[Fractint] FOTD 28-10-10 (Curious Coincidence [No Rating])

27 Oct 2010

      FOTD -- October 28, 2010 (No Rating)

Fractal visionaries and enthusiasts:

Today's image gets no rating.  Why?  Because it's a virtual 
repeat of yesterday's image.  The difference is that, while 
yesterday's image was sliced in the Oblate direction, today's 
is sliced in the Rectangular direction, which consists of the 
imag(Z) and imag(C) axes of the Julibrot.

IMO, what made yesterday's scene so curious is the fact that the 
fractal terminates on the left and right at values of plus and 
minus 1.618...., (the golden ratio), while the large valleys 
meet at plus and minus 0.618...., the reciprocal of the golden 
ratio).

This is curious enough, but when I checked the horizontal 
terminal points of today's image, I found a value of plus and 
minus 0.78615....  What an incredible coincidence it is that 
this is the reciprocal of the square root of the golden ratio.  
(The two large valleys in today's image meet at plus and minus 
0.521555...., though I have yet to find anything significant in 
this value.)

Mathematical interest aside, today's image is simply a variation 
on yesterday's image, with the same color palette.  Those who 
enjoyed yesterday's will enjoy today's almost as much.  Those 
who thought yesterday's image was boring will be no more 
impressed by today's.

The name "Curious Coincidence" refers to the mathematical aspect 
of the image.

The calculation time of only 3 seconds is mercifully brief.  The 
best way to view the image is to enjoy it already calculated on 
the FOTD web site at:

       <http://www.Nahee.com/FOTD/>

Wednesday began with clouds here at Fractal Central.  But the 
clouds broke up by midday, leading to a sunny afternoon with a 
temperature of 72F 22C.  These conditions were perfect for the 
fractal cats, who spent several hours in the sun that streamed 
in the southwest window.  My day was average; FL had a similar 
day.  The next FOTD will be posted in 24 hours, and is likely to 
be another mathematical curiositiy, though one with far more 
artistic impact.  Until then, take care, and why does the number 
epsilon equal minus one when it is raised to the power of 
imag(PI)?

Jim Muth
jamth@mindspring.com

START PARAMETER FILE=======================================

CuriousCoincidence { ; time=0:00:03.35-SF5 on P4-2000
  reset=2004 type=formula formulafile=basicer.frm
  formulaname=SliceJulibrot4 center-mag=0/0/1.398/1/\
  90/0 params=90/0/0/90/-1/0/0/0/2/0 float=y inside=0
  maxiter=300 inside=0 outside=real periodicity=6
  colors=000KZzJXwJVsJTpJRmJQjJPgJNeJMbJL`IKYHJVGISF\
  HPEGMDFJCDFBBCA8A958GQKJYKMeKPmKUmKZmKcmKhmKmmIrmF\
  vmCzjHzhNzeSzcXzbTzaPz`Lz_HzZDzZ9zSCzLFzEHz7Kz1Mz6\
  OzAPzFRzJSzNTzSSzXSzaRzfRzhRzjRzlRzmRzoRzqRzrRzzOz\
  zLzzIzzFzzKzzOzzTzzXzzVzzzzzzzzzzzzzzzzzzzzzzzzzzz\
  zzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz\
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  zzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz\
  zzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz\
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  zzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz\
  zzzzzzzzzzzzzzzzzzzzzzzzz }

frm:SliceJulibrot4   {; draws all slices of Julibrot
  pix=pixel, u=real(pix), v=imag(pix),
  a=pi*real(p1*0.0055555555555556),
  b=pi*imag(p1*0.0055555555555556),
  g=pi*real(p2*0.0055555555555556),
  d=pi*imag(p2*0.0055555555555556),
  ca=cos(a), cb=cos(b), sb=sin(b), cg=cos(g),
  sg=sin(g), cd=cos(d), sd=sin(d),
  p=u*cg*cd-v*(ca*sb*sg*cd+ca*cb*sd),
  q=u*cg*sd+v*(ca*cb*cd-ca*sb*sg*sd),
  r=u*sg+v*ca*sb*cg, s=v*sin(a), esc=imag(p5)+9
  c=p+flip(q)+p3, z=r+flip(s)+p4:
  z=z^(real(p5))+c
  |z|< esc }

END PARAMETER FILE=========================================