🚀 go-pugleaf

JRS:  In article <Im0c0MABmjc3Ew2K@merlyn.demon.co.uk> of Thu, 24 Jun
1999 14:58:25 in news:comp.lang.pascal.delphi.misc, Dr John Stockton
<jrs@merlyn.demon.co.uk> wrote:
>
>It will be interesting to see whether you've managed to avoid using a
>value for Pi, implicitly or explicitly, in the work.

Having now had an opportunity to access the rather nice Web page, I see
that you decided to do so.

As a presentation of the Needles, it looks excellent.

As a means of determining Pi, however, there is a logical difficulty
when the DirectionMethods are considered.

Method 1 is, rightly, both presented and rejected.

Method 2, which AIUI works, considers a semicircle with a circumscribing
semisquare, generates evenly-distributed random points within the
semisquare, and accepts for further use those within the semicircle.
This virtually calculates Pi anyway, as the proportion accepted is Pi/2.
The technique is, BTW, close to that of a good way of generating random
points with a defined but uneven distribution : <URL: http://www.merlyn.
demon.co.uk/pas-rand.htm#RandDist>.

Method 3 uses Sin & Cos of random and adequately large angles,
ostensibly not using Pi.  But these functions require Pi internally,
since practical evaluation will require subtracting multiples of TwoPi
from the argument (assumed in radians; otherwise Pi can be deduced as
below).

Method 4 uses half-angle and angle-addition formulae to generate sines
and cosines of evenly-spaced angles, starting with a "unit" of one
right-angle - these can then be randomly chosen, though it would be
better for the final Buffon value to systematically use all of them (and
to systematically evenly choose the needle-centres).  But, since, for
small x in radians, sin(x)=x, the angle-halving is itself essentially
capable of determining the radian as a fraction of a right angle, which
determines Pi.


I wonder whether there is a method of doing Buffon which does not use Pi
(implicitly or explicitly), does not have a subset-capability of
determining Pi, and does not, as Buffon did, use the Real World?


ISTR that it is established that more rapid convergence is obtained if
you glue the needles together in pairs, at right angles - centrally is
usual but I think not necessary; more rapid even if one counts needles
rather than throws.

It must be better still to glue large numbers, evenly distributed in
angle, together - centrally to get a disc, or endwise to get a ring.
Not too many, though; with an infinite number of needles, unless
carefully treated, there's a risk of getting infinity/infinity as the
result.


I guess that the true intention was to draw attention to the services
you provide at efg's Computer Lab (http://www.efg2.com/lab) - ISTM that
it is working.
 
-- 
 � John Stockton, Surrey, UK.  jrs@merlyn.demon.co.uk   Turnpike v4.00   MIME. �
  Web <URL: http://www.merlyn.demon.co.uk/> - TP/BP/&c. FAQqish topics & links.
  Timo's TurboPascal <A HREF="ftp://garbo.uwasa.fi/pc/link/tsfaqp.zip">FAQ</A>.
  <A HREF="http://www.merlyn.demon.co.uk/clpb-faq.txt">Mini-FAQ</A> of c.l.p.b.