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Lystad Fractal Info.
Below is a composite of 3 different write-ups I've made for different purposes
at different times. It needs a bit of polishing, but if you want to learn something
about fractals try reading what follows. I have several sections entitled:
Introduction
Complex Numbers
Julia Sets
What is a Fractal
A Little History
My Fractals
Godric's has more info.
Introduction
Around Oct. 1984 a friend pointed out the article in "Scientific American" about fractals, and I read my copy. There were some amazingly pretty pictures and a short history of fractals, which you can find elsewhere on the net (search for "fractals", there's a ton of it) Most of the pictures showed the Mandelbrot Set, named for Benois Mandelbrot. This set is sort of the "cannonical fractal" in some sense.
Complex Numbers
Those of you who are familiar with complex numbers will want to skip ahead to the next title. For those of you who are not, let me give you a little description of where my fractals live, mathematically.
A long time ago, the only numbers people had any knowledge of were the counting numbers: 1, 2, 3, 4, ... What other numbers could there be, after all? Well, there could be fractions, ratios, or rational numbers. A half, 1/2, was an easy idea to get a hold of. The Arabs extended our number system by adding zero. That's not hard to grasp either. Somebody, an accountant of some kind I'd guess, invented negative numbers. Now that was a bit of a stretch. You can't have minus 3 apples. Yet the idea was useful for keeping track of money, since people can have money or owe it, as well as other things.
There are lots of operations in mathematics, and, when possible, mathematicians find it handy for abstract manipulation to identify opposites for each of them. For addition there is an opposite, subtraction. Multiplication has division. There are lots of others. One operation that doesn't seem like much more than multiplication is squaring a number, that is, multiplying it by itself. The square of 3 is 9 (i.e. 3 x 3). The square of 2 is 4. The square of 1 is 1. The square of 0 is 0. The square of -1 is 1. The square of -2 is 4, and so on. It is an interesting fact that all squares are positive numbers.
So now, what's the opposite operation for squaring? It is taking the square root. If you start with some number, the square root of it is the number you could multiply by itself to get back to the number you started with. The square root of 4 is 2 but it is also -2. The square roots of 3 and 2 are numbers that are hard to write down. They cannot be written as fractions, though that was not known for a long time, but that's another story. The square root of 1 is 1 and also -1. The square root of 0 is 0. The square root of -1 is... Wait a minute, if all squares are positive, then there is nothing that can be multiplied by itself to give a negative number.
Well, there really aren't negative numbers either, in some sense, but that doesn't
keep people from using them. They are useful mathematical tools. Likewise, there is
no reason one couldn't imagine some special number that is totally unlike the numbers
we know that when multiplied by itself results in -1. That's just what mathematicians
have done. It is called Now just i by itself isn't much use, I guess. You need to be able to do some math
with it. If you had two of i, that is i 2 times, you would write it as 2 times i, or,
more briefly, 2i. It makes sense to multiply i by regular numbers therefore. You could
have 3i or 4i or even 3.1416i. What about multiplying 3i by 2i? Well, remember that
3i is 3 times i and 2i is 2 times i. When you multiply several things together, it
makes no difference what order you multiply them in, or which you multiply first.
Therefore 3 times i times 2 times i is the same as 3 times 2 times i times i. This is
the same as 6 times i times i and that's the same as 6 times -1 or finally -6. That's
really about it for multiplying with i.
How about addition? Well, as I said, i is a different sort of a number. It is
imaginary. Adding real numbers and imaginary numbers is like adding apples and
oranges. They don't really mix. If you had 5 apples and 3 oragnes you would describe
them just that way, 5 apples and 3 oranges. We do the same if we have the real number
5 and 3i. We just write 5 + 3i and that is it. Numbers like this, consisting of
a real number and a multiple of i are called complex numbers.
If we want to add 5 + 3i to 7 + 2i
we combine the apples and we combine the oranges, getting 12 + 5i.
