°º©o¿,,¿o©º°¨¨°º© Keep Those Smiles Rolling ©º°¨¨°º©o¿,,¿o©º°¨¨
John Ross
Greffindel wrote <5hhf8b$4...@raven.cc.ukans.edu>:
> (no joke: I heard one of the oxymorons known as Creation Scientists
> claim once that he had proved that the speed of light was undergoing a
> logarithmic decay, and that tracing the curve backwards it rises to an
> asymptote at right around 4004 B. C.... a good laugh was had by all)
Strange chap, this Greffindel. Next thing you know he'll suggest the
world isn't flat, the Earth goes round the sun, and man and ape are
cousins. :-)
--
John Ross
Southampton
Martin Ward
In article <5hhge0$4...@raven.cc.ukans.edu>, Greffindel
<ipsi...@eagle.cc.ukans.edu> writes
>Jonathan Cunningham (j...@sofluc.demon.co.uk) wrote:
-- snip --
>: Be careful if you are determining pi by measuring the ratio of the
>: circumference of a circle to its diameter: a one tonne weight in a 10 metre
>: diameter circle will affect the value of pi in about the 24th or 25th
>: decimal place (general relativity and curvature of space-time etc.). If
>: you wish to try this, it's worth knowing that the correct value of
>: pi should be around 3.14159265358979323846264338 in flat space, although
>: the last two or three digits will be different with the one tonne weight
>: in place. :-).
>
>: Jonathan
>
>you say about the set of weights such that under their influence, the
>"effective pi" would become rational? Algebraic? How big a weight
>would you need to get a "pi" of 4?
>
>--
>--Greffindel the Plaid
Jonathan obviously understands the physics of this better than I do, but
putting in a weight must *reduce* the local "pi", surely?
Since there are algebraic and rational numbers arbitrarily close to any
transcendental number like pi the same would be true of the set of
values of our central mass. If we can assign it a random real value
then most such would leave "pi" transcendental, but we could find
infinitely many values within any range >0 which make "pi" rational.
Of course quantum mechanics says, does it not, that masses cannot be
given an arbitrary real value but only one chosen from a set of discrete
values....
--
Martin Ward
_________________________________________________________________________
"When *I* use a word", Humpty Dumpty said in rather a scornful tone, "it
means just what I choose it to mean - neither more nor less".
"The question is", said Alice, "whether you *can* make words mean so many
different things".
"The question is", said Humpty-Dumpty, "which is to be master - that's all".
"Through the Looking-Glass" - Lewis Carroll.
_________________________________________________________________________
Turnpike evaluation. For Turnpike information, mailto:in...@turnpike.com
Vincent Johns
(posted & emailed)
Tony Buckland <buck...@unixg.ubc.ca> wrote:
>
> In article <5gmrkk$d00$1...@news-s01.ny.us.ibm.net>, <cha...@ibm.net> wrote:
> >In <AF53F4E8...@brierley.demon.co.uk>, ste...@brierley.demon.co.uk (Stephen Brierley) writes:
> >Another idea for an imaginary date--
> >How about any date in the year 0 (zero)?
>
> If only there really had been a year zero.
> Then we wouldn't have to be concerned about whether the new
> century and millenium begin Jan 1 2000 or Jan 1 2001.
It probably wouldn't have helped much. Would you then
have called year 0 the "first" one or the "zeroth" one?
If "first", then 1 would be the "second" year, etc., sort
of invoking an off-by-one error in naming.
If you called it the "zeroth" year, then AD 1 would still
be the "first" year (same as with the current numbering
system), etc., and AD 2000 would still be the 2000th year
(and 1901 would still have begun the 20th century).
--
-- Vincent Johns
Please feel free to quote anything I say here.
Vincent Johns
(posted & emailed)
Ron Bauerle <bau...@erie.ge.com> wrote:
>
> In article <5gk8lm$q...@opus.vcn.bc.ca>, land...@vcn.bc.ca (Scott Phung) writes:
> >
> >Isn't it ironic how the imaginary number raised to the imaginary
> >power equals a real number? (Involving the important constants pi,
> >and e) :-)
>
> Anybody got a proof for that? i^i = exp(-pi/2)?
