The Julia Set
The Mandelbrot Set
and the Julia
set
are both famous sources
of fractal images, partly because of the intricacy of the resulting
shapes, and partly because they're so easy to generate
using a computer.
Both types of image use
variations on the same method. We take a parameter z
, we square it, we add a constant, c, and
we then continue squaring the result and adding a constant
until the
result shoots off to infinity (or doesn't).
We can write
this repeating or iterating process,
where we
continually square z and add a
constant, as:
z
→ z^{2}
+ c
,
and we can check how quickly the process heads for infinity, as a
function of z or c.
To make things more interesting, we don't use conventional
numbers for z and c, but complex numbers, which combine a real
number component with an imaginary number
component.
Julia vs Mandelbrot
The Mandelbrot Set
and the usual Julia Set images use the same formula, but
apply it in different ways.
Since z and c
are complex numbers, they each have two components, giving us four
parts in total. For the purposes of these pages, we'll
label these four parts A, B, C,
and D.
A and B
are the real and complex components of the starting
value of z,
and C and D are the real and
imaginary components of the constant, c,
that we add every time the formula cycles,
z 
c 
/ 
\ 
/ 
\ 
real
 
imaginary
 
real
 
imaginary
 
A 
B 
C 
D 
... Mandelbrot Set
If
we start the initial values of z at zero,
and plot the values that we're using for the two
components of c on the horizontal and
vertical axes of a graph – if we
set AB to zero
– graphing
CD gives us the Mandelbrot Set.
Colouring schemes
Technically, the "set" is the "set
of points"
that are trapped by the formula. Points outside the Mandelbrot
Set
escape to infinity when we cycle the formula, points inside the set
don't. So we run the formula a large number of times, and count how
many cycles it takes to cross a threshold that indicates
escape.
We can then colour the diagram in two colours depending on whether the
centre of a pixel is "inside" or "outside" the critical set
boundary.
The
problem with this approach is that the set has a lot of fine detail
that's missing from the blackandwhite diagram above
– we may
want to know where the set has fine threadlike structures connecting
certain details, and these threadlike details are often too thin to
show up on a conventional plot. We can try calculating the set to finer
and finer detail, testing multiple points within
each pixel ("oversampling") and then using shading
or boundary weighting to force the subpixel detail to show up (e.g. the
"Minkowski sausage" technique), but this requires
extra computing time. Some of these fine threads will be arbitrarily
fine, and will still be missed no matter how strongly we oversample.
However, there's a simple method of indicating proximity
to the set boundary: We're already calculating the number of
"cycles for escape" for every pixel, so if we use this information to
colour the graph, we end up with an external "contour map"
that
indicates where fine boundary detail is located, even if none of that
detail has been calculated directly. Applying nice colour schemes to
these external contours is also a good way of generating evocative
colour images.
... Julia Set
If we try the
same exercise with the other two values – if we set C
and D to zero, and plot a graph of the dependency
of the formula on A and B
–
then the boundary of our "escape" values turns out to be a boring
circle:
HOWEVER ...
if we repeat the exercise with
different fixed values of C and D,
we get a range of different fractal images – graphing
AB
gives us a conventional Julia set image, and adjusting
varying C
and D then lets us control which Julia
image we
get. Here are a few examples:
Mapping the Julia Set
It can be handy to have a map showing where some
of the more interesting
Julia Set images can be found. Here's a rough map of how
these images vary
with our choice of C and D.
In the table below we're using increments of 0.1, but we can
specify C and D
with arbitrarilyfine precision to call up variations that lie between
the sample images in our table.

 
 
 
 
 
 
Julia Set
Map 
+ 
+ 
+ 
+ 
+ 
+ 


1.0 
0.9 
0.8 
0.7 
0.6 
0.5 
0.4 
0.3 
0.2 
0.1 
D 
0.1 
0.2 
0.3 
0.4 
0.5 
0.6 
0.7 
0.8 
0.9 
1.0 

0.7 

+ 
0.6 
+ 
0.5 
+ 
0.4 
+ 
0.3 
+ 
0.2 
+ 
0.1 
+ 
0 
C 
0.1 
 
0.2 
 
0.3 
 
0.4 
 
0.5 
 
0.6 
 
0.7 
 
0.8 
 
0.9 
 
1.0 
 
1.1 
 
1.2 
 
1.3 
 
1.4 
 
1.5 
 
1.6 
 
1.7 
 

1.0 
0.9 
0.8 
0.7 
0.6 
0.5 
0.4 
0.3 
0.2 
0.1 
0 
0.1 
0.2 
0.3 
0.4 
0.5 
0.6 
0.7 
0.8 
0.9 
1.0 

The outline of the darker region in the centre of
this table looks rather familiar. Here's
why ...
all original material
copyright © Eric Baird 2007/2008
