From:           isaacs@hpcc01.HP.COM (Stan Isaacs)
Newsgroups:     alt.fractals
Subject:        More on the area of the Mandelbrot Set
Date:           25 Feb 91 07:43:14 GMT
Organization:   HP Corp Computing & Services

At the Northern California Section of the MAA meeting this past
weekend, John Ewing gave an interesting talk called "Can We See the
Mandlebrot Set".  (He avoided using both the "C" word and the "F" word
after the introduction to the talk.  It was not about either chaos or
fractals, but about what the M set is, mathematically.)  Anyway, he
discussed two results which we've seen separately in this group in the
last month or so.  Namely, that by computing the area of the M-set
using lots of terms in a series (Laurent Series?), the upper bound of
the area seems to converge about at 1.72 (the graph gets quite flat,
and seems to have an asymptote there), and by counting pixals more and
more accurately, you seem to get a lower bound of very close to 1.52.
Both these bounds are close to the values the methods would produce in
the limit - that is, it is NOT the case that these numbers would get
closer if a finer grid were used, or more terms were taken in the
series.  So, why the difference of 10% or so?  No one knows.  One
possibility is that the pixal method misses "hairs" around the border.
(I know thats not described very well; it was late, and I wasn't
taking notes.)  There was also a vague theory for the higher number
being possibly wrong.  But basically, it is not known, at present,
which of these numbers represents the "real" area (although it is
believed that one of them does.)  I'm afraid he didn't give references
(I asked afterwards.)

  -- Stan Isaacs

From:           jasonp@wam.umd.edu (Jason Stratos Papadopoulos)
Newsgroups:     sci.math
Subject:        Re: centroid of the Mandelbrot's set
Date:           19 Jun 1996 15:52:41 GMT
Organization:   University of Maryland College Park

Zdislav V. Kovarik (kovarik@mcmail.cis.McMaster.CA) wrote:
: :
: :Is the Mandelbrot set measurable? I mean, does it *have* an area?
: :If so, what is it?

: A humble start: It is compact and has interior points, so its Lebesgue
: measure is well-defined, positive and finite.

: Good luck, ZVK (Slavek),

I know almost nothing about this, but see an article by Ewing and Schober,
something like "On the Coefficients of the Reciprocal of the Mandelbrot Set"
in "Mathematical Analysis and Applications". It seems the area of the Mandel-
brot set is bounded above by 

         Infinity
pi*( 1 - Sum      n* b(n)^2  )
         n=1

where b(n) are the coefficients of the power series mapping the unit circle
to the exterior of the Mandelbrot set. It converges very slowly (the 
terms of the series are very irregular, can be found recursively, and
asymptotically behave as O(n^(-5/4) ). I believe that adding a quarter 
million terms yields an area approximately of 1.72  (1.75? something like
that); this is interesting because huge scale Monte Carlo runs bound the
area at about 1.53 (or thereabouts). I've tried accelerating the power series
based on the first 20 or so terms, but haven't had any luck.

As for the centroid question...the centroid is the center of area, right?
If so, it'll be on the real axis (Mandelbrot is symmetric), and can be ap-
proximated roughly by finding the "center of mass" of the big cardioid, 
big circle, and the two largish (above/below the cardioid) bulbs all 
together.

Take all this cautiously. If anyone's interested, let me know and I'll
dig out a more complete reference list and post it here (these were the 
highlights).

jasonp

From:           Jeff Leader <JeffLeader@worldnet.att.net>
Newsgroups:     sci.math
Subject:        Re: centroid of the Mandelbrot's set
Date:           20 Jun 1996 01:26:14 GMT
Organization:   AT&T WorldNet Services

jasonp@wam.umd.edu (Jason Stratos Papadopoulos) wrote:

>million terms yields an area approximately of 1.72  (1.75? something like
>that); this is interesting because huge scale Monte Carlo runs bound the
>area at about 1.53 (or thereabouts). I've tried accelerating the power series
>based on the first 20 or so terms, but haven't had any luck.

There was a (fairly) recent article on this 1.5/1.75 in the Monthly or
Math. Mag. that was interesting.  Noone knows, but it's clearly an object
with a well-defined, finite (it's a subset of a disk of radius two) area.
Don't know about the centroid...