What about this?
i = e^(i*pi/2) = cos(pi/2) + i*sin(pi/2)
i^i = e^(i^2*pi/2)
= e^(-pi/2)
= exp(-pi/2)
Greffindel
John Ross (jo...@jross.demon.co.uk) wrote:
: Greffindel wrote <5hhf8b$4...@raven.cc.ukans.edu>:
Well, now, I wouldn't go *that* far. :)
: --
: John Ross
: Southampton
--Greffindel the Plaid; after al, we must never go to extremes... :)
/\/ \/ \/\
/ ( O O ) \ Art consists of drawing the line somewhere.
/ */\_/\_/\__\_ --G. K. Chesterton
/ *( o \
|*\ --v-v-v-v | Logic is like a one-legged duck,
\*\==\ Able to paddle only in circles.
\*\==\ --Ancient Chinese Secret
.sig |*|==|
V2.0 |*|==| Anything worth doing is worth doing well.
|*|==| --Lizzie Borden
Greffindel
Martin Ward (mar...@ward-hull.demon.co.uk) wrote:
: In article <5hhge0$4...@raven.cc.ukans.edu>, Greffindel
: <ipsi...@eagle.cc.ukans.edu> writes
: >Jonathan Cunningham (j...@sofluc.demon.co.uk) wrote:
: -- snip --
: >: Be careful if you are determining pi by measuring the ratio of the
: >: circumference of a circle to its diameter: a one tonne weight in a 10 metre
: >: diameter circle will affect the value of pi in about the 24th or 25th
: >: decimal place (general relativity and curvature of space-time etc.). If
: >: you wish to try this, it's worth knowing that the correct value of
: >: pi should be around 3.14159265358979323846264338 in flat space, although
: >: the last two or three digits will be different with the one tonne weight
: >: in place. :-).
: >
: >: Jonathan
: >
: >(thinks) (thinks) (thinks) You know, that makes sense. What could
: >you say about the set of weights such that under their influence, the
: >"effective pi" would become rational? Algebraic? How big a weight
: >would you need to get a "pi" of 4?
: Jonathan obviously understands the physics of this better than I do, but
: putting in a weight must *reduce* the local "pi", surely?
Better than I as well, I'm sure. It also depends on what one is
calling 'pi'. I'm sure the Leibniz series still converges to the same
result even near neutron stars. But yes, if space is curved by a
large weight, than the linear distance you'd travel to get from one
point on a circle to its antipode would be longer, therefore the
ratio of the circumference to that would be smaller. My error--
sorry. :)
: Since there are algebraic and rational numbers arbitrarily close to any
: transcendental number like pi the same would be true of the set of
: values of our central mass. If we can assign it a random real value
: then most such would leave "pi" transcendental, but we could find
: infinitely many values within any range >0 which make "pi" rational.
: Of course quantum mechanics says, does it not, that masses cannot be
: given an arbitrary real value but only one chosen from a set of discrete
: values....
: --
See, that's what I'm wondering. I suppose the answer is that since
mass is quantised, it makes a huge amount of sense to map the set of
allowable masses to the set of integers, but on the other hand, you
could specifically assign a transcendental 'number' to the smallest
possible unit of mass, and than the masses of all objects would be
transcendental. *but*, on the other hand, our distortion of pi is
based on measured distances in space, a lot depends (doesn't it?) on
the exact mathematical relation between mass and distortion.
: Martin Ward
: _________________________________________________________________________
: "When *I* use a word", Humpty Dumpty said in rather a scornful tone, "it
: means just what I choose it to mean - neither more nor less".
: "The question is", said Alice, "whether you *can* make words mean so many
: different things".
: "The question is", said Humpty-Dumpty, "which is to be master - that's all".
: "Through the Looking-Glass" - Lewis Carroll.