There's only one more operation you need to know to understand the basic operations
behind the fractals you will see on my Web pages. What if you have 5 + 3i and want
to multiply it by 7 + 2i? It's really not much different than multiplying 8 by 9 but
thinking of the 8 as 5 + 3 and thinking of the 9 as 7 + 2. To multiply one sum by
another you have to multiply each part of the one sum by each part of the other and
add the different products together. That is, we multiply 5 by 7 and by 2, and then we
multiply 3 by 7 and by 2. That is 5x7 + 5x2 + 3x7 + 3x2 and that's 72. Of course
8x9 is the same as 9x8. We could have thought of the result as 7x5 + 7x3 + 2x5 + 2x3
and that's the same thing. So when we want to multiply 5 + 3i and 7 + 2i we get
5x7 + 5x2i + 3ix7 + 3ix2i. That is 35 + 10i + 21i + 6ixi, or 35 + 31i -6, or 29 + 31i.
Likewise, there is one more convention you need to know. Real numbers can be
placed along a line, like a ruler. There is a place for 1, 2, 3, and all the whole
numbers, not to mention 0, -1, -2, etc. There's room for all the rational numbers
like 1/2 and 4.645 also. Numbers like the square roots of 2 and 3 have their place
too. i has no place on this number line. It isn't bigger than 1, say, not is it
smaller or the same either. Therefore mathematicians create a new line, at right
angles to the real number line, for i and its multiples. The lines intersect at 0.
No real, no imaginary, no anything, zero - it's all the same. This point is often
called the This arrangement allows one to graph complex numbers, much as one plots positions
on a map. Just as a map may use numbers to give an east-west position and a letter
of the alphabet for a north-south position, one can use the real part of a complex
number to get the horizontal position of a point and the imaginary multiple to get
the vertical position. The locations of all the complex numbers are the whole
map. This map of all the complex numbers is called the complex plane.
If you have 7 + 2i, it is seven units to the right of 0, the origin where the lines
(also known as axes) cross, and two units up. What is a unit? It's just a distance
you pick, or which the printer of your sheet of graph paper picked for you. In my
fractal pictures, the whole picture may be only 0.0001 unit horizontally and 0.0001
unit vertically. I label this size as the range of the picture. Fractals have
a literally infinite amount of detail, and so these extremely small areas are often
well worth looking at. Near the beginning of this century a couple of mathematicians, Julia and Fatou, and
possibly others, investigated some fractals that have become known as Julia sets.
Each set is associated with a particular complex number. Let's pick a particular
complex number and call it ZJ. To determine if a particular complex
number, which we will call Z0, is in the Julia set for ZJ, we need
to construct a sequence of complex numbers. Each number in the sequence is found by
squaring the previous number in the sequence and adding ZJ. Therefore,
beginning with Z0 we get There is an interesting fact about the product of two complex numbers.
Think of the line going from the origin (zero on each axis) to each of the two
complex numbers to be multiplied. The answer, the product, will be at the end
of a line whose length is the product of the lengths of the original lines.
Even more interestingly, the angle from the real axis to the line to the product is
the sum of the angles from the real axis to the original lines. Thus, when we
square a complex number, the answer is rotated twice as far around in the
counter-clockwise (or anti-clockwise) direction from the real axis as was the
number you started with.
As an example, suppose we start with 1.41 + 1.41i. It's length is
about 2 and it is at 45 degrees above the positive real axis (the real number
line to the right of the imaginary number line). If you square 1.41 + 1.41i you
will get a product about 4 long and at 90 degrees counter-clockwise from the positive
real axis. That is approximately 4i.
In the sequence for Z0 in the Julia set for ZJ, we are
repeatedly rotating and squaring the length and then shifting it horizontally and
vertically with the addition of ZJ. For our given ZJ, it may
be possible to find a Z0 where the sequence will stay around the origin
forever. Such Z0 points are defined to be the members of the Julia set
for ZJ. Other points will create a sequence of complex numbers that get
bigger and bigger, that is they get farther and farther from the origin as we go
farther and farther along in the sequence. Sequences for some numbers may bounce
around the origin and only after quite some time decide to head
out. One convenience for those trying to find out if the sequence is going to get
to bigger and bigger numbers, is that if the length from the origin to a point in
the sequence is bigger than 2 units, it will only continue to get bigger. Getting
bigger, that is farther from the origin, as the sequence continues, is refered to
as the sequence diverging. So one might say, "It can be shown that the sequence
will diverge, if any member of the sequence is more than 2 from the origin."
