🚀 go-pugleaf

In article <1993Dec13.194400.26337@ast.saic.com>, 
   hilljr@jupiter.saic.com (Jay R. Hill) writes:
|>        A warning to DEEPZOOM users.
|> 
|> Few can forecast the effects of technology on our futures as well 
|> as Arthur C. Clarke.  Since he has written of at least one who 
|> zoomed too deep once too often, we might want to be careful.  In 
|> his 1990 novel _The Ghost from the Grand Banks_, he describes one 
|> Edith Craig at the Torrington Clinic for psychological disorders, 
|> who is mentaly locked in a 40 digit deep zoom into the Mandelbrot 
|> Set.
|> 
|> Hmmmm.  There are now about 70 who have gotten my DEEPZOOM program.  
|> In the novel Clarke notes that "there are no less than sixty-three 
|> examples of Mandelmania now in the data banks."
|> 
|> Warmly,
|> Jay "Not tonight honey, just another factor of ten, just..." Hill

In an email I was asked:
> A naive question at a late date: what's Mandelmania?

In the novel, MSet pictures were everywhere by the turn of the 
century.  Some people just couldn't handle the infinite detail
and beauty.  They had to see it all, experience it all.  Some 
let it get to the point of an incurable Mandelmania.  No cure.  
Well, about 80% make an 'adjustment' back to normality by use 
of medications and for some, as Clarke indicated, implants - duuh.  


The real line portion of the Mandelbrot Set provides an exhibit
of several types of iteration behavior.  Outside of the interval
(-2..0.25) iteration diverges.  An example of a Misiurewicz point
is -2.  Then inside the buds and cardioids the iterations become
periodic.  On the edge of a bud or cardioid iteration convergence
is very slow since the Lyapunov exponent is 0.  Other points are
chaotic with Lyapunov exponents greater than 0.  But the Lyapunov
exponent is defined in the limit of infinitely many iterations.
Without carrying the calculation to infinity, can we determine if 
an arbitrary point is chaotic?  I doubt it.  Aha!  An excuse to ZOOM!

The Lyapunov exponent for c=1-1.8 is approximately 0.4046 which  
certainly looks chaotic.  I wondered what the MSet looks like if I 
repeatedly zoom in on such a chaotic point.  So I used DEEPZOOM to 
do just that.  With -1.8 in the center, each increase in magnification
shows the point -1.8 missing midgets repeatedly, with small ones 
near -1.8 always moving just out of reach.  Finally, at 10^68 
magnification I moved the focus over a pixel or so to a midget
right next to -1.8.  The final view at 10^70 magnification is shown 
in 18000000.gif which I just posted to alt.binaries.pictures.fractals.  
The gif took more than two days to compute on my 486/33.  The deepzoom
parameters were

maxiter = 3200076
x = 
-1.79999999999999999999999999999999999999999999999999999999999999999941756
y  =  962e-72
dx = 1280e-72
dy =  960e-72

My warning post (see above) got a few interesting replies.  Alas, it 
looks like it's too late.  One reply was, in part, "...seeing as how my 
brain is already warped and locked into a permanent ZOOM state...".  

And this. "You can add me to the 'mentally locked up Mandelbrot Zoomers 
Set'.  I have recently downloaded DEEPZOOM, and have started playing 
with it.  It's great.  This program is just another great reason to 
leave my computer on all night."

The current version, 0.10, has an improved color scheme using 108
rainbow colors before repeating.  Also, I have added a sleep mode
which aids TSR screen grabbers.  In four days, DEEPZOOM will be two
months old and is still FREE.  ;-)  Maybe by then (Real Soon Now) I 
will add gif-ability.  Then automatic generation of a gif sequence 
which can be used for an mpeg ought to really jam my machine.  If we 
magnify by a factor of two, there are obvious reuse of calculations 
from the last view.  There should be information reuse even if the 
magnification jump is 15%.  Any ideas?

Warmly,
Jay "Not tonight honey, I'm stuck at magnification 10^70" Hill
-- 
{ hilljr@jupiter.saic.com } begin writeln(3*ln(640320)/sqrt(163):17:15) end.
void main(){double sqrt(), y=1/sqrt(2.), a=.5, m=1,z; int n=0;
for(;m*=2,z=sqrt(1-y*y),y=(1-z)/(1+z),a=a*(1+y)*(1+y)-m*y,n<4;n++);
printf("%17.15lf\n",1/a);}

Some time ago BJ06%C53000.PETROBRAS.ANRJ.BR@UICVM.UIC.EDU <Fausto 
A. A. Barbuto (Totxo)> sent me a nice email with permission to quote:

Dr. Jay R. Hill played his best this time with DEEPZOOM and HILLZOOM.