: _________________________________________________________________________
: Turnpike evaluation. For Turnpike information, mailto:in...@turnpike.com
--Greffindel the Plaid
/\/ \/ \/\
/ ( O O ) \ Art consists of drawing the line somewhere.
/ */\_/\_/\__\_ --G. K. Chesterton
/ *( o \
|*\ --v-v-v-v | Logic is like a one-legged duck,
\*\==\ Able to paddle only in circles.
\*\==\ --Ancient Chinese Secret
.sig |*|==|
V2.0 |*|==| You can lead a horse to water, but you can't rollerskate
|*|==| in a buffalo herd. --Various
Earle Jones
In article <5hmksr$v...@raven.cc.ukans.edu>, ipsi...@eagle.cc.ukans.edu
(Greffindel) wrote:
>Martin Ward (mar...@ward-hull.demon.co.uk) wrote:
>: In article <5hhge0$4...@raven.cc.ukans.edu>, Greffindel
>: <ipsi...@eagle.cc.ukans.edu> writes
>: >Jonathan Cunningham (j...@sofluc.demon.co.uk) wrote:
>: -- snip --
>: >: Be careful if you are determining pi by measuring the ratio of the
>: >: circumference of a circle to its diameter: a one tonne weight in a 10
metre
[...clippit here...]
>: >
>: >(thinks) (thinks) (thinks) You know, that makes sense. What could
>: >you say about the set of weights such that under their influence, the
>: >"effective pi" would become rational? Algebraic? How big a weight
>: >would you need to get a "pi" of 4?
>: >--Greffindel the Plaid
Consider this: The mass and/or temperature required to produce pi = 4 is
unthinkable. You are going the wrong way.
In the frozen north (Fairbanks, Alaska) where everything shrinks with the
lowered temperature, it is known that pi = 3.
We call it "Eskimo pi".
earle
__
__/\_\
/\_\/_/
\/_/\_\ earle
\/_/ jones
Jason Woolever
> so i^i = (e^(i*pi/2))^i
> = e^(i*i*pi/2)
> = e^(-pi/2)
Not entirely true. You did not proof uniqueness. There are
infinately many solutions.
- Jason
The Silicon Surfer
Martin Ward wrote:
>
> In article <5hhge0$4...@raven.cc.ukans.edu>, Greffindel
> <ipsi...@eagle.cc.ukans.edu> writes
> >Jonathan Cunningham (j...@sofluc.demon.co.uk) wrote:
> -- snip --
> >: Be careful if you are determining pi by measuring the ratio of the
> >: circumference of a circle to its diameter: a one tonne weight in a 10 metre
> >: decimal place (general relativity and curvature of space-time etc.). If
> >: you wish to try this, it's worth knowing that the correct value of
> >: pi should be around 3.14159265358979323846264338 in flat space, although
> >: the last two or three digits will be different with the one tonne weight
> >: in place. :-).
> >
> >: Jonathan
> >
> >you say about the set of weights such that under their influence, the
> >"effective pi" would become rational? Algebraic? How big a weight
> >would you need to get a "pi" of 4?
> >
> >--Greffindel the Plaid
>
> Jonathan obviously understands the physics of this better than I do, but
> putting in a weight must *reduce* the local "pi", surely?
>
> transcendental number like pi the same would be true of the set of
> values of our central mass. If we can assign it a random real value
> then most such would leave "pi" transcendental, but we could find
> infinitely many values within any range >0 which make "pi" rational.
>
> Of course quantum mechanics says, does it not, that masses cannot be
> given an arbitrary real value but only one chosen from a set of discrete
> values....
> --
>
>
>
Hmmmm, hadn't thought of Pi as dependent on local conditions, but of
course it's true.
Suppose the one ton weight was a singularity with an mass x and an
effective diameter of 1 millimetre. This could be easily drawn and
visualised but how would you work out the radius/circumfrence.
Also, if my memory serves, object travelling at very near "c" will be
"stretched" along the axis of travel, so the orientation of a circle to
said axis would presumably affect pi, or maybe not as it would no longer
be a circle.