Julia sets are fractals. They each have a very intricate shape. As you go from
one
Julia set to another, by picking different values for ZJ, you will find
quite a diversity in their appearances. For two values of ZJ that are
very close together, the Julia sets will be very similar.
As you can probably imagine, trying to calculate whether a particular point is in
a particular Julia set can involve a lot of calculations. Trying to get an idea of
the shape of an entire Julia set with only a pencil and paper would be a very large
job. Examining and comparing a large number of Julia sets without a computer would
probably take years.
In more recent times, the late 1970s if I remember correctly, Benois Mandelbrot
got an idea for making a sort of map of the Julia sets. He had his computer draw
each point in the part of the complex plane that he was examining in its own Julia
set. Mathematically speaking, he drew each point Z0 in the Julia set
of Z0. That is, for each point he let ZJ be Z0.
Since for small changes of ZJ there are only small changes in the
resulting Julia set, the picture he got, at each point, looked just like the Julia
set for that point in a small area around the point. The result is known as the
Mandelbrot set. It not only serves as a map to the Julia sets, but it is also one
of the most intricate and pleasing mathematical pictures ever created. You will
see pictures of it these days almost anywhere that the subject of fractals comes up.
Most, if not all, of the fractals I have displayed at this Web site are a slight
variation on Mandelbrot's formula. I have often not squared each number in the
sequence, but cubed it or taken the fourth or fifth power. More importantly, I have
not taken the sequence for each Z0 in its own Julia set, but instead, in
a Julia set related to Z0. In many of the fractals where I cubed each
complex number to get to the next number in the sequence, I added in 3 times
Z0. Where I used the fifth power in the sequence, I used 2 times
Z0. The exact multiplier of Z0 is not important so long as
it is not smaller than 1 or extremely big. The results are qualitatively different
from the Mandelbrot set.
A rough definition of a fractal is, a shape that is pretty much the same
whether you look at the whole thing or a part of it, or a part of the part, and
so on. A shape having that property is said to be "self-similar" or "invariant
under scaling". One of the early mathematicians to think about such things was
named Cantor. When I think of Cantor, I remember that he proved that there are
different "sizes" of infinite sets. But he also described a fractal known as the
Cantor Set. To imagine the Cantor set, imagine a line, let's say it is 9 inches
long, though the length doesn't really matter. The set can be defined by saying
which part of that line is not in the points between 0 and 9 inches do NOT belong
to the set: The middle third is not part of the Cantor set, that is every point
greater than 3 inches and less than 6 inches is out. Also the middle third of
each of the end pieces is out. So everything greater than one inch and less than
2, everything greater than 3 and less than 6, and everything greater than 7 but
less than 8 is out. But that is not all there is to the definition. Each time a
piece is left, you remove the middle third of it. If you were to actually have to
remove the pieces by hand you would never finish, for there are an infinite
number of middle thirds to remove. In fact, each time you remove the middle third
of one piece that leaves two new ones that need their middle third removed. One
interesting thing about this set to a mathematician is that is is "totally
disconnected." What that means is, that given any two points in the set you can
find two open sets, one containing each of the points, but the sets have no points
in common. Such a set is commonly called a "dust". Some, but by no means all, of the
Julia sets are dusts. Can you think of some points that are in the Cantor set?
How about the ones at 0, 1/3, 1/2, 7/9, 6. All but one of those is in the set.
The Cantor set is one example of a fractal. If you look at either half of it
you see an exact copy of the whole set only smaller. Either half of either half
also looks exactly like the whole set, and so on. There are other such
mathematical "curiosities". Some of them are nicely illustrated at
Godric's web site
in the U.K. Mathematicians of the
ninteenth century generally considered these constructions to be intuition-defying monsters and the mathematicians gave them a minimum of attention.
In what follows, I will describe various sequences. Often, such sequences
are given a single letter name and then are given a subscript to indicate a
particular member of the sequence. Therefore, in a sequence of numbers P
representing population size, the first will be P0, the second P1,
the third P2, and so on. When I
want to represent an arbitrary member of the sequence, I will use a variable. So
when you see Pn, you can think of it as just any member of the P sequence
that has been singled out for discussion. When I write a formula involving
Pn, I'm
really saying this is true, no matter what value you might pick for n. Then, when
you see Pn+1, you will know that I'm talking about the next element of the
sequence. So, if you want to think of n as being 6, then Pn is
P6 and Pn+1
is P7. But, it might just as well be that n is 1047, in which case
Pn+1 is P1048.