When I thought the MS had already been exhaustively explored with 
the programs available around this little Earth Jay presented me 
his ZOOM tools.  I should learn to stay silent.

I explored ZOOM tools at home, in my 486 DX2-66. The MS view 
suggested in the manuals is really terrific. A nice view of an 
isolated MS surrounded by lines resembling to lightnings; inside 
the lonely MS violet/purple circular areas at both the cardioid 
and the main circle were shown.  Shadow  effects in gray, east of 
the MS, gave me a visual illusion as if it was  suspended in the 
air, croping out from the screen. And all these surrounded by 
colour levels produced with a very sophisticated technique I do 
not dare to wonder on how it was developed. Arrow keys allowed me 
to select a new region to be explored. Then I chose a journey to 
the neighbourhoods of the Seahorse Valley (executing 
HILLZOOM 256 -0.73125 0.19167 0.001 0.001 640 480).

A thrilling trip indeed. Details were shown in deep and - even 
with a low number if iterations (256) - the outcroping effect 
was still there.

Then I focused my lenses to the Elephant's Valley Region.  
(executing...  HILLZOOM 256 0.30625 0.000 0.05 0.05 640 480).  What 
could I say?  New surprising views I had never seen before.

At MS's Antenna Region (HILLZOOM 256 -1.7625 0.00 0.10 0.10 1024 768) 
one can have a complete idea on  the sophistication of  the colouring 
features.  The outcroping effect gave me the sensation of observing 
a MS detail through an eletronic microscope.

Finally, I cast a global look to the whole MS.  (HILLZOOM 256 0.0 0.0 
2.0 2.0 640 480). Nice aereal view. So I realized that ZOOM tools 
are  didactic too! I saw Periods 1 and 2 (Cardioid and Main Circle) 
coloured in  slightly different blue scales - and the Period 3 Bud 
as well. I use these period's detection to cut down CPU time when 
computing the MS Area.

As a matter of fact, HILLZOOM is faster than DEEPZOOM and FRACTINT 
is faster than either, but ZOOM tools are worth the time one spends 
to run them.  The price is right.

May I suggest... [some nice ideas followed]

  Best regards,
              Fausto. (Bj06@C53000.Petrobras.Anrj.Br)



Fausto, you are too kind.

Monday I wrote that DEEPZOOM is just about 2 months old and is 
still FREE.  ;-)  And maybe (Real Soon Now) I will add gif-ability.  
[This is now in version 0.11 of HILLZOOM.  But DEEPZOOM has a flu 
bug ;-) and produces faulty GIF files.]  Now automatic generation 
of a gif sequence which can be used for an mpeg ought to really 
jam my machine.  ...  I then asked:
  Any ideas?

fargher@unixg.ubc.ca (Robert Fargher) writes:
  Yeah, what an opportunity for another GIPPP.  [GIPPP, coined by 
Jon Noring, stands for Grand Internet Parallel Processing Project]  
Farm out the gif generation to produce the worlds greatest, deepest 
fractal zoom animation.

mcdavey@maths.tcd.ie (Matthew Davey) said it all:
  Wild.  Absolutely.

Now let's see.  70*ln(10)/ln(1.15)=1153 frames.  At 4 frames per 
second we would have about 5 minutes of non-stop Mandelmania!  Is 
320x200 a reasonable image size?  

I observed that in his novel, Clarke notes that "there are no less 
than sixty-three examples of Mandelmania now in the data banks."
There are 83 ZOOMers now and if the those 63 who have contracted
Mandelmania [mentally locked up at magnification 10^40 Mandelbrot 
Zoomer] take this on, using an average of 28 hours for each image, 
the task would be done in three weeks.  We'll have to discontinue 
our medications and for some, disconnect our implants  ;-). 

We should carefully consider strategy and tackics while choosing a
focal point.  We could revive something like the recent 'have we 
seen it all' thread or the 'best of fractint' thread to plan on 
what we would like to see in these images.  