Newsgroups:     sci.math,sci.fractals
From:           "Jay R. Hill" <JAY.R.HILL@cpmx.saic.com>
Subject:        Re: centroid of the Mandelbrot's set
To:             JeffLeader@worldnet.att.net,jhill@nosc.mil
Organization:   SAIC
Date:           Thu, 20 Jun 1996 23:10:07 GMT

Jeff Leader <JeffLeader@worldnet.att.net> wrote:
>jasonp@wam.umd.edu (Jason Stratos Papadopoulos) wrote:
>
>>million terms yields an area approximately of 1.72  (1.75? something like
>>that); this is interesting because huge scale Monte Carlo runs bound the
>>area at about 1.53 (or thereabouts). I've tried accelerating the power series
>>based on the first 20 or so terms, but haven't had any luck.
>
>There was a (fairly) recent article on this 1.5/1.75 in the Monthly or
>Math. Mag. that was interesting.  Noone knows, but it's clearly an object
>with a well-defined, finite (it's a subset of a disk of radius two) area.
>Don't know about the centroid...
>
>

Ah, the old MSet area problem. I compute the area is at least 1.505936.

Three years ago a bunch of us (66) who read sci.fractals computed upper 
and lower bounds. We found it is more than 1.5031197 and no greater than 
1.5613027. See Y. Fisher and J. Hill, Bounding the Area of the Mandelbrot 
Set, Numerische Mathematik,. (Submitted for publication). Available via 
World Wide Web (in Postscript format) 

http://inls.ucsd.edu/y/Complex/area.ps.Z

As for the centroid? Just guessing, but how about -1/3?

Jay "Not to night honey, it's that Mandelbrot project again" Hill
-- 
int main(){float g,s,f,r,i;char*_="!/-,;<:!lnb/bh`r/ylqbAmmhI/S/x`K\n";int m,u,
e=0;_[32]++;for(;e<3919;){u=(256*(s=(r=.0325*(m=e%80)-2)*r+(i=.047*(e/80)-1.128
)*i)-96)*s+32*r<3?25:16+32*r+16*s<1?31:0;if(u==(s=f=0))do g=s*s-f*f+r;while((f=
2*s*f+i)*f+(s=g)*g<4&&++u<27);putchar(_[++e>3840&&m<25?31-m:m>78?32:u]^1);}}

From:           mert0236@sable.ox.ac.uk (Thomas Womack)
Date:           21 Jun 1996 18:56:03 GMT
Newsgroups:     sci.math,sci.fractals
Subject:        Re: centroid of the Mandelbrot's set

Jay R. Hill (JAY.R.HILL@cpmx.saic.com) wrote:
: Jeff Leader <JeffLeader@worldnet.att.net> wrote:
: >jasonp@wam.umd.edu (Jason Stratos Papadopoulos) wrote:
: >
: >>million terms yields an area approximately of 1.72  (1.75? something like
: >>that); this is interesting because huge scale Monte Carlo runs bound the
: >>area at about 1.53 (or thereabouts). I've tried accelerating the power series
: >>based on the first 20 or so terms, but haven't had any luck.
: >
: >There was a (fairly) recent article on this 1.5/1.75 in the Monthly or
: >Math. Mag. that was interesting.  Noone knows, but it's clearly an object
: >with a well-defined, finite (it's a subset of a disk of radius two) area.
: >Don't know about the centroid...
: >
: >

: Ah, the old MSet area problem. I compute the area is at least 1.505936.

: Three years ago a bunch of us (66) who read sci.fractals computed upper 
: and lower bounds. We found it is more than 1.5031197 and no greater than 
: 1.5613027. See Y. Fisher and J. Hill, Bounding the Area of the Mandelbrot 
: Set, Numerische Mathematik,. (Submitted for publication). Available via 
: World Wide Web (in Postscript format) 

: http://inls.ucsd.edu/y/Complex/area.ps.Z

: As for the centroid? Just guessing, but how about -1/3?

It's somewhere around -0.288, by straight Monte Carlo. The area is about
1.52, by the same argument.

--
Tom

"The first Ariane 5 flight did not result in validation of Europe's new
launcher"

Newsgroups:     sci.math,sci.fractals
From:           "Jay R. Hill" <JAY.R.HILL@cpmx.saic.com>
Subject:        Re: centroid of the Mandelbrot's set
To:             JAY.R.HILL@cpmx.saic.com,jhill@nosc.mil,marblanc@mail.mcnet.ch,Dt812n.8J6@undergrad.math.uwaterloo.ca,1C1992A.1FCA@mail.mcnet.ch,lhf@csg.uwaterloo.ca
Organization:   SAIC
Date:           Fri, 21 Jun 1996 23:54:57 GMT

Sylvestre Blanc  <marblanc@mail.mcnet.ch> wrote:
>I know that my question is little odd, but does anybody know where is 
>the centroid of the Mandelbrot's set ?

