I have been wondering about this myself - and that wikipedia page is curiously silent about this issue.
It does give one hint, which is "Using Liu Hui's algorithm (which is based on the areas of regular polygons approximating a circle)" though the link supplied for that algorithm doesn't really help me. Still, we can define two regular polygons which bound a unit circle. The outer polygon touches the circle in the middle of each edge, the inner polygon touches the circle at each corner. Since the unit circle has a radius of 1, we know how to define these polygons without using pi, and we can approximate pi as the average of their perimeters divided by 2 (or their sum divided by 4). I don't quite see how that approach has anything to do with the sequences mentioned on the wikipedia page, but that's more disappointing than surprising. And, maybe, a comment on my lack of insight. Thanks, -- Raul On Wed, Jan 31, 2018 at 6:56 AM, Martin Kreuzer <[email protected]> wrote: > Seems that up til now I have been totally unaware of the existence of the > Stern-Brocot tree of fractions or the Mediant ... Thanks for opening up that > door for me. -- > > After some reading I now accept that going through the tree I could > approximate any real number with fractions (taking mediants on the way). But > doesn't that mean I have to know the target in the first place..? > > In the case of pi approximation (in Ancient Chinese Math) the real number > (target) isn't known ... > > If I started out with the interval [3;4] written as fractions 3/1 and 4/1 > (or 'coefficients' 3 1 and 4 1 as they called it, stating > circumference-diameter pairs) > > piacm 3 1;4 1;0 > 3 4 > 1 1 > > and repeatedly take the mediant of the lower interval boundary and the upper > (replacing the upper with the mediant), after 7 steps I'll be past the > (famous) first reasonable approximation 22/7 > > piacm 3 1;4 1;7 > 3 4 7 10 13 16 19 22 25 > 1 1 2 3 4 5 6 7 8 > > Taking the last two thus obtained values as interval boundaries and continue > the process for another 15 steps > > piacm 22 7;25 8;15 > 22 25 47 69 91 113 135 157 179 201 223 245 267 289 311 333 355 > 7 8 15 22 29 36 43 50 57 64 71 78 85 92 99 106 113 > > I'll come up with the (famous) better approximation 355/113, which coincides > with the partial sum of the continued fraction [3;7,15,1] > > ecf 3 7 15 1 > 3.14159 > x: ecf 3 7 15 1 > 355r113 > > [Almost all of these values are shown in the animated picture contained in > the Wikipedia article.] > > Now my question is: > In that algorithm, what would be the indicator (or break-off condition) for > having hit a reasonable result, and which new 'summand' to use in improving > it..? > > -M > > > > At 2018-01-29 21:01, you wrote: > >> Guess you are right. >> Brocot (from the Stern-Brocot tree of fractions) came up with the algo to >> produce from two fractions p/q and r/s the fraction (p+r)/(q+s), which is in >> between the two parent fractions. Starting with 0/1 and 1/0 you can converge >> to any positive real number x, where (p+r)/(q+s) becomes the new 'weak' or >> 'strong' fraction dependent of its position w.r.t. x. >> >> >> R.E. Boss >> >> >> > -----Original Message----- >> > From: Chat [mailto:[email protected]] On Behalf Of >> > Roger Hui >> > Sent: maandag 29 januari 2018 18:50 >> > To: Chat Forum <[email protected]> >> > Subject: Re: [Jchat] weak | strong fraction ... >> > >> > Guess: p<x and x<q. p and q are rational. p is a "weak fraction" and q >> > is a >> > "strong fraction". If you have a process where at each step the weak >> > becomes less weak and the strong becomes less strong, then you are in >> > business. >> > >> > >> > >> > On Mon, Jan 29, 2018 at 9:05 AM, Martin Kreuzer <[email protected]> >> > wrote: >> > >> > > Hi - >> > > >> > > This question goes to you Math people: >> > > >> > > Looking at this article >> > > https://en.wikipedia.org/wiki/Mil%C3%BC >> > > I found (in the last paragraph) the terms "weak" and "strong" >> > > fractions. >> > > >> > > I've never before seen this mentioned anywhere and I'm in doubt >> > > whether it is a valid concept. >> > > [nb: I do own the Martzloff (1996) book.] >> > > >> > > Any enlightenment welcome ... >> > > >> > > -M >> > > >> > > ---------------------------------------------------------------------- >> > > For information about J forums see >> > http://www.jsoftware.com/forums.htm >> > ---------------------------------------------------------------------- >> > For information about J forums see http://www.jsoftware.com/forums.htm >> ---------------------------------------------------------------------- >> For information about J forums see http://www.jsoftware.com/forums.htm > > ---------------------------------------------------------------------- > For information about J forums see http://www.jsoftware.com/forums.htm ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