In 1845, P.F. Verhulst created a simple equation that was designed to model
population growth for any living thing from bacteria to people. The change in
population from one generation to the next is proportional to the size of the
population, when the population is well below the maximum the environment can
support. However, he thought, if the population is near the maximum, the change
should decrease. He came up, therefore, with an equation of the form [this is
from memory and I need to check it.]
The results given by this simple equation were not simple at all. It took
the invention of computers and more than a century before the behavior of the
Verhulst equation was fully understood. For values of R below 2, the population
given by the equation is stable, that is, as more generations go by, the size of
the population approaches some limit and stays there. However, just above 2, the
population begins to bounce back and forth between two sizes from one generation
to the next. At 2.45 it begins to bounce among 4 sizes, and as R increases
further, 4 is replaced by 8, then 16, then 32 and so on. For most values of R
greater than 2.57, no pattern can be found in the changing population sizes from
one generation to the next. The graph of population sizes versus R is itself a
fractal, and amazingly enough, very closely related to the now famous Mandelbrot
Set.
In the mid 1970s, it was discovered that the distances between the values of R
where the number of population sizes changes shortenes by a factor that
approaches 4.669201660910.... closer and closer the larger R gets. It turns out
that this number appears over and over in fractal studies, just as 3.14159....
appears over and over again in the study of conventional geometry and calculus.
If you have studied "Newton's method" from elementary calculus for finding
the root of an equation (i.e. finding x such that f(x) = 0), you are familiar with
the fact that successive approximations are "drawn toward" one of the solutions.
If f(x) involves x squared there will generally be two solutions, for x cubed
there will be three, etc. In the late 1800s Lord Arthur Cayley tried to
investigate the range of infuence of the different solutions, that is, which
approximate solutions would lead to which correct solutions. The boundary
between such regions of influence is a fractal, and its complexity caused Lord
Cayley to give up. Electronic computers or calculators had not been invented, so
trying to get a picture of what was going on was a very difficult task.
During the first world war, a couple of mathematicians named Fatou and Julia,
mentioned above, still without computers, studied the equation
As with the Verhulst equation, the above equation represents a sequence of
numbers, each obtained by squaring the previous and adding a given constant. One
difference though is that the values were not necessarily real numbers, they were
complex numbers and therefore represented points in the complex plane.
It turns out that for a given value of c, the sequence Z0, Z1,
Z2, Z3,
.... will often just give bigger and bigger (in absolute value) complex numbers.
It depends on your choice of Z0. Such a sequence starting at Z0
is said to
"go to infinity" or to "diverge". There are certain choices for Z0 that
never
diverge, however. These are the members of (and so define) the Julia set for the
particular choice of c. While the equation that defines the Julia set for a
given c is simple, the shape of Julia sets is very complicated and, for certain
choices of c, it is quite beautiful. They are fractals. The work of Julia and
Fatou had been largly forgotten, until it was expanded upon by Benois Mandelbrot
in the 1970s.
One other pioneer, along the way to the discovery of fractals, was Edward N.
Lorenz, a mathematician turned meteorologist, at MIT. He tried to make a simple
model of the earth's weather using one of the first computers, one that had
vacuum tubes rather than transistors. There were two very surprising, at the
time, qualities of his model. The first was that the weather patterns in his
model never exactly repeated themselves, though they were often similar for
stretches of time. For the most part, things that had been studied by
mathematicians and scientists of the past repeated themselves exactly when they
repeated at all. The second is the now-famous story involving the "butterfly
effect".
Lorenz's computer printed out the values of the variables in his weather
model at each step. One day he wanted to rerun part of the simulation over again
from a point that was on the computer printout. He set the variables in his
program to what had been printed out and went off to get a cup of coffee. When
he returned he was amazed to see that, while the weather in his model had started
off the same as in the previous run, after a short time it became completely different. At
first he thought his computer wasn't working correctly, but shortly, he realized
that his computer represented numbers with 6 digits of accuracy (e.g. 24.3682),
but he had only entered the 3 digits on the printout (e.g. 24.4).