"Jay R. Hill" <JAY.R.HILL@cpmx.saic.com> wrote:
[snippity-snip]
>As for the centroid? Just guessing, but how about -1/3?

The area of the cardioid is 3pi/8 with centroid at -1/6.
The circle has area 1/16 centered at -1. The combined centroid
is -2/7.  Pixel counting (I can't believe I wrote that) gets me
the centroid near -0.288, still close enough to -2/7 to wonder....

>Jay "Not to night honey, it's that Mandelbrot project again" Hill

Jay "Not again or is it still..." Hill
-- 
int main(){float g,s,f,r,i;char*_="!/-,;<:!lnb/bh`r/ylqbAmmhI/S/x`K\n";int m,u,
e=0;_[32]++;for(;e<3919;){u=(256*(s=(r=.0325*(m=e%80)-2)*r+(i=.047*(e/80)-1.128
)*i)-96)*s+32*r<3?25:16+32*r+16*s<1?31:0;if(u==(s=f=0))do g=s*s-f*f+r;while((f=
2*s*f+i)*f+(s=g)*g<4&&++u<27);putchar(_[++e>3840&&m<25?31-m:m>78?32:u]^1);}}

Newsgroups:     sci.math,sci.fractals
From:           "Jay R. Hill" <JAY.R.HILL@cpmx.saic.com>
Subject:        Re: centroid of the Mandelbrot's set
To:             jhill,@,nosc.mil,marblanc,@,mail.mcnet.ch,Dt812n.8J6,@,undergrad.math.uwaterloo.ca,,1C1992A.1FCA,@,mail.mcnet.ch,lhf,@,csg.uwaterloo.ca
Organization:   SAIC
Date:           Mon, 24 Jun 1996 16:21:26 GMT

"Jay R. Hill" <JAY.R.HILL@cpmx.saic.com> wrote:
>Sylvestre Blanc  <marblanc@mail.mcnet.ch> wrote:
>>I know that my question is little odd, but does anybody know where is 
>>the centroid of the Mandelbrot's set ?
>
>"Jay R. Hill" <JAY.R.HILL@cpmx.saic.com> wrote:
>[snippity-snip]
>>As for the centroid? Just guessing, but how about -1/3?
>
>The area of the cardioid is 3pi/8 with centroid at -1/6.
>The circle has area 1/16 centered at -1. The combined centroid
                     ^
    make this pi/16 _|

>is -2/7.  Pixel counting (I can't believe I wrote that) gets me
>the centroid near -0.288, still close enough to -2/7 to wonder....

From an over the weekend run: -.28781.

>
>>Jay "Not to night honey, it's that Mandelbrot project again" Hill
>

Jay
 
-- 
int main(){float g,s,f,r,i;char*_="!/-,;<:!lnb/bh`r/ylqbAmmhI/S/x`K\n";int m,u,
e=0;_[32]++;for(;e<3919;){u=(256*(s=(r=.0325*(m=e%80)-2)*r+(i=.047*(e/80)-1.128
)*i)-96)*s+32*r<3?25:16+32*r+16*s<1?31:0;if(u==(s=f=0))do g=s*s-f*f+r;while((f=
2*s*f+i)*f+(s=g)*g<4&&++u<27);putchar(_[++e>3840&&m<25?31-m:m>78?32:u]^1);}}

From:           fc3a501@GEOMAT.math.uni-hamburg.de (Hauke Reddmann)
Newsgroups:     sci.math,sci.fractals
Subject:        Re: centroid of the Mandelbrot's set
Date:           25 Jun 1996 09:24:49 GMT
Organization:   University of Hamburg -- Germany

Jay R. Hill (JAY.R.HILL@cpmx.saic.com) wrote:
: 
: From an over the weekend run: -.28781.
: 
Was this a Monte Carlo run?
(Maybe it is better to add up the pieces: the circle, the
cardiodid, the small circle... until the defining formulae
are to awkward to integrate. Anyone try?)
-- 
Hauke Reddmann <:-EX8 
fc3a501@math.uni-hamburg.de              PRIVATE EMAIL 
fc3a501@rzaixsrv1.rrz.uni-hamburg.de     BACKUP 
reddmann@chemie.uni-hamburg.de           SCIENCE ONLY

