Example outputs from Genfract

The full size pictures on this page average 70 kb.

    This is a re-creation of Beauty Of Fractals map 48.
It is a detail from the basic Mandelbrot for Z  =  Z^2 + C, showing a satellite set.

    Note that two palette sections have been allocated to the escape fractal. All eleven sections could be used, if desired.

    This is the full picture for BOF fig 58. (Note that the book is 90 degrees out.)
    Z  =  C(Z^2+1)^2  /  Z(Z^2-1)
    The internal colouring of the Mandelbrots shows that it does not possess L-R symmetry. In the orange body to the L, there is an attractor of period two. But in the green body to the R, there are two attractors of period one.
    Similarly, the mauve body top L has two attractors of period 3, while the yellow at top R has one attractor of period 6.

    This is BOF map 6, which is a detail in a Julia for 'Magnet 1'.
    The 'beads' in the spiral tendril are in alternate colours because they belong to different attractors.
Green has the attractor Z = 1 whereas plum-red has the attractor infinity.
A third attractor lives in the orange region, which includes Z = 0.

    Here is BOF fig 45, a detail in the Mandelbrot for the Newton formula for
Z^3  +  (C-1)Z  -  C  =  0.
    In the green areas, the critical value is attracted to one of the roots of the cubic. But in the 'Mandelbrot'-like shape, and the infinitely many smaller ones, the critical value goes to a periodic attractor.

    This is a detail from the 'Julia' of a point in the main body of the 'Mandelbrot' above.
    The points in the cyan area, which include the critical point, go to an attractor of period 2.
    The other three colours are parts of the basins of attraction of the three roots of the cubic for the value of  C chosen.

  Here is BOF map 77-78 , which is the Newton for
Z^3 + (C-1)Z - C = 0 when C = 1/2.
  So its attractors are the roots of the cubic equation
    ( Z - 1 )( Z^2 + Z +1/2 ) = 0

  This is a detail showing part of the basins of attraction of each of the three roots.

  Z = Z^3 + (L-1)Z - L, where
L=1 - 3*C^2, has itself an interesting Mandelbrot.
Note L is represented like this so that the critical values are +/- C.
  For +C you get four Mandelbrot-like sets. The small corner ones look like the normal set with the common slight distortion, but the bigger two are abnormal.

  For -C the map is rotated 180 degrees. A combined Mandelbrot should be used. It is interesting to compare the result with the separate and combined Mandelbrots for Z = Z^3 + (C-1)Z - C . The latter shows the full picture and this transforms into each half of the plane of the former via C=+/-sqr((1-C)/3).

The Mandelbrot of the mapping
   Z = Z^3 - 3* C^3 *Z    is another where the apparent  90 degree symmetry is proved false by the colouring.
The orange buds on the x-axis contain an attractor of period 2, but the green buds on the y have two attractors of period 1.
 

The Julia to the R is for a point in the green bud at the top.

The Mandelbrot on the left is for Z = Z^3 + C.

The main body has two lobes and the buds are double ended.
All of the satellite sets are of this same unusual shape.

The body of the Mandelbrot for Z = Z^n + C has n-1 lobes, and is a hypocycloid. All the satellites have the same basic shape as the main body.

This is the Mandelbrot for Z = Z^3 +CZ.
The L-R symmetryof the shape is again revealed as false.
As previously,the orange bud to the L has one attractor of period 2 and the green to the R has two attractors of period 1.

The Julia here is from a point in the (rather small) mauve bud to the left of the cyan top bud on the main body.
This contains points leading to two cyclic attractors, each of period 3 and highlighted in yellow.
   The corresponding yellow bud on the right gives just one attractor of period 6.

    BOF fig 24 shows the Mandelbrot for
Z  =  exp(Z) + C. The black divergent areas are here coloured brown and the white interior has been coloured by the period of the attractor.

    Higher iteration values do not reduce the solidity of the brown divergent areas, although these are penetrated by microscopic filaments growing in from the coloured fringes as can be seen by zooming in.

  The colours for Mandelbrots relate to the last digit of their associated period :-
      1 sea-green    2 orange        3  mauve          4 cyan      5 plum
      6 yellow          7 magenta      8 lime-green    9 red        0 blue

    Home page

    A basic explanation of Julia and Mandelbrot fractals

    Critical values and compound Mandelbrots

    An explanation of Volterra- Lotka fractals

    Description and examples of Volterra-Lotka fractals

    A short description of what Genfract can do

    Down-load Genfract  (version 3.1, zip 291 kb)

    e-mail :-brian.edwards@felicite-parmentier.freeserve.co.uk