Way back in early 2013 Ed Algra wrote to me and asked if I would like to try out his new formula the then unpublished "Lucky". I explained to him that I had only been working in UF for a few months but would willingly help if I could. The formula used here for this tutorial is an earlier one of his and has less variables. As such it is easier to use and more intuitive. Several people have asked me "What is a Ducky?" The easiest explanation is that it is a formula designed to make use of the black area or "inside" of the mandelbrot.
1. Go to ea.ufm and open Ducky Revisted J. If it isn't already then set it to Ducky Lite from the mode drop down menu. It will look like this
2. Now go to the inside tab and find Lyapunov in dmj.ucl. Click on that and It will now look like this
3.We will need to zoom in. To zoom in hold down shift and drag with the left mouse button. Centre it by holding down Ctrl and dragging with the left mouse button. It should now look like this
4.If we go to the inside colouring tab the Lyapunov variable can be changed.
This is imaginary part of z
and this is magnitude of z
with the seed changed to -0.32762/-0.14762
5. One way of generating new patterns is to change the main function. For the purpose of this exercise we will use 'flip' So click on the formula tab and change main function to flip
Your fractal should now look like this but blurrier. To make it less blurry we need to drop the iterations down to say 40. Play with the seed to see if you can get a more distinct image. Also add some colours to your gradient. This next image has the seed changed to -0.32762/-0.05238 and a brown added in to the gradient at position 300. So that you can follow along copy and paste these parameters into a new fractal window then drag the layer on to the top of your layer stack.
title="Fractal1" width=640 height=480 layers=1
caption="Layer 6" opacity=100
center=0.033253175/0.040910805 magn=0.70730426 angle=-0.0531
maxiter=40 filename="ea.ufm" entry="Ducky_Rev_J" p_mode="Ducky Lite"
p_bailout=100000 p_seed=-0.32762/-0.05238 p_variant="*" f_fn1=flip
f_fns=ident p_q=1/0 p_power=1/0 p_q1=1/0 f_fn2=zero p_q2=0/0 p_t=0/0
transfer=linear filename="dmj.ucl" entry="dmj-Lyapunov"
p_trackvariable="magnitude of z" p_negative="absolute value"
p_power=2.0 p_bailout=1E20 p_smooth=no
smooth=yes index=0 color=8716288 index=34 color=307448 index=101
color=8651008 index=200 color=9560063 index=300 color=1118551
smooth=no index=0 opacity=255
Your layer will now look like this
Now for the clever part. Click on the inside tab and then click on the little browse button. Scroll through the .ucl colouring modes in Standard or public and select one you like the look of. Now go back to the formula tab. click on the seed (s) and select explore. You may already be familiar with this tool but for those who aren't it looks like a little target or compass. Move your cursor slowly around in this window till you find a variation you like and just click. All of these fractals below were made using this method.
Finally experiment by adding different colours to your gradient and changing the maximum iterations. Remember in the case of inside colouring less iterations will give a more sharply defined image.
Finally here's one to play with :-