OK, my head hurts.
JB
Chris Dearlove
Greffindel (ipsi...@eagle.cc.ukans.edu) wrote:
: (Feel free to shoot me down if I've got a detail slightly bobbled, but...)
: I read once (and subsequently verified) that with 3.1415927 as an
: approximation for pi, you can compute the circumference of the earth
: (or rather, a perfect sphere having the same diameter as the earth)
: and be off by only 6 or 7 inches... So it really begs the question--
: when, if ever, are we likely to need more than 10 digits of pi?
There's more to pi than circles, and numerical calculations which use pi
in other contexts may need more accuracy.
A colleague described to me once a problem he once found (in someone
else's code, and not while he worked here) where pi had been accidentally
coded to too few places. (In some Fortran - all the places were there,
the "D0" after the "3.1415926535..." was missing). This caused a noticeable
ripple (maybe 140 dB down) when numerically simulating a signal of some
sort. This was sufficiently large to matter.
Of course here just a few more digits would have been enough, maybe 10
would have done. (I have seen cases where a few more than that are a
good idea in general precision requirements.)
--
Christopher M. Dearlove, | chris.d...@gecm.com
GEC-Marconi Research Centre, | Tel: 01245 242194 Int: +44 1245 242194
Gt. Baddow, Chelmsford, CM2 8HN, UK | Fax: 01245 242124 Int: +44 1245 242124
cha...@ibm.net
In <adinkin-ya0231800...@news.usa1.com>, adi...@commschool.org (Aaron J. Dinkin) writes:
>In article <199703211...@isdn78.hrz.uni-siegen.de>,
>holg...@studm.hrz.uni-siegen.de (Kuddel) wrote:
>
>>Martin Ward <mar...@ward-hull.demon.co.uk> wrote:
>>
>>> Pi day
>>>
>>> I saw a posting in one of the humour groups to the effect that 14 March
>>> is Pi Day because 3/14 was a good approximation to pi. Someone had
>>> replied that 3/14 was closer to 0.2142857...
>>
>>Pi can be approximated by 22/7 (that's what the anciet Greeks did, when
>>they build wheels) So we 22nd of July should be the Pi day, shouldn't
>>it?
>
>This isn't just a humor group thing - Pi Day was actually observed in
>numerous elemantary schools.
>
>And it's 3.14, people. Not 3/14. And the 22d of July is 7/22.
>
>-Aaron J. Dinkin
>Dr. Whom
Some places on this earth use date/month/year--even year/month/date,
not the US (and some other places) standard month/date/year.
Therefore, 22 July would be 22/7 in those places.
Robert R. Chapman, Jr.
cha...@ibm.net
Lynnwood, WA 98036 USA
**********************************************************
It's interesting how superstition thrives in unsettled times,
and how many are prepared to listen to it, at least with
half an ear.
--Dietrich Bonhoeffer
Letters and Papers from Prison, The Enlarged Edition
(New York: Macmillan Paperbacks Edition 1972), page 153
**********************************************************
Mike McCarty
In article <E7s8D...@cwi.nl>, Dik T. Winter <d...@cwi.nl> wrote:
)In article <333C26...@caiw.nl> "R.W.Dirksen" <R.W.D...@caiw.nl> writes:
) > > > Speed of light: 3x10E8 m/s
)...
) > > Are you sure? I think the error in the speed of light and in the number
) > > of days in a year is much too large to warrant this assertion.
)...
) > The speed of light is known very exact because it is a constant value.
) > They actually made it a definition:
) >
) > 1 meter = the distance that light travels in 1/299792458 seconds
) > Therefore, the light speed c is defined as 299,792.458 km/s.
)
)I know. So the assumed speed of light had an error exceeding that for
)pi. And what is that to the discussion?
This speed of light is neither assumed, nor has an error. It is -exact-.
Mike
--
----
char *p="char *p=%c%s%c;main(){printf(p,34,p,34);}";main(){printf(p,34,p,34);}
This message made from 100% recycled bits.