From:           rice@servo.eng.sun.com (Daniel Rice)
Newsgroups:     sci.math,sci.fractals
Subject:        Re: centroid of the Mandelbrot's set
Date:           26 Jun 1996 23:50:42 GMT
Organization:   Sun Microsystems Computer Corporation

In article <4qob91$8q4@rzsun02.rrz.uni-hamburg.de>,
Hauke Reddmann <fc3a501@GEOMAT.math.uni-hamburg.de> wrote:
>Jay R. Hill (JAY.R.HILL@cpmx.saic.com) wrote:
>: 
>: From an over the weekend run: -.28781.
>: 
>Was this a Monte Carlo run?
>(Maybe it is better to add up the pieces: the circle, the
>cardiodid, the small circle... until the defining formulae
>are to awkward to integrate. Anyone try?)

  Another approach might be to find the centroids (or areas, or whatever)
of the various iterates of z^2 + c.  Is there an analytic technique to
determine the centroid of such a figure (defined by |p(c)| == 2 for some
polynomial p)?  Is the desired centroid the limit of the resulting sequence?

						Dan

Subject:        Mandelbrot Set Area, new result
From:           Munafo@prepress.pps.com
Date:           1996/12/23
Organization:   PrePRESS SOLUTIONS
Newsgroups:     sci.fractals

I have continued to compute the area of the Mandelbrot Set using
my method of averaging 20 runs by the pixel-counting method,
with slightly different grid placements and grid spacings. The
latest result (after about 3.7 trillion Mandelbrot iterations)
is 1.50659230 +- 0.0000006.

Here are the results of a set of such averaged runs. For each
grid size, 20 runs were computed and averaged together.

grid  dwell    average        standard
size  limit    area           deviation
----  -------  -------------  -------------
32    8192     1.511 4        0.037 2
64    16384    1.502 6        0.019 4
128   32768    1.504 84       0.005 68
256   65536    1.506 34       0.002 26
512   131072   1.506 88       0.001 37
1024  262144   1.506 783      0.000 493
2048  524288   1.506 674      0.000 152
4096  1048576  1.506 585 0    0.000 074 1
8192  2097152  1.506 593 8    0.000 027 5
16384 4194304  1.506 588 0    0.000 012 6
32768 8388608  1.506 591 54   0.000 004 48
65536 16777216 1.506 592 30   0.000 001 90

The standard deviation measures how much (on average) each
run differs from the average. This means that the average
itself almost certainly varys even less from the true area.
For 20 samples, the standard error of the mean is about 0.23
of the standard deviation. A safe estimate of the error
would be about 1/3 of the standard deiation.

(My old estimate, from April 13th 1993, was 1.506595
+- 0.000002. That estimate was based on an average of 40
runs, and used a dwell limit of 524288. I should have
subtracted 0.000005 to account for the error in using such
a low dwell limit; I also was a little too "liberal" in my
calculation of the expected error.)

- Robert Munafo

From:           edgar@math.ohio-state.edu (G. A. Edgar)
Date:           Fri, 05 Feb 1999 15:33:25 -0500
Newsgroups:     sci.math
Subject:        Re: Mandelbrot set

In article <36BA4B97.F6A08E45@erols.com>, John VanSickle
<vansickl@erols.com> wrote:

> Just an idle question, but has the area of the Mandelbrot set been
> calculated (or estimated to any particular degree of precision)?

The area of the Mandelbrot set is between 1.5 and 1.71.  Here are
some messages on the topic collected from usenet two years ago.
Those not identified are from me.

From:           edgar@math.ohio-state.edu (G. A. Edgar)

The area of the Mandelbrot set is 

    A = (1 - sum (n=1 to infinity) n (b_n)^2) Pi

or approximately 2.089.

Here the numbers  b_n  are the coeffients of the Laurent series about
infinity of the conformal map  psi  of the exterior of the unit disk
onto the exterior of the Mandelbrot set:

  psi(w) = w +  sum (n=0 to infinity)  b_n w^(-n)
  
         = w - (1/2) + (1/8) w^(-1) - (1/4) w^(-2) + (15/128) w^(-3)
                 
           + 0 w^(-4) - (47/1024) w^(-5) + ...
                 
