The Beauty of Fractals.
                                  by
                            Richard Hansen

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    EDITORS' NOTE: Richard sent us a disk full of Mandelbrot pictures 
		   which he created on the two programs mentioned in his
		   letter, and they are wonderful images. He has agreed to
		   let us include the disk as one of our Theme Disks, so
		   that any interested users can also have a look. We've
		   set them up as a slideshow, and to be viewed with 
		   cycling colours separately. So if you're interested
		   see MDSERVICES on how to get it.


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         Not so long ago it was only the Mathematicians that knew anything 
    about fractals.  Those wonderfully weird looking geometrical constructs 
    tucked away in the silent of world of mathematics.  Then they were  
    brought to light with the aid of computers and whose actual structure
    was for all purposes unknown.   Mathematicians watched with awe as the
    aesthetically pleasing patterns became visible on the screen.  Many 
    more became entranced by their strange beauty, and have since  
    published their works in this field in a number of available books. 

         Technology has made it possible for you and I to partake in this 
    exiting field of exploration, and add to the ever increasing number of
    "exploits".  The variation of these patterns is endless and no two
    people  are ever likely to produce exactly the same picture, unless 
    they deliberately choose to do so. 

        You see what you are exploring is the seemingly infinite domain of
    a mathematical world from another dimension.  If you read up on the 
    subject, or perhaps dabble in it for yourself you will soon know what I 
    mean.

         The long and the short of it is that either it leaves you 
    unimpressed, or like others you find yourself bewitched by the magic 
    beauty of the fractal kingdom. 

         Beside the appropriate hardware and a good software program you will
    need an ounce of patience and some dedication to produce the 
    State-Of-The-Art pattern or trace.  For the pictures on this disk I used 
    the Amiga 1000,  and the two excellent software programs currently
    available on Public  Domain, i.e  one by Wilcox and the other by French
    and Mical. 

         A quite good and impressive pattern can be generated in about 10 
    minutes, i.e the time it takes the computer to do the amount of number  
    crunching inherent in the picture you want to produce.   Remember that  
    for each pixel plotted on your screen (regardless of the resolution or 
    size of the screen) the computer has to calculate anywhere between fifty
    and one thousand times to test if it should plot the pixel, or give it a 
    miss.  Thus on a lo-res screen it can be up to over sixty million 
    computations if you use the recommended maximum of one thousand 
    iterations (computations) per pixel.  You use the higher number of
    iterations only when you truly have "zoomed" in on a pattern, or else
    you will not be able to get a trace.

         Patterns of less "magnification" require fewer computations and are 
    thus quicker to produce.  Besides not all pixels need the maximum amount
    of iterations before being plotted, only the ones that lie in the darker
    and deeper recesses of this magnificent 'universe' of fractals.  

         For really good hi-res pictures which take some time to produce, I 
    start setting up and tracing before bedtime; switch off the screen and 
    the next morning - voila I have the trace!!

         For the pictures on this disk I have used either of the two
    programs that are available on  Public  Domain.   Each has its pros and
    cons, but  each  will give you the trace you want.

         The Wilcox Program is well documented and easy to use.  The program 
    'senses' the number of iterations for each pixel-plot, and seizing of 
    both, region and window can be done by mouse.  Its only weakness I found
    is the somewhat limited control it gives you over the color set,  and  
    color off-set.  It also does a second trace on pictures that require 
    higher iteration counts for plotting.  The first trace seems to take care
    of pixels requiring a lesser number of computations, whereas the second 
    trace takes care of the rest, filling in the missing details.

         However, it offers two additional mathematical functions which I 
    have not yet explored to any extent,  but of which I included two samples 
    of in pic.21, and pic.22.   The pictures I have produced with this  
    program  are #9, #10, #11, and #19.

         The rest of the pictures are produced by using the slightly more 
    complex, but more sophisticated  program  by  French & Mical.
      
         Of course once you have a picture you can transfer it into any one 
    of the two paint programs available (like  Deluxe Paint or Aegis), and 
    fiddle with it to your hearts content.  There are three such pictures in
    the collection, #6 to which I have added the tiny white blossoms to the 
    'twiggy' trace and now call it "Spring Blossoms",  #21 which is a collage 
    and to which in addition I have added the eyes and now call it "Space  
    Babies".  The original tracing was done with the mathematical function 
    z^3+z(c-1)-c using the Wilcox  Program.  The last pic.22 is one "Space 
    Babie", and its immediate environ which I used as a brush sharply pulling
    it down the screen.  The effect is as seen and it very much now looks 
    like a reflection on a Japanese lantern or vase..whichever. 

         To some of the other pictures I have added borders to the free 
    spaces on either side to improve their setting.  As you can see the  
    variations on a theme are endless and choosing different colors on the 
    same trace can not only bestow a different mood on a picture, but quite 
    often profoundly change its meaning,  for which especially pic.#11, 12 
    and 13 are examples. 

         I hope that the pictures in this modest 'Artgallery' give you some 
    idea of what can be achieved with this new art form.  I am no expert
    (yet..), and what you see here is perhaps some distance away of the truly
    ideal and perfect.

         Perhaps, some day a benevolent sponsor can stage a competition for 
    the best tracing achieved on an Amiga Screen, and for first price offers
    a fully sync. screen of 1000 x 1000 pixel resolution and the rest of the
    hardware to go with it.

         Just think what you could do with that... the mind gyrates!

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