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.

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=========================================