These coefficients can be computed recursively, but a closed form
is not known.  The above approximation comes from using the first
72 terms of the series.

From:           scott@ferrari.LABS.TEK.COM (Scott Huddleston)
Newsgroups:     alt.fractals
Subject:        the Mandelbrot area formula
Date:           7 Dec 90 00:51:07 GMT
Organization:   Computer Research Laboratory, Tektronix, Inc., Beaverton OR

A formula for computing the area of the Mandelbrot set was discussed on
this newsgroup recently.  Gerald Edgar posted an area estimate based on
72 terms.  Yuval Fisher pointed out that a 256-term estimate gives a
much lower value, and he described how to compute coefficients in the area
series.

Using Yuval's suggestion, I computed over 100,000 terms of the series.
The estimate converges very slowly -- a summary appears below.  An
inportant point is that every truncation of the series is an 
*upper bound* on the area of M.  Based on the trend I'd say the true 
area is probably less than 1.70, but I won't bet my firstborn on that.

# terms: area estimate
    72: area < 2.09288
   128: area < 2.02781
   180: area < 2.01237
   256: area < 1.97752
   360: area < 1.94961
   512: area < 1.92751
   720: area < 1.91255
  1024: area < 1.89534
  1440: area < 1.87172
  2048: area < 1.85461
  2880: area < 1.84576
  4096: area < 1.83452
  5760: area < 1.81649
  8192: area < 1.80616
 11520: area < 1.79642
 16384: area < 1.78636
 23040: area < 1.7747
 32768: area < 1.76683
 46080: area < 1.75936
 65536: area < 1.75337
 92160: area < 1.74321
115232: area < 1.73847
--
Scott Huddleston
scott@crl.labs.tek.com

From:           isaacs@hpcc01.HP.COM (Stan Isaacs)
Newsgroups:     alt.fractals
Subject:        More on the area of the Mandelbrot Set
Date:           25 Feb 91 07:43:14 GMT
Organization:   HP Corp Computing & Services

At the Northern California Section of the MAA meeting this past
weekend, John Ewing gave an interesting talk called "Can We See the
Mandlebrot Set".  (He avoided using both the "C" word and the "F" word
after the introduction to the talk.  It was not about either chaos or
fractals, but about what the M set is, mathematically.)  Anyway, he
discussed two results which we've seen separately in this group in the
last month or so.  Namely, that by computing the area of the M-set
using lots of terms in a series (Laurent Series?), the upper bound of
the area seems to converge about at 1.72 (the graph gets quite flat,
and seems to have an asymptote there), and by counting pixals more and
more accurately, you seem to get a lower bound of very close to 1.52.
Both these bounds are close to the values the methods would produce in
the limit - that is, it is NOT the case that these numbers would get
closer if a finer grid were used, or more terms were taken in the
series.  So, why the difference of 10% or so?  No one knows.  One
possibility is that the pixal method misses "hairs" around the border.
(I know thats not described very well; it was late, and I wasn't
taking notes.)  There was also a vague theory for the higher number
being possibly wrong.  But basically, it is not known, at present,
which of these numbers represents the "real" area (although it is
believed that one of them does.)  I'm afraid he didn't give references
(I asked afterwards.)

  -- Stan Isaacs

From:           shallit@graceland.waterloo.edu (Jeffrey Shallit)
Subject:        area of the Mandelbrot set
Organization:   University of Waterloo
Date:           Tue, 10 Mar 1992 14:52:52 GMT

There was some discussion a while back about the area of the
Mandelbrot set.  I just noticed the following article:

        J. H. Ewing and G. Schober, The area of the Mandelbrot set,
        Numer. Math. 61 (1992), 59-72.

It may be of interest.

Jeff Shallit
shallit@graceland.waterloo.edu

From:           edgar@math.ohio-state.edu (G. A. Edgar)

The article computes the area estimate using 240,000 terms.  The result
is 1.7274...  The behovior of the approximations suggests that the limit
is between 1.66 and 1.71.  However, the estimates of the area from below,
using pixel counting show that the area is at least 1.52.  The large
gap between the lower bound 1.52 and the upper bound 1.71 may possibly
be an indication that the boundary of the Mandelbrot set has
positive area...

The Ewing and Schober paper cited also explains the computation of
the coefficients b_m in the series.

