Pi and fractal sets
The Mandelbrot set - Dave Boll - Gerald Edgar
A big surprise !
let us consider the point c=(-0.75,X) of the
complex plane, that is a point straight over the "neck" of the Mandelbrot
set.
Let n be the number of iterations from which the characteristic quadratic sequence of the Mandelbrot
set Z_{n+1}=Z_{n}^{2}+c with Z_{0}=-0 diverges (*Z*_{n}2). With X being smaller
and smaller we have:
What relation is there between Pi and the Mandelbrot
set ?
This pretty picture at the top of the page
is the Mandelbrot set (which is ... by the way). It is made up of the points
c=x+i*y of the complex plane (with coordinates (x,y) in the catesian plane)
such that the sequence Z_{n+1}=Z_{n}^{2}+c with Z_{0}=0 does not diverge.In practice to build up this set and
dits graphical representation, one shows that when the modulus of Z_{n} is greater
than 2 it will diverge. No need to reach a big value ! The points c of this
Mandelbrot set are represented in black (So the points for which Z_{n} stays
bounded), and the colours around represent the points according to the value
of n from which we consider the sequence diverges.
In 1991 David Bolle tried to verify if the
narrowing we can see at (-0.75,0) was actually infinitely thin. That is to
say that that however wide a non-zero width vertical line would be passing
through that point it would meet the fractal set before the x-axis.
And D Bolle then had the idea of using the
point c=(-0.75,X) for the quadratic iteration and to make X tend to 0. And
there, what was his surprise when he counted the number of iterations before
which the series diverged and by discovering the following table .:
*X* |
*iterations* |
*1.0* |
*3* |
*0.1* |
*33* |
*0.01* |
*315* |
*0.001* |
*3143* |
*0.0001* |
*31417* |
*0.00001* |
*314160* |
*0.000001* |
*3141593* |
*0.0000001* |
*31415928* |
Yes, it was Pi that was appearing magnificientely ! As he could not manage
to prove this he posted it in 1992 on the sci.maths newsgroup. Gerald Edgar
from a university of Ohio answered it on 27 march 1992 by bringing an intuitive
explanation of this result. This has been put lower down in the "Trial"
section.
After this David Bolle approached the problem
with another point c=(0.25+X,0) which were those of the... well... butt part of
the set on the right.
Once again, same surprise, this time it is X^{1/2}*n which tends to Pi.
*X* |
*iterations* |
*1.0* |
*2* |
*0.1* |
*8* |
*0.01* |
*30* |
*0.001* |
*97* |
*0.0001* |
*312* |
*0.00001* |
*991* |
*0.000001* |
*3140* |
*0.0000001* |
*9933* |
*0.00000001* |
*31414* |
*0.000000001* |
*99344* |
*0.0000000001* |
*314157* |
A few words about Mandelbrot, who started all
of it !
Benoît Mandelbrot was born in Poland in 1924
and emigrated to France in 1936 with his family, of which Szolem Mandelbrojt,
mathematician and professor at the *college de France.*
Very quickly Benoit is seen as an original mind not really following the trends and ideas of the time. His stay at the
ENS Ulm was short (one day!) and his entered *Polytechnique* in 1944.
As opposed to the Bourbakist atmosphere of the French mathematical school
he will then spend most of his career in the US (Applied maths and economy
in Havard, but also engineering sciences and Physiology at the Yale and Einstein
college of medcine !).
His uncle Szolem introduced him in 1945 to
the forgotten work of Gaston Julia (1918) about the set of complex points
derived drom succesive iterations. Benoit prefered to follow his own path,
guided by a wonderful geometrical intuition, but actually gets to meet thework
of Julia in the seventies.. He then creates the theory of fractals in "Les objets fractals, forme, hasard
et dimension" (1975) and over all "The fractal theory of nature"
(1982). Thanks to the research labs that IBM lets him use, he inspires deeply
the geometrical vision of fr`ctals and creates the first programs of graphical
creation on computers.
A trial to explain the phenomena
It is given by Gerald Elgar and I don't know
of any proofs (on internet) of the result. Edgar uses c=(0.25+X,0)
We consider the sequence given by Z_{n+1}=Z_{n}^{2}+1/4+X, with Z_{0}=0.
