Transcript The Mandelbrot Set - Frederick H. Willeboordse

Taming Chaos
GEM2505M
Frederick H. Willeboordse
[email protected]
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The Mandelbrot Set
Lecture 4
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Today’s Lecture
Fractal Dimension
Complex Numbers
Prisoners and Escapees
Julia Set
Mandelbrot Set
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Fractal Dimension
Usually when we talk about dimension we think of lines
or surfaces.
Line: 1 – Dimensional
Plane: 2 - Dimensional
But what does this actually mean? It is (to a certain
degree) related to self similarity!
Previously, we saw that the meter stick is perfectly
self-similar. Consequently, one can rescale it easily.
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Fractal Dimension
For example:
1dm = 10cm
1m = 10dm
Magnify by a factor 10.
To turn this around, when magnifying by a factor 10,
the new stick contains 10 times the number of original
sticks.
I.e. Magnify 1dm by a factor 10 to obtain
1m which contains 10 pieces of 1dm.
1m = 10 dm
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Fractal Dimension
That’s quite obvious of course. Now what happens if we
do the same trick with a square?
1 sq. m = 100 sq. dm
If we magnify the small red square by a factor 10. What
we see is that the new big square does not contain 10 times
as many small squares but 102 times as many squares!
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Fractal Dimension
This leads us to an important observation:
a: number of pieces
S: scaling factor (i.e. magnification)
D: dimension
In other words, the dimensions is:
Defined in this way, D is called the self-similarity
dimension.
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Fractal Dimension
Can we apply this to the Cantor Set? Yes!
If we take a chunk and make it 3 times bigger, how
many copies of the original do we have? 2!!!
So for the Cantor Set
we have:
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Complex Numbers
In order to talk about the Mandelbrot set we need to know
what complex numbers are.
Complex sounds like this is very complex but in fact
it’s not.
We all know that it’s quite easy to solve an equation
like this one:
Seeing that, it’s not particularly far-fetched to wonder
what the solution is of:
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Complex Numbers
Ah, easy enough!
ergo:
Oops! There’s no root for -1 (even if we try very hard
indeed). We’re stuck and it would really be useful to
be able to solve such equations.
The solution is simple, if there’s no number in
existence which is the root of -1, we can just introduce
one. Clever solutions need not be complicated!
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Complex Numbers
All right, we can define
Twisting history a bit we can say: Since we imagined
this solution let’s call it ‘imaginary’.
Then the solution to
or
is just:
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Complex Numbers
Excellent, well if we can do the trick once, we can do it
twice! How about the solution to:
That must be:
Stuck again! Now we don’t know what that is… well
perhaps we can think of some equation whose square
is
Indeed:
or
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Complex Numbers
This brings us quite closely to the general notation of a
complex number:
with a and b real numbers (like 0.231, 1.949, 2.000)
Consequently, a complex
number can be drawn as a
point in a plane where the xaxis is the ‘real’ part a, and
the y-axis the ‘imaginary’
part b.
a
b
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Complex Numbers
If we have:
Then the modulus is:
And the conjugate is:
conjugate
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Complex Numbers
Addition:
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Complex Numbers
Subtraction:
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Complex Numbers
Multiplication:
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Complex Numbers
Multiplication as rotation:
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Prisoners and Escapees
Now let us consider the map:
Prisoners
What happens if we start with r = 0.8 and f = 10O:
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Prisoners and Escapees
Escapees
Next, let’s consider what happens if we start with r = 1.5
and f = 50O:
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Prisoners and Escapees
Guards?
And lastly let’s set r = 1.0 and f = 10O:
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Prisoners and Escapees
Graphically
Prisoner
Boundary
Escapee
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Julia Set
Definition
The Julia Set is the boundary between the escapees and
prisoners of a complex iterative map.
In the case of
this means the unit circle
Often the inside of the Julia Set is
filled in and the escaping points are
colored according to how long it
takes for the point to become larger
than a certain value.
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Julia Set
Adding a constant
The situation changes drastically when a constant is added
to the iterative map.
Connected Julia Sets
Disconnected Julia
Sets
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Julia Set
Some famous Julia sets of the complex quadratic map
Dendrite Fractal
c at the boundary of the
Mandelbrot Set (for this
picture, c = i)
Rabit Fractal
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Another Chaos Game
1. Make a drawing of the complex plane
2. Pick any point and call it w choose a complex constant
c
3. Roll a dice, if it is 1,2 or 3 move to
otherwise
move to
.
4. Repeat 3 for a while!
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Another Chaos Game
Result
A Julia Set!
Dendrite Fractal
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Another Chaos Game
Explanation
The Julia Set is the boundary
between escapees and prisoners.
Hence all points not exactly on
top of it move away from it.
If we go backwards in the
iteration, we will get closer to it.
Forward iterate
Dendrite Fractal
Backward iterates
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Mandelbrot Set
There are two types of Julia Sets, connected Julia
Sets and disconnected Julia Sets.
The Mandelbrot Set is defined
as the set of parameters c that
lead to a connected Julia Set.
Alternatively:
M = {c C | c c2+c  … remains bounded }
I.e. parameters c for which the orbit of z0 is bounded.
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Mandelbrot Set
The difference
Julia Set
Initial conditions z0
Find boundary between
prisoners and escapees.
Mandelbrot Set
Parameters c
Find connected Julia Sets.
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Mandelbrot Set
Escape?
How can we know that an orbit escapes to infinity?
Answer: if |z| > r(c) = max(|c|,2), an orbit will escape.
Proof: Take a number such that |z| > r(c) is true.
Iterating once we obtain:
|z2 + c|
Applying the inequality we get:
|z2| = |z2 + c – c|  |z2+c| + |c|
Ergo:
|z2+c| |z2| - |c| = |z|2 - |c| |z|2 - |z| = (|z|-1)|z| = (1+e)|z|
So when iterating z it grows and thus eventually escapes.
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Mandelbrot Set
Prisoners: black
How can we make the pictures?
Escapees: use this
Im Mininum
Im Maximum
A screen is an array of pixels:
Re Mininum
Re Maximum
For each of the pixels,
calculate whether it
escapes and if so how
many steps it takes to
reach e.g. 2. Color the
pixel according to table
above.
Note: Color assignment
is of course arbitrary.
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Mandelbrot Set
The Heart
Intersects real axis
from -0.75 – 0.25.
Julia set c = 0 + 0i
Associated with a Julia
set that has a period 1
attractor.
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Mandelbrot Set
The Buds
Period 4
Period 5
Associated with Julia
sets that have higher
period attractors.
Period 2
c = -1 + 0i
P.3
c = -0.134 – 0.742i
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Mandelbrot Set
Some Features
The Mandelbrot Set is connected
The boundary is a fractal and infinitely long
The dimension is 2
It is quasi-self-similar
And some more pictures It’s better to use the applet though
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Key Points of the Day
Fractal Dimension.
Julia Set
Mandelbrot Set
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Think about it!
What is our dimension?
Julia,
Romeo,
Singing,
Dire Straits
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References
http://mathworld.wolfram.com/
Dave Short’s course on complex numbers
Peter Alfeld’s Mandelbrot Applet
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