Date:           Fri, 18 Sep 92 10:31:51 EST
From:           kbriggs@mundoe.maths.mu.oz.au (Keith Briggs rba8 7088)
Subject:        Re: Conformal maps
To:             edgar@mps.ohio-state.edu

  I have done some refined pixel counting and
my best result is
 Total Mset area=  1.499936.
On this basis I conjecture that the exact value is 3/2.
However, this number is not universal in the way that the feigenvalues are,
since it varies with reparameterization of the map z^2+c.   I have also
done z^d+c (d=3,4,5..), with no clear pattern yet.
Keith.

Newsgroups:     sci.fractals
From:           hilljr@jupiter.saic.com (Jay R. Hill)
Subject:        My Mandelbrot Flower
Keywords:       Mandelbrot area
Organization:   SAIC
Date:           Mon, 15 Mar 1993 17:06:44 GMT

          My Mandelbrot Flower
          (C) by Jay Hill, 1993

[poem omitted because of the copyright notice]

              (-:  ---  ;^)

So let's get started. We can for each period, p, count around
the Cardioid.  Let's name the buds (i,p), the i-th bud with
period p.

n:=0;
for p:=1 to inf do
  for i:=1 to p do
    n:=n+1;

But we find we already counted some. For example, (2,4) is
the same bud as (1,2).  The modified loop is

n:=0; m:=0;
for p:=1 to inf do
  for i:=1 to p do
    if gcd(i,p)=1 then
      n:=n+1
    else
      m:=m+1;

where m will count the duplicates. To count the buds on buds,
we must make this algorithm recursive. When we are done, we
can also count the island Mandelbroties, l. They will be

l = 2^p - (n+m) - Mandelbrotie buds.

Table I shows the results of this calculation.  We can also
keep a running total of the number of buds.  The total for
each doubling of the period is shown in Table II.  The last
column shows the ratio of the total with each doubling which
approaches 5.56.  An approximate formula for the total is

  Total buds, T = 5.56^(ln2(p/2)) = 5.56^(-1+1.442695*ln(p))
                = exp( 2.46*ln(p)-1.7 )

The density of buds is approximately

  dT/dp = (0.45/p)*exp( 2.46*ln(p) )

Now if we use Milnor's bud radius formula r=sin(pi*i/p)/(p*p),
we can estimate the area of the Cardioid plus attached buds.
The formula is not good for buds on buds, however if I use it
anyway, the area approaches 1.507818.  I must point out the
formula is quite inaccurate, with error as large as 100%.
Therefore, measured radii for p less than 11 were substituted
in the revised area estimate, Table III, which approaches
1.504106.  This is approximate since the bud radii (area) for
periods beyond 2 are still unknown.

The importance of the bud area as a function of p can be
gauged the total area of all buds and cardioids as a function
of p.  An estimate based on 'pixel counting' is shown in Table
IV.  The values for p=1,2 can be compared to the exact values.

A(1) = 3*pi/8 = 1.178097245 (calculated=1.17809694),

A(2) =  pi/16 = 0.196349541 (calculated=0.19634926).

Their error is about 3e-7.  The calculation used 8192x4096
samples in region (-2,-1.125),(0.5,1.125) with an iteration
limit of 8388608.  Period detection was used for early exit.
When a period was found, an additional set of iterations were
used with a fuzzy period test to distinguish the real period
from a multiple.  A simultaneous graphic display showed the
errors in the period selection to be very few.  The undecideds
after 8388608 iterations were counted.  Only 73 samples out
of 8987185 were unable to decide to exit or find a period.
We can see from Table IV that accurate estimates of bud radii
will be needed up to p=48 if we want 5 or 6 digit accuracy.
That is 2417 buds on the main Mandelbrot, not an impossible
task.  8^)   |-0   >^]   |-(  (Are we having fun yet?)  ;^)