Unfortunately DEEPZOOM is only an MSDOS product since it depends on 
an assembly language C++ class for extended precision arithmetic.
Until a work station version is written, work station folk would 
certainly be welcomed in a search for a good zoom focus.  The last
GIPPP which determined bounds for the Mandelbrot Set involved 
mostly workstations and only 2 PC class machines.  This for now 
would have to be mostly a PC project.  ;-)

<Climbing up on soap box and shouting toward the back of the crowd>
If you don't have the lastest DEEPZOOM and have not recieved my
email instructions on how to get it, just send me email.  Now it
may seem I'm being a pain here, but I know of fine programs placed
at mirror sites containing pleas for money or just fame.  Authors
hear mostly no feedback and have no idea if anyone is even looking
at the program.  I at least now have an idea how many folk have 
any interest and a number have sent me nice remarks.  Thank you 
very much.  The GIPPP idea may well have never occured without 
this novel approach.  DEEPZOOM is mainly a toy, a novelity, a 
research tool.  So, if some of us can have some fun, fine.  [There 
have been some discussions with the FRACTINT folk about putting 
these capabilities into FRACTINT.  Now that would be great.]
<Falling off soap box and limping back to my 486/33>

Warmly,
Jay "Not tonight honey, I'm with the guys planning a GIPPP" Hill
-- 
{ hilljr@jupiter.saic.com } begin writeln(3*ln(640320)/sqrt(163):17:15) end.
void main(){double sqrt(), y=1/sqrt(2.), a=.5, m=1,z; int n=0;
for(;m*=2,z=sqrt(1-y*y),y=(1-z)/(1+z),a=a*(1+y)*(1+y)-m*y,n<4;n++);
printf("%17.15lf\n",1/a);}

In article <1993Dec17.221053.12835@ast.saic.com>, 
  hilljr@jupiter.saic.com (Jay R. Hill) writes:
|> Monday I wrote that DEEPZOOM is just about 2 months old and is 
|> still FREE.  ;-)  And maybe (Real Soon Now) I will add gif-ability.  

Version 0.12 is now ready with internal gif and pcx writing ability.

|> Unfortunately DEEPZOOM is only an MSDOS product since it depends on 
|> an assembly language C++ class for extended precision arithmetic.

Several have emailed about porting to Windows and workstations.  Maybe
I can figure out what we can do after the holidays.

|> <Climbing up on soap box and shouting toward the back of the crowd>
|> If you don't have the lastest DEEPZOOM and have not recieved my
|> email instructions on how to get it, just send me email.  

I guess I better lower my voice,  ;^) I got two dozen emails over the 
weekend.  Now there are 107 potential Mandelmaniacs who will no
doubt cause a run on medications and implants in a hopless attempt
to return to normal.  ;-)

One wrote: "You couldn't sent a more effective sales brosure.  I was 
thinking about getting FRACTINT, but, your program sounded much 
better."

One simple fact: FRACTINT has no equal!  GET IT too!

|> <Falling off soap box and limping back to my 486/33>

I'll be OK, I'll just sit here and watch this little white dot...

Warmly,
Jay "Not tonight honey, my eyes are moving in a raster pattern" Hill
-- 
{ hilljr@jupiter.saic.com } begin writeln(3*ln(640320)/sqrt(163):17:15) end.
void main(){double sqrt(), y=1/sqrt(2.), a=.5, m=1,z; int n=0;
for(;m*=2,z=sqrt(1-y*y),y=(1-z)/(1+z),a=a*(1+y)*(1+y)-m*y,n<4;n++);
printf("%17.15lf\n",1/a);}

In article <1993Dec13.194400.26337@ast.saic.com>,
   hilljr@jupiter.saic.com (Jay R. Hill) writes:
|>        A warning to DEEPZOOM users.
|>
|> Few can forecast the effects of technology on our futures as well
|> as Arthur C. Clarke.  Since he has written of at least one who
|> zoomed too deep once too often, we might want to be careful.  In
|> his 1990 novel _The Ghost from the Grand Banks_, he describes one
|> Edith Craig at the Torrington Clinic for psychological disorders,
|> who is mentaly locked in a 40 digit deep zoom into the Mandelbrot
|> Set.

In the novel Clarke also describes a small lake on the grounds of a 
wealthy Mandelmaniac which had been contoured into the shape of the 
Mandelbrot Set.  Now we don't know to what level of detail the lake
was created, but it was large enough for one to enjoy boating.  

Now let's suppose we want to build one of these lakes using bricks to
line the edge.  For starters, let's just design the contiguous interior
composed of the cardioid and tangentially attached buds.  We will not
consider the filements and detached cardioids with their buds.  Now
how many bricks will we need?  This is a question not unlike the one
Dr. Mandelbrot asked about the length of coastline around England.  

What is the fractal dimension of the contiguous Mandelbrot Set?  