These small differences had caused a dramatic change in the calculations
after a fairly short time. This was a surprise, because mathematicians and
scientists were accustomed to believing that small changes in the starting point
of an experiment ALWAYS lead to small changes in the result. When a person is
dealing with "linear" equations, that is, ones that don't involve squares of the
variables or any higher powers, that belief is correct. When the equations (or
physical laws) are not linear, that belief may be incorrect. Non-linear equations
or experimental situations are very hard to solve and had traditionally been
either approximated by linear systems where possible or simply ignored. Therefore
Lorenz's surprise was to be expected. People just didn't look at such things.
This great sensitivity to starting conditions, as I mentioned, is known as
the "butterfly" effect. I have read that this refers to the thought that the air
currents from the wings of one butterfly can, after enough time has passed, alter
the weather patterns of the entire world. I am also reminded of a science
fiction story, by Robert Heinlein possibly, where time travel is possible and
people pay great sums of money to be taken back in time to take photographs and
see the sights in the prehistoric world. The tourists walk on elevated paths so
as not to disturb anything and so change the future. But eventually one tourist
strays from the path and, in so, doing steps on a butterfly. No other harm is
done, but when he returns to his own time, the future has been changed
radically.
All or most of the common themes of the above are that of similarity of a
part, or a part of a part, to the whole thing, and that of great sensitivity to
initial conditions. That may not be obvious in some cases, unless you think about
it. In looking at Newton's method and Lord Cayley, which answer you get becomes
very sensitive to slight changes in the initial guess, when you are in the area of
the border between the regions of influence of the various solutions. There is
not just a single border point that one crosses, but a whole infinite sequence
of going from one to another, to the first, to the second, and so forth. It is
often an infinite fractal pattern in the points that lead to one solution or
another.
Benois B. Mandelbrot studied a variety of natural and mathematical phenomena
that had to do with self-similarity. He found that such things as the price of
cotton varied in a way that looked much the same, whether you were looking at
records for a single month or for 100 years. He saw the same self-similarity in
the patterns of interference to signals transmitted long distances over wires. He
realized that similar mathematical shapes, such as the Koch curve can have their
complexity characterized by thier "Hausdorff dimension", a number that describes
how thoroughly lines fill 2 dimensions, and how thoroughly complex surfaces fill
three dimensional space. His well known question, "How long is the coast of
England?" and the answer that it depends on the size of the unit you use to
measure with, "the size of your ruler", illustrate the non-intuitive nature of
complex shapes in nature. The coast of England is a fractal, as is any coast
line. The complexities of the shape of the whole coast line are reflected in the
complexity of the details on much smaller scales. If you measure 50 miles at a
time you miss all the little bays and peninsulas. If you measure a mile at a
time you will measure these and get a longer measure, but not as long as if you
measured with a yard stick and took into account every protruding rock and other
small wiggle in the coast. If you measured on the level of an inch you would get
a still longer length for the coast.
He also looked at the Julia sets and in 1980 created a sort of map that
describes what sort of set you will get for any choice of c. The formula
Fractals occur many places in nature and have been used to create very
realistic-looking pictures in computers. Fractal trees and other plants are made
so that each branch is a smaller version of the tree itself, each sub-branch is a
further scaled-down version, etc., until one decides to stop and add the leaves.
Fractal mountains can be made by adding some large bumps to a flat plane,
dividing up the result into smaller pieces and adding the same pattern of scaled-down bumps, dividing each piece into smaller pieces and adding the same pattern
of even further scaled-down bumps, etc. Fractal clouds can be created similarly
for overall shape, and a similar technique can be used for modeling density of
clouds or other collections of particles (water droplets in the case of clouds).
There is self-similarity in many pictures if you look hard enough for it, and
this fact has been used to develop techniques to compress pictures on computers.
Saving disk space on computers is still an important goal, as well as making
pictures smaller so that they will travel over the Internet more quickly. You
can download a free plug-in for Netscape that will let you view pictures that
have been compressed using a fractal technique. Research in this field is
on-going.