                  Table I

Period  Buds                  Islands    Island buds

     1     0                        0              0
     2     1                        0              0
     3     2                        1              0
     4     3                        3              0
     5     4                       11              0
     6     6                       20              1
     7     6                       57              0
     8     9                      108              3
     9    10                      240              2
    10    12                      472             11
    11    10                     1013              0
    12    22                     1959             29
    13    12                     4083              0
    14    18                     8052             57
    15    24                    16315             26
    16    27                    32496            117
    17    16                    65519              0
    18    38                   130464            286
    19    18                   262125              0
    20    44                   523209            517
    21    36                  1048353            120
    22    30                  2095084           1013
    23    22                  4194281              0
    24    78                  8384100           2262
    25    36                 16777120             44
    26    36                 33546216           4083
    27    50                 67108068            490
    28    66                134201223           8241
    29    28                268435427              0
    30   104                536836484          17417
    31    30               1073741793              0
    32    81               2147417952          32847
    33    60               4294964173           2036
    34    48               8589803488          65519
    35    72              17179868739            294
    36   158              34359469848         135274
    37    36              68719476699              0
    38    54             137438429148         262125
    39    72             274877894595           8178
    40   156             549754764132         525192
    41    40            1099511627735              0
    42   156            2199021133728        1064937
    43    42            4398046511061              0
    44   110            8796088826787        2098153
    44   110            8796088826787        2098153
    45   152           17592185993904          33724
    46    66           35184363700180        4194281
    47    46           70368744177617              0
    48   270          140737471477920        8455890
    49    78          281474976710172            342
    50   140          562949919864320       16779140
    51    96         1125899906645935         131054
    52   132         2251799746572177       33558501
    53    52         4503599627370443              0
    54   230         9007199120130408       67370674
    55   120        18014398509476663           4162
    56   234        36028796750519292      134226594
    57   108        72057594037141413         524268
    58    84       144115187538984904      268435427
    59    58       288230376151711685              0
    60   456       576460751228083221      537943113
    61    60      1152921504606846920              0
    62    90      2305843007066210240     1073741793
    63   228      4611686018424240100        2098752
    64   243      9223372032559775420     2147516493

                  Table II

Period     Buds      Total   T(n)/T(n/2)

     2        1          1
     4        3          6       6.00
     8        9         31       5.17
    16       27        166       5.35
    32       81        879       5.30
    64      243       4826       5.49
   128      729      26537       5.50
   256     2187     147542       5.56
   512     6561     817561       5.54
  1024    19683    4537366       5.550
  2048    59049   25234399       5.561
  4096   177147  140622586       5.573

                  Table III

         p  Area of attached buds
         2  1.37444678594553454
         3  1.43048719821709103
         4  1.45354470458157996
         8  1.48460977988076338
        16  1.49712977124274433
        32  1.50154588264492087
        64  1.50316313962822509
       128  1.50373319034984287
       256  1.50402541594610357
       512  1.50407075095490535
      1024  1.50409528851486800
      2048  1.50410386745418746
      4096  1.50410686280880857

              Table IV

          p   Total Area
          1   1.17809694
          2   0.19634926
          3   0.05651246
          4   0.02323865
          5   0.01310426
          6   0.00892924
          7   0.00502947
          8   0.00446101
          9   0.00293483
         10   0.00265840
         11   0.00137865
         12   0.00211441
         13   0.00084858
         14   0.00113071
         15   0.00094799
         16   0.00089401
         17   0.00038674
         18   0.00079376
         19   0.00027878
         20   0.00060651
         21   0.00039814
         22   0.00033678
         23   0.00015858
         24   0.00049604
         25   0.00018691
         26   0.00021256
         27   0.00019345
         28   0.00025916
         29   0.00007979
         30   0.00028716
         31   0.00006403
         32   0.00017836
         33   0.00011114
         34   0.00009857
         35   0.00010108
         36   0.00020602
         37   0.00003872
         38   0.00007141
         39   0.00007040
         40   0.00013813
         41   0.00002883
         42   0.00012220
         43   0.00002397
         44   0.00007527
         45   0.00007761
         46   0.00004191
         47   0.00001844
         48   0.00011047
         49   0.00002933
         50   0.00005817
         51   0.00003218
         52   0.00004794
         53   0.00001290
         54   0.00006118
         55   0.00002933
         56   0.00005917
         57   0.00002447
         58   0.00002212
         59   0.00001106
         60   0.00007845
         61   0.00000888
         62   0.00001676
         63   0.00003386
         64   0.00003302
       Total  1.50659429

Warmly,

Jay

From:           munafo@gcctech.com
Newsgroups:     sci.fractals
Subject:        Mandelbrot Set center-of-gravity is -((ln(3)-1/3)^Feig)
Date:           Wed, 04 Mar 1998 14:36:24 -0600
Organization:   Deja News - The Leader in Internet Discussion

Recently I restarted the computing effort I undertook
about a year ago to
compute the area of the Mandelbrot
Set. After reviewing the notes from that
time I saw that there had
been some interest in the past of computing the
Mandelbrot Set's
"center of gravity" as well. I have also done some
optimizations of
the code in the mean time and fixed a few bugs, and have
started
doing a long run.