Let P be the perimeter of this contiguous Mandelbrot Set lake and let
p denote the perimeter of the smallest bud to be constructed.  The 
size of p is some function of the brick size, noting that it will take
six or eight bricks minimum to line a nice looking bud.  Now let's
start by computing the perimeter of the cardioid and neighboring buds.
Then we shall sort this list by the perimeters from large to small, 
so we can ignore those smaller than whatever cutoff size we want.  We 
shall also compute the area of each component.  That will let us know
how much water we will need to fill this lake.  The list of components
will look something like this.

period    real(center)      imag(center)    perimeter        area
   1   0.00000000000000  0.00000000000000 4.0000000000 1.17809724509617
   2  -1.00000000000000  0.00000000000000 1.5707963268 0.19634954084936
   3  -0.12256116687665  0.74486176661974 0.5938968503 0.02806800174027
   3  -0.12256116687665 -0.74486176661974 0.5938968503 0.02806800174027
   4  -1.31070264133683  0.00000000000000 0.3708738371 0.01094567455674
   4   0.28227139076691  0.53006061757853 0.2764613682 0.00608217682639
   4   0.28227139076691 -0.53006061757853 0.2764613682 0.00608217682639
   5  -0.50434017544624  0.56276576145298 0.2437413687 0.00472768603336
   5  -0.50434017544624 -0.56276576145298 0.2437413687 0.00472768603336
   6  -1.13800066665096  0.24033240126210 0.1666690123 0.00221054754153
   6  -1.13800066665096 -0.24033240126210 0.1666690123 0.00221054754153
   5   0.37951358801592  0.33493230559750 0.1490826608 0.00176866021683
   5   0.37951358801592 -0.33493230559750 0.1490826608 0.00176866021683
   6  -0.11341865594944  0.86056947250157 0.1401668781 0.00156343898703
   6  -0.11341865594944 -0.86056947250157 0.1401668781 0.00156343898703
   7  -0.62243629504129  0.42487843647563 0.1307578033 0.00136058402674
   7  -0.62243629504129 -0.42487843647563 0.1307578033 0.00136058402674
   7   0.12119278610591  0.61061169221075 0.1052875813 0.00088215405359
   7   0.12119278610591 -0.61061169221075 0.1052875813 0.00088215405359
   8  -0.35910239011245  0.61735345339883 0.0963333110 0.00073848743594
   8  -0.35910239011245 -0.61735345339883 0.0963333110 0.00073848743594
   8  -0.99944238720657  0.26538753246841 0.0953172405 0.00072299127841
   8  -0.99944238720657 -0.26538753246841 0.0953172405 0.00072299127841
   6   0.38900684056977  0.21585065087082 0.0889638069 0.00062982058930
   6   0.38900684056977 -0.21585065087082 0.0889638069 0.00062982058930
   8  -1.38154748443206  0.00000000000000 0.0814869765 0.00052840454564
   8   0.32481970146546  0.56381562214033 0.0668384002 0.00035550214864
   8   0.32481970146546 -0.56381562214033 0.0668384002 0.00035550214864
   7   0.37600868184677  0.14474937132163 0.0571213948 0.00025964965170
   7   0.37600868184677 -0.14474937132163 0.0571213948 0.00025964965170
   8   0.35903106283661  0.10093487686430 0.0387604563 0.00011955504299
   8   0.35903106283661 -0.10093487686430 0.0387604563 0.00011955504299

In this table, perimeters of components whose periods greater than 3 are
approximate, good to about 6 digits.  The fractal dimension formula is

D = -ln(P/4)/ln(p/4)

Using the values in this table, when I sum perimeters up to p=0.03876046 
I get P=12.08072520.  So D = 0.23838700 and Area = 1.49206225.  Now, if 
the lake is to be about 200 meters across (unity scaled up to 100 meters) 
we will have about 1200 meters of boundary to build.  The smallest bud 
will be about 5/4 meters in diameter.

I got the impression that Clarke's lake is somewhat larger than this.
How about making it about 5 times larger or about a kilometer across.
Running the table on out to p = 0.0075 I get P = 18.4559, A = 1.50146 and
D = 0.2436.  In the limit of very small p I expect D approaches 1/4.

P = sqrt(8/sqrt(p)),  approximately.

My wife's favorite TV show is Wheel of Fortune (the one featuring Vanna).
Their theme song is something along the lines of "I'm a Wheel Watcher,
I'm a Wheel Watcher..." (funny, for the longest time I thought they were
singing "I will watch her").  Well, I prefer a different screen to watch.

Warmly,
Jay "I'm a cardioid watcher, I'm a cardioid watcher..." Hill
-- 
{ hilljr@jupiter.saic.com } begin writeln(3*ln(640320)/sqrt(163):17:15) end.
void main(){double sqrt(), y=1/sqrt(2.), a=.5, m=1,z; int n=0;
for(;m*=2,z=sqrt(1-y*y),y=(1-z)/(1+z),a=a*(1+y)*(1+y)-m*y,n<4;n++);
printf("%17.15lf\n",1/a);}