One last thing to notice is that most equations that occur in nature are
non-linear. As I mentioned these have, until recently, been either approximated
or ignored, in most cases, because of their complexity. It is only with the advent
of the computer that we are beginning to make headway on these most commonly-occuring relationships among natural phenomena.
It is an interesting fact that the Mandelbrot Set can be proven to be
connected. That was shown by Adrien Douady and John Hubbard. Intuitively, that
means that it is all one piece, but there is a precise mathematical definition of
connectedness also, available in beginning books on topology.
Though I don't
have a proof, the pictures of my fractals seem to indicate that my fractals are
not connected. Most interesting to me are the areas of roughly constant
divergence rates that meet other areas of roughly constant but different
divergence rates and blend together there, in often very interesting shapes.
(Divergence rates are shown by color in my pictures.)
Sometimes you will see one of those "bay" shaped features in the light
purple that are normally black, but that particular one is green, or
blue, with tiny bits of other colors inside. (Look at "The Snake" below
and to the right of the lower of the two "eyes". This is in my Set #1.)
Such things occur in
these funny areas of different divergence rates. It's as if one had a
fractal with a slow divergence rate and put another, unrelated fractal
with a higher divergence rate on top of the first one. The first one
shows through in places. But it isn't completely that way, because where
those two "different fractals" meet their features blend together in
ways that are sometimes unlike the features of either of the two.
As in the case of the picture I call "Sylvia" (Sylvia is in Set #1 also.) there
is a pattern that is repeated over and over again. Then, there is a shape unique
to that part of the fractal such as Sylvia. Then, some other
pattern begins repeating on the other side. I hope to have the time to
investigate this in detail some day.
In case you are wondering just what the computer actually does when making
one of my pictures or a picture of the Mandelbrot Set, here's what happens. I
pick a little square in the complex plane and if, say, I want the picture to be
100 pixels by 100 pixels I divide the width to get 100 equally spaced real
(horizontal) coordinates and do the same thing in the imaginary direction
(vertically). For each of the resulting 10,000 points I start calculating the
sequence Z0, Z1, .... Z0 is always the current
point I'm going to color in the picture and the formula gives Z1 and
the rest. As mentioned above, it can be shown that if Zn ever gets
outside a circle of radius 2 centered at 0 + 0i for any of these
sequence points the sequence will diverge. So if that happens I use the value
of n, that is how far out in the sequence I've gotten, to choose the color. White
is chosen if I get outside right away. This fades into purple if it takes a
little longer and then I go through the spectrum (rainbow order) to red for the slowest.
Sometimes I go through this sequence of colors a few times for slower and slower
rates of divergence (i.e. number of terms to get outside the circle of radius 2.)
What about the points that never diverge? Well, eventually I have to give up. I
pick some number and say, that's enough. If the sequence has not diverged by that
member of the sequence, then I give up, color it black, and go on. Not all people
use the same choice of colors. In fact you will find as many different choices
of coloring schemes as you will find people drawing fractals. Also, while I
color based on rate of divergence, since that conveys information I am interested
in, some people color based on the absolute value (i.e. length of a line from the
point to the origin/center, 0 + 0i) of the first point that goes
outside the circle of radius two. That makes the picture look somewhat
different, though the shape of the actual fractal will, of course, be the same.
There is much more that could be said about fractals. You may find a lot
more on the Web. If you do find something historical, mathematical, or just
interesting or pretty, I'd appreciate your telling me the URL (net address) by
email.
The bulk of what I have told you above comes from two books, the rest from my
memory. The books are "Chaos, Making a New Science" by James Gleick, Viking
Press, and "The Beauty of Fractals" by H.-O. Peitgen and P.H. Richter,
Springer-Verlag. If you have corrections please email them to me at
lystadmas@centurytel.net
Julia Sets
Z0
Z1 = (Z0)2 + ZJ
Z2 = (Z1)2 + ZJ
Z3 = (Z2)2 + ZJ
Z4 = (Z3)2 + ZJ
etc. So, What is a Fractal?
A Little History
Pn+1 = R * Pn * (1 - Pn)
Zn+1 = (Zn * Zn) + c.
Zn+1 = (Zn * Zn) - Z0
My Fractals
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