Julia Sets

Pascal Kraft is a researcher at the Institute for Applied and
Numerical Mathematics of the Karlsruhe Institute of Technology
(KIT) and he introduces us to Julia Sets which he investigated
for his Bachelors Thesis. It is natural for us to think something
like this: If I take two simple things and put them together in
some sense, nothing too complex should arise from that. A
fascinating result of the work of mathematicians like Gaston
Julia and Benoît Mandelbrot dating back to the first half of the
20th century show that this assumption doesn't always hold.


In his bachelor's thesis under supervision of Jan-Philipp Weiß,
Pascal Kraft worked on the efficient computation of Julia Sets.
In laymans terms you can describe these sets as follows: Some
electronic calculators have the functions of repeating the last
action if you press "=" or "enter" multiple times. So if you used
the root function of your calculator on a number and now you want
the root of the result you simply press "=" again. Now imagine
you had a function on your calculater that didn't only square the
input but also added a certain value - say 0.5. Then you put in a
number, apply this function and keep repeating it over and over
again. Now you ask yourself if you keep pressing the "="-button
if the result keeps on growing and tends to infinity or if it
stays below some threshold indefinitely.


Using real numbers this concept is somewhat boring but if we use
complex numbers we find, that the results are astonishing.


To use a more precise definition: for a function f(z), the Filled
Julia Set is defined as the set of values z, for whom the series
(fn(z))n stays bounded. The Julia Set is defined as the boundary
of this set. A typical example for a suitable function f(z) in
this context is f(z)=z2+c. We now look at the complex plane where
the x-axis represents the real part of a complex number and the
y-axis its imaginary part. For each point on this plane having a
coordinate (x,y) we take the corresponding complex number z=x+iy
and plug this value into our function f(z) and the results over
and over again up to a certain degree until we see if this
sequence diverges. Computing a graphical representation of such a
Julia Set is a numerically costly task since we have no other way
of determining its interior points other then trying out a large
amount of starting points and seeing what happens after hundreds
of iterations.


The results, however, turn out to be surprising and worth the
effort. The geometric representations - images - of filled Julia
Sets turn out to be very aesthetically pleasing since they are no
simple compositions of elementary shapes but rather consist of
intricate shapes and patterns. The reason for these beautiful
shapes lie in the nature of multiplication and addition on the
complex plane: A multiplication can be a magnification and
down-scaling, mirroring and rotation, whereas the complex
addition is represented by a translation on the complex plane.
Since the function is applied over and over again, the intrinsic
features are repeated in scaled and rotated forms over and over
again, and this results in a self-similarity on infinite scales.
In his bachelor's thesis, Pascal focussed on the efficient
computation of such sets which can mean multiple things: it can
either mean that the goal was to quickly write a program which
could generate an image of a Julia Set, or that a program was
sought which was very fast in computing such a program. Lastly it
can also mean that we want to save power and seek a program which
uses computational power efficiently to compute such an image,
i.e. that consumes little energy. This is a typical problem when
considering a numerical approach in any application and it arises
very naturally here: While the computation of Julia Sets can
greatly benefit from parallelization, the benefits are at loss
when many tasks are waiting for one calculation and therefore the
speedup and computational efficiency breaks down due to Amdahl's
law.


The difference of these optimization criteria becomes especially
obvious when we want to do further research ontop of our problem
solver that we have used so far. The Mandelbrot Set for example
is the set of values c, for whom the Filled Julia Set is not
equal to the Julia Set (i.e. the Filled Julia Set has interior
points). One detail is important for the computation of either of
these sets: If we check one single point we can never really say
if it is inside the Filled Julia Set for sure (unless we can
prove periodicity but that is not really feasible). What we can
show however is, that if the magnitude of a point in the series
of computations is above a certain bound, the results will tend
to infinity from this point on. The approach is therefore to
compute steps until either a maximum of steps is reached or a
certain threshold is exceeded. Based on this assumption, we see
that computing a point which lies inside the filled Julia Set is
the bigger effort. So if computing a Julia Set for a given
parameter is a lot of work, its complex parameter c most likely
lies inside the Mandelbrot Set (as we find many points for whom
the computation doesn't abort prematurely and it is therefore
likely that some of these points will be interior). If we want to
draw the Mandelbrot Set based on this approach, we have to
compute thousands of Julia Sets and if the computation of a
single image was to take a minute this would not really be
feasible anymore.


Since the computation of a Julia Set can even be done in a
webbrowser these days, we include below a little tool which lets
you set a complex parameter c and compute four different Julia
Sets. Have fun with our Interactive Julia Sets!

References and further reading

J. Dufner, A. Roser, F. Unseld: Fraktale und Julia-Mengen,
Harri Deutsch Verlag, 1998.

H.-O. Peitgen, P. H. Richter: The beauty of fractals: images
of complex dynamical systems, Springer Berlin Heidelberg, 1986.

B. B. Mandelbrot: Fractal aspects of the iteration of
zλz(1−z) for complex λ and z, Annals of the New York Academy of
Sciences 357.1: 249-259, 1980.

P. Kraft: Paralleles Rechnen auf GPUs - Julia Mengen und das
magnetische Pendel Fraktal, Bachelor Thesis. 2012.

J. Gaston: Mémoire sur l’itération des fonctions
rationnelles, Journal de Math´ematiques pures et appliqu ´ees 4
(Rep 1968), pp. 47-245 / 121-319, 1918.

P. Blanchard: Complex analytical dynamics on the Riemann
sphere, Bulletin of the American Mathematical Society 11, pp.
84-141, 1984.