I don't speak for DSC. <- They make me say that.
cha...@ibm.net
In <19970327....@jross.demon.co.uk>, jo...@jross.demon.co.uk (John Ross) writes:
>Mike Hall wrote <thehalls-260...@news.elite.net>:
>
>> <jo...@cartmell.demon.co.uk> wrote:
>
>> > Quite true. Making sausage rolls the pastry needs to be three widths
>> > of sausage and a bit. The hardest thing to teach is when an
>> > approximation is appropriate.
>>
>> Don't I recall that the Temple of Solomon had a golden "bath" which was 10
>> cubits in diameter, and a circle of 30 cubits "encompassed it round
>> about"??? It comes as a shock to many to find that the Bible says pi is
>> equal to 3 ...
>
>Perhaps He created the universe more curved than we know it. Now He's
>loosening or losing his grip. Either way space is becoming less
>
>--
>John Ross
>Southampton
Actually, in English translation we miss something.
There are slight differences in the sizes for the sea (bath) in the
Hebrew original in the two biblical books. When these numbers are set in
a ratio, you end up with something like 3.14--not 3!
Unfortunately, I don't remember where I picked this up. I am not
learned in ancient Hebrew. I picked the following up in reading a journal
article somewhere. Is there a Hebrew scholar or historian or a
mathmatician out there that can help what is probably an over 10-year
memory?
Thanks.
Bruce Cox
cha...@ibm.net writes:
> Actually, in English translation we miss something.
>
> There are slight differences in the sizes for the sea (bath) in the
> Hebrew original in the two biblical books. When these numbers are set in
> a ratio, you end up with something like 3.14--not 3!
>
> Unfortunately, I don't remember where I picked this up. I am not
> learned in ancient Hebrew. I picked the following up in reading a journal
> article somewhere. Is there a Hebrew scholar or historian or a
> mathmatician out there that can help what is probably an over 10-year
> memory?
There is someone who has a copy of a similar article :)
\begin{quote}
The key to an alternative reading of the verse 1 Kings 7:23 is to be found
in the very ancient Hebrew tradition to differently write (spell) and read
some words of the Bible; the reading version is usually regarded as a
correct one (in particular, it is always correct from the point of view of
the Hebrew grammar, and this is why it could be easily either remembered or
reconstructed from the written version), whereas the written version slightly
deviates from the correct spelling.
...
In our case there is such a disparity for the word "line": in Hebrew, it is
written as "QVH (Qof, Vav, Hea)", but it has to be read as "QV (Qof, Vav)".
...
Tradition asserts that not only does this disparity testify to an approximate
character of the given length of the line circling around the "sea" (tank),---
a much more accurate approximation to \pi is hidden in the choice of the
written version.
\end{quote}
This is then followed by some numerology, where the standard numbering of
the Hebrew letters gives numerical equivalents QVH = 100 + 6 + 5 = 111
and QV = 100 + 6 = 106. This is then used to compute the value:
\pi_{Hebrew} = 3 * 111/106 ~ 3.1415094...
In the parallel passage (2 Chron 4:2) it is observed that the text is
almost verbatim the same as the above, except that there is no disparity
between the reading and written versions. A suggested explanation is
that the much later chronicler was 'more preopccupied with rebuilding the
Temple and preserving the spirit of the Torah, than with the "correct" value
of \pi hidden in the descriptions of dimensions of the sacred object in the
First Temple'. The other paper presumably makes the same computation
based on the difference in spelling in the written version.
The title of this paper is "On the Rabbinical Exegesis of an Enhanced
Biblical Value of \pi", Shlomo Edward G. Belaga. A footnote declares
"An earlier version has appeared in the Proceedings of the XVIIth Canadian
Congress of History and Philosophy of mathematics, Queen's University,
Kingston, Ontario, May 27--29, 1991, pp 93--101." The paper which you
remember is probably A.S. Posamentier, N. Gordan, 1984: "An astounding
revelation on the history of \pi", The Mathematics Teacher 77 (1).