The method I use involves counting pixels on a
grid. There is error
in the estimate resulting from grid placement and the
fact that the
grid squares on the boundary get counted as completely in or
completely out when in fact only part of their area is in. It is hard
to
estimate the magnitude of this error, so I use statistical methods
to measure
it. I use a bunch of grids of slightly different grid
spacings with slight
vertical and horizontal offsets, and take the
mean and standard deviation of
all the runs.

These are the figures I've compiled so far. Each run is the
average
of 20 grids for which the standard deviations are shown.

  grid 
iter  area and  standard  iterations
  size  limit  center-of-       deviations 
time (seconds)
  gravity  (20 runs)  performance (MFLOPs)

  256  65536 
1.506 38  0.002 31  18,623,118
  -0.286 741  0.000 912 9.4  13.918

  512 
131072  1.506 87  0.001 35  82,659,183
  -0.286 881  0.000 417 40.2  14.376

  1024  262144  1.506 782  0.000 493  372,533,718
  -0.286 802    0.000 173 
176  14.839

  2048  524288  1.506 674       0.000 151  1,716,452,044
  -0.286 811
0  0.000 074 8  788  15.246

  4096  1048576  1.506 584 0  0.000 074 4 
7,933,339,783
        -0.286 765 8  0.000 024 8  3,580  15.502

  8192  2097152 
1.506 593 6  0.000 027 7  36,987,339,092
  -0.286 769 51  0.000 009 73 
16,300  15.852

 16384  4194304  1.506 588 1  0.000 012 7  171,493,547,507
 
-0.286 767 27  0.000 003 20  74,400  16.138

 32768  8388608  1.506 591 50 
0.000 004 51  814,136,316,927
        -0.286 768 37  0.000 001 77  347,000  16.442

 65536 16777216  1.506 592 31  0.000 001 90  3,844,750,451,387
  -0.286 768
423  0.000 000 510  1,630,000  16.529

131072 33554432 1.506 591 734  0.000
000 624  18,173,551,931,685
  -0.286 768 317  0.000 000 295  7,590,000 
16.754

For each average, the standard deviation gives the expected error
in
each of the 20 individual grids. The expected error of the mean
is 1/sqrt(19)
of the standard deviation (this is called "the standard
error of the mean").
I use 1/3 as a conservative estimate. So, my
best estimate of the
center-of-gravity is -0.28676832 +- 0.00000010.
For the area, we have
1.50659173 +- 0.00000020.

The Inverse Symbolic Calculator is a resource at
the URL

  http://www.cecm.sfu.ca/projects/ISC/ISCmain.html

which can be
used to guess what expression produces a given numeric
value. I plugged in
the values 1.50659173 and 0.28676832. I saw
nothing interesting for the area
but for the center of gravity it
gives

  -((ln(3)-1/3) ^ Feig1).

Feig1 is
the larger Faigenbaum constant, whose value is
4.6692016091029906718532038,
and shows up in the Mandelbrot Set as
the ratio of the radii of successive
mu-atoms on the real axis.
Plugging in this value we get the following
hypothetical precise
value of the Mandelbrot Set's center of gravity:

-0.2867682633829350268529586

A few other observations:

- My program
currently runs at about 2.39 million Mandelbrot iterations
  per second (each
iteration requires 7 floating-point operations).
  The latest model Macs are
about 4 times faster. Intel machines get
  nowhere close because they don't
have enough floating-point registers
  to keep the pipeline full.

- Each
time I double the grid size it takes almost 4.7 times as long.
  The current
rate of progress in computer performance is
  about 1.56 per year; at that
rate it takes about 3.5 years
        to increase computing power by 4.7.

- Each
time the gridsize is doubled the error goes down to about 0.4
        of the
previous value (but it varies from 0.3 to 0.55). It would take
  about 8 to
10 years for personal computers to improve enough to give
  us one more digit
in the area estimate (given the same run time).

- Robert Munafo
  Malden
Massachusetts
        4 March 1998
  rpm%mrob.uucp@spdcc.com