This sequence increases slowly towards 1/2 (the limit if X=0 following the
theorem of the fixed point) then, after this step, diverges faster. We are
interested in the point 1/2 by shifting the sequence, i.e. by taking Zn=Yn+1/2.
Our equation becomes Y_{n+1}+1/2=(Y_{n}+1/2)^{2}+1/4+X, i.e. :
Y_{0}=-1/2
Y_{n+1}=Y_{n}^{2}+Y_{n}+X
The Yn increase towards 0 slowly around 0,
so we can consider that Y_{n} is in this case a function of n, taken as a continuous
variable, and - clever part ! - we can consider that Y_{n+1}-Y_{n} is very close
to Y_{n}', i.e. Y_{n} differentiated.
So our equation becomes: Y_{n}'=Y_{n}^{2}+X, thus :
Hey, we know how to solve this!, it gives
:
But, as we are studying Y_{n} as it goes near
0, the last point of the iteration before Y_{n} escapes corresponds to the moment
when tan approaches 0 negatively, i.e. b=0 and a.n close to i.e :
where n is the number of iterations
before Yn escaped, which is exaclty the experimental result found by D. Bolle.
This metthod is quite surprising as a method to compute Pi. Imagine yourself
not calculating iteration after iteration the decimals of Pi but instead its
Pi's decimals which will give you the required precision for X ! If we imagined
that we could do as many iterations as we wanted in a reasonable time, it
could have become a fun game: you can try and found out what the next decimal
of Pi is by calculating how many oterations you want thanks to your prevision
!
Other cases ?
Lets come back to the first case c=(-0.75,X).
We use Z_{n+1}=f(Z_{n})=X_{n}^{2}-3/4 woth a fixed point 1/2 with value -1, so we use
f(f(z)) and Z_{n}=-1/2+Y_{n} to get Y_{n} close to 0. By the same process as previousely
we can find the differential equation :
The integral which comes out of this is not
easely sovable, but we can derive from it that the required number of iterations
for diverence is ... to */(2X)*. As this concerns the composition f(f(z)), we need
this to diverge for f(Z_{n}) to be doubled, i.e */X*, which confinrms once again D. Bolle's result for
these points.
One of the good principles of this calculation with the Mandelbrot set is
that a good number of narrowings exist, we can see it graphically. So several
appoaching angles can be used, like the points (-1.25,X). However theree is
for this case a chaotic behaviour at the beguining, and my calculator's memory
dod not resist ! David Petry, who did the calculation, indicates that *n*X/* is always very close to an integer or of half an
integer for small X. These aapproaching angles are quite obvious because they
are vertical or horizontal, but the multiple bulges can give raise to interesting
relations.
Even better, we can replace Z_{n}^{2} by Z_{n}^{3} or Z_{n}^{4}, this gives new Mandelbrot sets,
as the ones below, and here come some new narrowings ! I dont know what points
we can try on, if someone can give them to me I can try...
(*Z*_{n}^{3}) (*Z*_{n}^{4})
The broad subject of Julia sets
And if you want to go deeper into fractal theory, the yahoo.com directory
is quite full. Also there are quite a few Java applets around to draw the
Julia and Mandelbrot sets, these are in various sites about fractals.
This theory seems quite fashion nowerdays,
especially in france where the *TIPE* of *preparatory schools* about
dynamic systems during 2 years mention this quite a lot.
To cpmplete the Mandelbrot set and the theory
of complex points obtained by iteration let us just briefly talk about the
Julia sets which are intimately bounded.
They come from a work by Gaston Julia, brilliant
French mathematician between the two wars and professor at the *Ecole Polytechnique*.
We take once again the quadratic iteration but this time its Z_{0} that we are
going to vary. And we take once again the points for which Z_{n} is bounded to
Z_{0} variable to construct the Julia set. THere is tus a Julia set for each
point c. ATHis gives new images and new deep properties. For example, if we
take a point within the Mandelbrot set, the Mandelbrot set associated is ...,
otherwise no. Here are a few examples:
Fractal Dendrite, *c=(0,1)=i*
DouadysRabbit Fractal *c=(-0.123,0.745)=-0.123+0.745i*
San Marco Fractal*c=(-3/4,0)=-3/4*
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