Cheers,
bruce
--
Bruce Cox br...@maths.usyd.edu.au
School of Mathematics and Statistics F07 +61 2 9351 3814
University of Sydney 2006
AUSTRALIA
Steve Ryder
>Hmmmm, hadn't thought of Pi as dependent on local conditions, but of
>course it's true.
>
>Suppose the one ton weight was a singularity with an mass x and an
>effective diameter of 1 millimetre. This could be easily drawn and
>visualised but how would you work out the radius/circumfrence.
>
>Also, if my memory serves, object travelling at very near "c" will be
>"stretched" along the axis of travel, so the orientation of a circle to
>said axis would presumably affect pi, or maybe not as it would no longer
>be a circle.
Someone pointed out to me that if pi was not equal to
3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510, then it
wouldn't be a circle, would it? corect me if I'm wrong...
--
Steve Ryder
<sry...@jersey.net>
http://www.jersey.net/~hryder/
go ahead and remove the "NOSPAM" to email me!
Glenn Baddeley
Chris Dearlove wrote:
> A colleague described to me once a problem he once found (in someone
> else's code, and not while he worked here) where pi had been accidentally
> coded to too few places. (In some Fortran - all the places were there,
> the "D0" after the "3.1415926535..." was missing). This caused a noticeable
> ripple (maybe 140 dB down) when numerically simulating a signal of some
> sort. This was sufficiently large to matter.
>
> Of course here just a few more digits would have been enough, maybe 10
> would have done. (I have seen cases where a few more than that are a
> good idea in general precision requirements.)
Pi as a constant in a program can be accurately assigned during
initialisation by calculation rather than coding the digits.
eg. Pi = arctan(1) * 4
Just make sure the real precision of the variable and arctan
function is sufficient.
eg. In Visual Basic
dim Pi as double
Pi = atn(1) * 4
Print Pi
3.14159265358979
This is the atn() example in the Language Reference book!
HTH,
Glenn.
Dick Smith
In message <oulo706...@maths.usyd.edu.au> Bruce Cox wrote:
[snip]
] \begin{quote}
[snip]
] \end{quote}
Spot the LaTeX user!
--
=============================================================================
Dick Smith Acorn Risc PC di...@risctex.demon.co.uk
=============================================================================
Mark Twells
In article
<adinkin-ya0231800...@news.usa1.com>,
adi...@commschool.org (Aaron J. Dinkin) wrote:
>And the 22d of July is 7/22.
Not in the UK, it isn't.
----
Mark
Greffindel
Steve Ryder (antagron@[NOSPAM]rocketmail.com) wrote:
: >Hmmmm, hadn't thought of Pi as dependent on local conditions, but of
: >course it's true.
: >
: >Suppose the one ton weight was a singularity with an mass x and an
: >effective diameter of 1 millimetre. This could be easily drawn and
: >visualised but how would you work out the radius/circumfrence.
: >
: >Also, if my memory serves, object travelling at very near "c" will be
: >"stretched" along the axis of travel, so the orientation of a circle to
: >said axis would presumably affect pi, or maybe not as it would no longer
: >be a circle.
: Someone pointed out to me that if pi was not equal to
: 3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510, then it
: wouldn't be a circle, would it? corect me if I'm wrong...
Well, that's true in two dimensions. In three or four dimensions,
things become more complicated. After all, think of meridian lines on
a globe-- they're parallel by many tests; yet they intersect at the
poles. But then, you can draw a triangle on a globe whose internal
angles add to 270 degrees... on a curved surface or in a curved space,
geometry gets non-euclidean, and life gets funky.
: --
: Steve Ryder
: <sry...@jersey.net>
: http://www.jersey.net/~hryder/
: go ahead and remove the "NOSPAM" to email me!
--
--Greffindel the Plaid
/\/ \/ \/\
/ ( O O ) \ Art consists of drawing the line somewhere.
/ */\_/\_/\__\_ --G. K. Chesterton
/ *( o \
|*\ --v-v-v-v | Logic is like a one-legged duck,
\*\==\ Able to paddle only in circles.
