**Fractals
for the compleat idiot**

**by
Kathy Roth**

**August
21, 2000**

**Here is
my effort to explain the basics of fractals-please feel free to
comment.**

**Fractals
start with a process called iteration. An operation is repeated over
and over again. Often a very simple formula gives rise to an
incredibly complicated-appearing image. Most people who initially
feel put off by looking at this passed high school algebra and did
things that were much more difficult than this.**

**You can
start with the Mandelbrot formula:**

**z=z^2 +c
(z=z squared + c). **

**This is
really:**

**z(new) =
z(old) ^2+ c. **

**For the
Mandelbrot equation, z is set at the beginning and most often is
(0,0). The c corresponds to the pixel- picture the computer screen
with an x-axis horizontally and a y- axis vertically. **

**So a
point on the x-axis would be (2,0) and a point on the y-axis would be
(0,2) and a point in the lower left would be (-2-,2). **

**(The
second number is actually multiplied by i, the square root of minus
1- more on this later.) (like very later when someone else is doing
the writing)**

**So to see
what color a particular pixel turns out, you put it into the equation**

**z= z^2 +
c**

**z(new) =
z(old) ^2 + c.**

**Start
with the point (2,0) on the x axis. The second number is multiplied
by i,**

**the
square root of minus one, so this number is 2+ 0i= 2. In the
Mandelbrot equation, the z is set at the beginning and the pixel is
going in at the c value.**

**For z=0
and c=pixel=(2,0)**

**z= 0^2
+2= 2**

**Then you
put the 2 in for the old z and get a new new z- **

**this is
the iteration part.**

**z= 2^2
+2=6**

**z=6^2+2=38
etc.**

**Try
another pixel- (-0.2,0.2)**

**z=0^2 +
(-0.2 +0.2i)= -0.2 + 0.2i**

**z=(-0.2 +
0.2i)^2 +(-0.2 +0.2i)**

**etc.**

**So where
does the beautiful image come from? **

**You run a
point corresponding to the pixel through the formula. You choose a
bailout value- most often the circle corresponding to r^2 = 4 (points
at (2,0) (0,2) (0,-2) (-2,0). **

**If the
point runs out of this it is "outside " and if it is less
than this it is "inside". (Most often "outside"
gets a color and "inside" is black- more on this later)**

**The path
that the number traces as it goes through the result of each
iteration is called the "orbit".**

**The orbit
could go on forever- you can plug it into the iteration formula
endlessly but sooner or later there's a clear trend. Is it going to
be outside the bailout circle or not?**

**The
number of times you plug it into the formula is called the "maximum
iterations" or maxiter- one of the things you can vary in the
fractal program.**

**Here is
an example. If you are using Ultrafractal, you can set the "inside"
to black and the "outside" to white- the most simple
coloring method. This is the Mandelbrot fractal with that coloring, a
bailout of 4 , and a maxiter (i.e. maximum iterations, the number of
times you plug it into the equation) of 100.**

Fractal2
{

fractal:

title="Fractal2" width=320 height=240
author="kathy roth"

created="August 20, 2000"
numlayers=1

layer:

caption="Layer 1" visible=yes
alpha=no

mapping:

center=-0.5/0 magn=1
angle=0

formula:

filename="Standard.ufm"
entry="Mandelbrot" maxiter=100 percheck=normal

p_start=0/0
p_power=2/0 p_bailout=4

inside:

transfer=none
repeat=yes

outside:

transfer=linear
repeat=yes

gradient:

smooth=yes numnodes=4 index=0
color=16777215 index=89 color=16777215

index=238 color=16777215
index=334 color=16777215

}

**So where
do all the colors come from? The basic coloring method is "escape
time" coloring. The simplest situation is that the pixels whose
orbits never "escape" i.e. are inside the bailout circle,
are colored black. The pixels that are "outside" are
colored in lots of different ways. The escape time coloring is
simple.**

**(Yeah- I
don't really understand the others. Well, sometimes sort of.) Escape
time coloring colors the pixels in terms of how many iterations it
takes to "escape" the bailout circle.**

Fractal3
{

fractal:

title="Fractal3" width=320 height=240
author="kathy roth"

created="August 20, 2000"
numlayers=1

layer:

caption="Layer 1" visible=yes
alpha=no

mapping:

center=-0.5/0 magn=1
angle=0

formula:

filename="Standard.ufm"
entry="Mandelbrot" maxiter=100 percheck=normal

p_start=0/0
p_power=2/0 p_bailout=4

inside:

transfer=none
repeat=yes

outside:

transfer=linear
repeat=yes

gradient:

smooth=yes position=-14 numnodes=9 index=0
color=34303 index=1

color=58879 index=13 color=65535 index=23
color=11160234 index=91

color=7667829 index=166 color=6946890
index=288 color=6291488

index=374

color=1048576 index=385
color=0

}

**I think
it is easier to find really lovely images with the Julia sets than
with the Mandelbrot set.**

**In the
early 1900's, before computers, before cars were common, around the
time that the theory of relativity was being developed and way before
DNA was discovered, Julia came up with the Julia set.**

**(Julia
was a man-sorry women, as far as I know Sylvie Gallet was the first
woman in fractals, followed by Janet Parke Preslar and Wizzle- hey
this is not my field!) **

