;     Date: Mon, 09 Sep 2013 16:40:38 -0400
;     From: Jim Muth <jimmuth@earthlink.net>
;  Subject: [Fractint] FOTD 09-09-13 (Not Critical [A-3,M-8])
;       Id: <1.5.4.16.20130909164147.2a772ff8@earthlink.net>
; ---------
; 
; FOTD -- September 09, 2013 (Rating A-3,M-8)
; 
; Fractal visionaries and enthusiasts:
; 
; I have received a couple inquiries over the past few days about 
; critical points and their relation to fractals, and also how I 
; use Fractint to graphically find critical points.  This will be 
; the topic for the next several FOTD discussions.  Hopefully, my 
; limited math knowledge will be equal to the task.
; 
; To begin, a critical point is a point on the graph of a function 
; where the rate of change is zero and the tangent is therefore 
; parallel to the X-axis.  This point can appear as the tip of a 
; parabola or a point of inflection, where the direction of 
; curvature changes.
; 
; In fractals, when Z is initialized to a critical value, the 
; fractal will consist of undistorted Mandelbrot stuff, such as 
; appears all around the edges of the classic Mandelbrot set, and 
; the fractal's depths will be filled with intact minibrots.  In 
; the classic M-set, the critical value is zero, but in today's 
; fractal, in which Z is initialized to zero, the result is 
; distinctly non-critical.  No perfect minibrots lie in its 
; depths, and no undistorted buds lie along the edges of the large 
; bays.  For those who cannot wait, if any, today's fractal has 
; three actual critical points.  These are:
; 
;   +0.2246790860663939  -0.2602855645470608  -0.9753887460956279
; 
; If one of these values is entered as the value of real(p5), the 
; resulting fractal will closely resemble today's non-critical 
; version, but it will have many critical areas, especially along 
; the negative X-axis, with intact buds, and minibrots in their 
; depths.
; 
; The name "Not Critical" is a simple truism.
; 
; Since the math, which rates a lofty 8, is the chief topic of 
; today's image, I put little effort into the coloring, which 
; rates a lowly 3.  The fractal is a parent fractal, this is true 
; by definition, but it holds little of interest.  The thin 
; vertical column in the center of the main bay is created by the 
; 0.01 part of (1/Z) in the iterated expression.
; 
; I will resume this topic tomorrow, when I will describe my 
; rather convoluted technique of finding critical points 
; graphically with the Fractint program.  
; 
; The calculation time of a fireball 20 seconds will cause no 
; grief.  The web sites can bring extra assurance of satisfaction.
; 
; The finished image is posted at:
; 
;       <http://www.crosscanpuzzles.com/Archives.html>
; 
; <http://www.emarketingiseasy.com/TESTS/FOTD/jim_muths_fotd.html>
; 
;        <http://www.Nahee.com/FOTD/>
; 
;   <http://user.xmission.com/~legalize/fractals/fotd/about.html>
; 
; An un-notable day passed with little fanfare here at Fractal 
; Central today.  The sky was mostly cloudy, but the clouds were 
; thin enough to give the impression of a mostly sunny day.  The 
; morning temperature of 45F +7C made sweaters comfortable, but 
; the afternoon high of 73F 23C was near perfect.  The fractal cat 
; slept through most of the afternoon, while the fractal humans 
; accomplished what needed to be done.
; 
; The next FOTD will be posted in the appropriate time.  Until we 
; see what is appropriate, take care, and, "I think that crocodile 
; is still alive!" Tom said off-handedly.
; 
; 
; Jim Muth
; jimmuth@earthlink.net
; 
; 
; START PARAMETER FILE=======================================

Not_Critical { ; time=0:00:20.00 SF5 at 2000MHZ
  reset=2004 type=formula formulafile=basicer.frm
  formulaname=MandelbrotMix3a function=ident passes=1
  center-mag=-0.12/0/0.8 symmetry=xaxis
  params=0.1/2/1/3/0.5/6/0.01/-1/0/0
  float=y logmap=1 maxiter=500 inside=0 periodicity=6
  colors=000D1iD1lD1nH3oK4pN5pR6qU7qX8r_9rcAsfBtiCtm\
  DupEusFvvGvsJrpLomNlkPihSeeUbcW_`YXY`TWbQTdNQfKOiG\
  LkDImAGo7Fn8Fn8Fn8Fn9Fn9Fn9Fn9EnAEnAEnAEnBEnBEnBEn\
  BbqXXp`SpdMphHplKmiNjgPgeSdcVaaX___XYbUWdRUgOSiMQY\
  jPXfQXcRX_SXXTWUUWQVWNWWKWXMVXNUXOTYPTYQSYSRZTQZUQ\
  ZVP_WO_YN_ZN`_M``L`aK`bKZ`JX_JVYITXIRWIPUHNTHzRGzQ\
  GzPGzNFzMFzKEzJEzIEzLJzONzQRzTWzW_zYczeVzlNmsFmz7m\
  sAmlCzeEzZGzSIzVKzXLzZNz`OzcQzeRzgTziUzh_ozSnzSmzS\
  lzSOzJQzLRzNTzOUzQWzSXzUZzW_zXazZbz`dzbezcbzd_zeXz\
  fUzgSzgPzhMziJzjGzkEzkBzl8zm5zn2zo0zo4zn7znAzmDzmG\
  zlJzlMzkQzkTzjWzjZziazidzhgzhjzhhzffzddzcbza`z_ZzY\
  YzWWzUUzTSzRQzPOzNMzLKzKIzIHzGFzEDzCBzA9z97z75z53z\
  31z10zo0zg0z`0zT0zM0zE0z7wzXrzUmzRizPdzM_zKWzHRzFN\
  zCIzADz79z54z2IzBHzAGzAFz9Fz9Ez8Dz8Dz8Cz7Bz7Bz6Az6\
  9z59z58z57z46z46z35z34z24z23z22z12z11z00z0JzlHzgFz\
  bDzYBzT9zOJzLKzJKzHLzFLzD }

frm:MandelbrotMix3a {; Jim Muth
z=real(p5), c=fn1(pixel), a=real(p1), b=imag(p1),
d=real(p2), f=imag(p2), g=real(p3), h=imag(p3),
j=real(p4), k=imag(p4), l=imag(p5)+100:
z=(a*(z^b))+(d*(z^f))+(g*(z^h))+(j*(z^k))+c,
|z| <=l }

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