\*\==\ --Ancient Chinese Secret
.sig |*|==|
V2.0 |*|==| Confucius say: Glass arways gleenel where
|*|==| honolabre dog make doody.
http://eagle.cc.ukans.edu/~ipsifend/dragon.html *NO E-MAIL ADVERTISEMENTS
TOLERATED* *REMOVE THIS ADDRESS FROM ALL COMMERCIAL MAILING LISTS*
Greffindel
The Silicon Surfer (surf...@mail.zynet.co.uk) wrote:
: Martin Ward wrote:
: >
: > In article <5hhge0$4...@raven.cc.ukans.edu>, Greffindel
: > <ipsi...@eagle.cc.ukans.edu> writes
: > >Jonathan Cunningham (j...@sofluc.demon.co.uk) wrote:
: > -- snip --
: > >: Be careful if you are determining pi by measuring the ratio of the
: > Jonathan obviously understands the physics of this better than I do, but
: > putting in a weight must *reduce* the local "pi", surely?
: >
: > Since there are algebraic and rational numbers arbitrarily close to any
: > transcendental number like pi the same would be true of the set of
: > values of our central mass. If we can assign it a random real value
: > then most such would leave "pi" transcendental, but we could find
: > infinitely many values within any range >0 which make "pi" rational.
: >
: > Of course quantum mechanics says, does it not, that masses cannot be
: > given an arbitrary real value but only one chosen from a set of discrete
: > values....
: > Martin Ward
: Hmmmm, hadn't thought of Pi as dependent on local conditions, but of
: course it's true.
: Suppose the one ton weight was a singularity with an mass x and an
: effective diameter of 1 millimetre. This could be easily drawn and
: visualised but how would you work out the radius/circumfrence.
Hmm. Well, Suppose you draw the circle on "flat" (in a 3-dimensional
sense) space. All nice and Euclidean, c = pi*d. Now, you drop your
neutron star or whatever smack in the centre (Oh, what an ugly
calculation if the mass is off-centre!), and because of the mass,
space stretches, sort of like this:
-------------_ _-------------
\ /
\ /
*
Now, if you do this with a rubber sheet (the most common model that I
know of), you notice that all of the rubber stretches *some*-- but the
outer edges don't stretch very much. The part right under the weight,
of course, stretches the most, but the variance is going to be
continuous-- there isn't a sharp line between stretched and
not-so-stretched rubber. So there's some continuous function that
tells you how much space has been stretched at any given point based
on the "distance" from the weight. I'd have to ask a physicist what
it was; it would make an interesting calculus problem. (OK, maybe a
boring calculus problem)
So our "circle" will get pulled inward a little bit (because the space
outside our circle stretches too-- just not as much as the space
inside). Therefore, the circumference is reduced a little bit [*]. But
the diameter changes far more drastically, since the diameter line
passes right through the centre of the circle (where the weight was
placed) and hence through the most stretched-out bits of space in the
whole experiment. That line, being stretched, gets longer. So,
taking pi as the ratio of circumference to diameter, you have a
smaller numerator, bigger denominator, and hence a smaller "pi".
-----------
[*] 'But how can the circle's circumference get smaller (squashed) if
space is getting stretched out?' Because space is getting stretched
along any axis which intersects the centre of the circle (where the
weight is). Along a perpendicular axis (the 'circle'-- or rather, any
tangent to the circle-- is always perpendicular to a radius of that
circle intersecting the point of tangency), space can and does get
squashed. Complicated!
-----------
: Also, if my memory serves, object travelling at very near "c" will be
: "stretched" along the axis of travel, so the orientation of a circle to
: said axis would presumably affect pi, or maybe not as it would no longer
: be a circle.
Squashed, actually, in the direction of travel-- it's called, or at
least was once called "Fitzgerald Contraction" (named for
mathematician Albert Contraction). But the result is an ellipse, and
the circumference is pi times the average of its major and minor axes.
This also reduces the circumference and therefore reduces "pi".
: OK, my head hurts.
: JB
--