**The
equation for the Julia set is the same as the Mandelbrot set, except
that z=pixel and c is set as a constant in the beginning- i.e. c is
the "Julia Seed." **

**z=z^2 +
c.**

**If you
start a Mandelbrot set and hit "switch" it gives you the
Julia set for whatever value of c you have chosen. **

**It has
been described that the Mandelbrot set is the "map" to the
Julia sets- that is if you take a value of "c" for the
Mandelbrot set and apply it to the Julia set you get a variety of
interesting images. You can try this- bring up a Mandelbrot set in
Ultrafractal and put the pointer on one of the bays or somewhere near
the edge- if you hit "switch" you get a different Julia
set.**

**Really,
the whole thing is part of a multidimensional "Julibrot"
image. If the computer screen is a graph of the pixels, the
Mandelbrot set is a graph of real c and imag c. The Julia set is a
graph of real z and imag z. There are several other possible ways to
view this. Jim Muth has been doing an exploration of the
multi-dimensional Julibrot- his daily Fractal of the Day is on the
fractal philosophy and the fractint list. There is real c and real z,
real c and imag z, imag c and real z, imag c and imag z. Jim Muth
calls these the oblate, rectangular, parallel and something else
dimensions-**

**Addendum
by Julian Smith**

**Here's an
interesting exercise if you want to understand a bit of what is
happening when a fractal is calculating. This is a bit of an exercise
you can do quite simply on excel or other spreadsheet packages. I
presented a demonstration of this to some students recently. It was a
bit beyond most of them (some were only twelve), but not by much. If
we had been able to spend more time on it, I think most of them would
have got it.**

**Here's
some instructions for those interested.**

**Start off
with a new spreadsheet.**

**In cell
A1 type**

** 0.3**

**In cell
B1 type**

** 0.3**

**This is
the coordinate we will start with -- (0.3,0.3) We can change**

**this
later.**

**In cell
A2 type**

** =A1+A1*A1-B1*B1**

**In cell
B2 type**

** =B1+2*A1*B1**

**This is
the formula which will produce a new coordinate.**

**Select
cells A2and B2. There should be a small square visible on the bottom
right corner of B2. Click on this and drag down to B20. This copies
the formula twenty times so that it will be repeated (iterated)
twenty times.**

**Now,
here's the fun part. Use your mouse to highlight A1 to B20. Click at
a button at the top of the screen -- the one that is labelled "Chart
Wizard". This will produce a graph of the orbit of your point.
You will need to choose XY scatter graph type. You then have a number
of sub options. Click one of the ones that has points connected
together. **

**Click
"next" and "finish" to produce the graph.**

**You
should now see an erratic zigzag sort of a path, starting at the
point (0.3,0.3) and finishing in a blob of points very close together
near (0,0) This is the orbit of (0.3,0.3) using that particular
formula.**

**Now what
you can do is change cells A1 and B1 to find different orbits. **

**You
should find two types -- **

** those
where the points zigzag around a bit, and then settle down**

**somewheer
near (0,0).**

** those
where the points fly off to infinity at a high rate.**

**You can
find some places where a very small change in the starting coordinate
will produce huge changes in the shape of the orbit. This is called
chaos theory.**

**What
Mandelbrot did in the eighties was to paint different starting points
different colours depending on how the orbit behaved. The ones that
flew off to infinity, he painted white, and the ones that stayed
around zero he painted black. The black region he drew became known
as the Mandelbrot set. All points are either inside the set, or
outside. (UF uses these terms) **

**(Ultrafractal
needs some guidelines to tell it when to stop calculating the orbit,
decide if it is an inside or an outside point, and start calculating
the next point. The first guide is the Bailout value. If either of
the coordinate values gets bigger than this value, then UF calls it
an outside point. The other guide is maximum iterations. This was set
to 20 on your spreadsheet. If a point has not bailed out after a
certain number of iterations, then UF calls it an inside point.)**

**The
interesting thing to note is that the mandelbrot set has a perimeter
of infinite length -- one that twists and turns and folds back on
itself an infinite number of times. It encloses a finite space
though. This is the definition of a fractal (in simple terms).**

**What
ultrafractal does is enably you to colour the points with different
colouring algorithms. Not just black and white the way Mandelbrot
did, but a range of different colours depending on the shape and
behaviour of the orbit. These are colouring formulas or UCLs.**

**The other
thing that ultrafractal does is allow you to use different formulas
on your points -- not just the simple one that you typed into the
spreadsheet. As Kathy has pointed out, the Julia formula is related
to the mandelbrot formula, but slightly different, and produces
different shapes -- a lot of spirals. Other formulas come from
different fields of maths and science -- Volterra-Lotka from biology,
Newton methods from engineering and physics, and so on. Also, clever
people on this mailing list write formulas for artistic purposes --
those which will produce a particular desired effect.**

**And
because Frederik has produced such a wonderful piece of software,
your computer can produce thousands of calculations per second --
working out all these orbits. This allows the fractal artist to
forget about all the maths and get on with worrying about the way the
image appears.**

**Hope this
is helpful to someone. The spreadsheet exercise was a bit of an eye
opener for me -- although I knew the theory, I did not appreciate the
behaviour of orbits until I actually saw one.**