HONR 300/CMSC 491
Computation, Complexity, and Emergence
Mandelbrot & Julia Sets
Prof. Marie desJardins
February 18, 2016
Based on slides prepared by Nathaniel Wise
Chapter 8:
The Mandelbrot Set & Julia Sets
There once was a young man from Trinity
Who took
.
But the number of digits
Gave him the fidgets;
He dropped Math and took up Divinity.
A New Kind of Fractal



The mathematical fractals we've looked at are generally self-identical,
in that you can look at them at different scales and they look exactly
the same.
The Mandelbrot and Julia sets are only self-similar: they have a kind
of pattern that's instantly recognizable, at every scale, but no two
scales are exactly the same.
We'll see this kind of “different sameness” again when we start to look
at chaos theory and chaotic systems.
The Mandelbrot Set





Benoit Mandelbrot (1924-2010) is known as the “father of fractal geometry.”
He invented the term “fractal,” and used the new field of computation and
digital computers to explore complex mathematical objects that had
previously only been studied in the abstract.
The Mandelbrot set is defined using an iterative function:
xt+1 = xt2 + c,
where x0= 0.
Reminder: the magnitude of a complex number a + bi, is the Euclidean
distance of that point from the origin of the complex plane, i.e., a2 + b2
For a given value c, it turns out that the magnitude of xt+1 will do one of two
things:

It will always be smaller than 2 (no matter how large t gets), or

It will eventually diverge (i.e., xt will go to ∞ as t goes to ∞).
The Mandelbrot set is defined as the set of values c for which xt+1 remains
smaller than 2.
Computing the Mandelbrot Set


The Mandelbrot set contains those values of c for which the magnitude
xt remains smaller than 2 for all t.
But we have no easy way to know whether the Mandelbrot series
diverges for a given value of c!


If we compute the Mandelbrot series for some value c and the
magnitude of xt ever becomes greater than 2, that value c is
definitely not in the Mandelbrot set. (It is a property of the series
that if xt is greater than 2, then subsequent values will always
increase.)
But a Mandelbrot series may remain below 2 for arbitrarily long
before diverging, and the only way to tell if it will diverge is to
compute the sequence for long enough.
Exercises for the Reader

Are these values of c in the Mandelbrot set?

c=0

c=i

c=1

c = -1

c = a + bi (audience choice!)
Properties of the Mandelbrot Set


The Mandelbrot set is connected (that is, you can
move through the Mandelbrot set from any point to
any other point).
But the distribution of the values of c that are in the
Mandelbrot set, when plotted, forms an infinitely
intricate shape that is nearly incomprehensible in its
mystery!
The black area corresponds to points
in the Mandelbrot set. The colored
area represents points not in the
Mandelbrot set, where the brightness
of the color is proportional to the
number of iterations before
divergence (i.e., the smallest value of
t for which xt ≥ 2).
Julia Sets

Long before Mandelbrot, Gaston Julia (1893-1978) had studied a
similar function. (In fact, Mandelbrot started out by studying the Julia
set...)
Here, c is a fixed complex number (so we talk about “the Julia set for c
= some value”) and x1 is the point being examined (i.e., the point that
is plotted in a display of the Julia set as belonging to that Julia set (or
not)).


Julia examined what happens to the series for a given c and x1 as i
increases. As with points in the Mandelbrot set, each such series either
diverges, or it does not.
Without the aid of computers, Julia could only sketch relatively crude
drawings of these shapes. Today, we can compute the Julia set for any
value, to an arbitrary degree of resolution.
Julia Sets
The central black areas are points that converge and are a part of the set.
The different colors represent how many iterations before that point
diverges.
c = -0.375 + 0.61875i
c = 0.21875 - 0.575i
Julia Sets
The central black areas are points that converge and are a part of the set.
The different colors represent how many iterations before that point
diverges.
c = -1.16875 - 0.2875i
c = 0.325 + 0.06875i
Julia Sets
The central black areas are points that converge and are a part of the set.
The different colors represent how many iterations before that point
diverges.
c = -0.04375 + 0.9875i
c = -0.3875 - 0.69375i
The Mandelbrot Set






Some Julia sets consist of infinitely many disconnected regions; others
are a single contiguous region (although they may be connected only
by arbitrarily fine “filaments”).
The Mandelbrot set serves as a “map” of all the Julia sets.
If a point is inside the Mandelbrot set (colored black), then the
corresponding Julia set is contiguous.
The closer a point is to any border area of the Mandelbrot set, the
more complex that Julia set will be.
Julia sets often seem to share similar visual characteristics to the
corresponding point in the Mandelbrot set.
The NetLogo model posted on the course page lets you explore the
Mandelbrot set and corresponding Julia sets:
http://www.csee.umbc.edu/~mariedj/complexity/2016/Mandelbrot.nlogo
The Mandelbrot Set




The Mandelbrot set is perhaps the most complex object in
mathematics.
One could spend a lifetime exploring it and never see all of it.
It contains infinitely many imperfect copies of the set within it, none
of them matching any other copy.
YouTube user ckorda spent 5 months with about 15 PCs all rendering
a video of a Mandelbrot zoom to a depth of 2316 (about 1095):
http://www.youtube.com/watch?v=_QskAoLIzuI

One can zoom as far as your computing power and patience holds up:
the NetLogo model can do up to a one-billion zoom, depending on the
region.