🚀 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);}