**The Mandelbrot Set**

**BENOIT MANDELBROT HAD A SET...**

**If you were persistent enough on the
Iteration of z2+c display, you detected**

**certain regions where the iterations
settled down, or at least did not diverge**

**out of bounds. On the next display the
program will outline for you the area**

**where the function z2+c does not diverge.
The set of all points in the complex**

**plane where the function z2+c does not
diverge under iteration, is called the**

**"Mandelbrot set", after Benoit
Mandelbrot. The outline of the set resembles the**

**insect known as a ladybug, after being
run over and possibly struck by**

**lightning.**

**The outline of the Mandelbrot set is
produced by a trick called the escape time**

**algorithm. We visit each pixel on the
screen, taking its corresponding location**

**in the complex plane to be the constant c
in z2+c. Then we iterate the function**

**using that value for c. If the function
diverges quickly we leave the pixel**

**black and go on to the next one. If the
function converges quickly we also leave**

**the pixel black and move on. In those
cases where the function is indecisive**

**about which way it is going to go we
treat that pixel differently.**

**In some cases one might iterate for a
very long time before it would be**

**conclusive that the function would
ultimately diverge. We establish a limit,**

**called the depth, on the number of
iterations we will allow. If that limit is**

**reach without divergence, we leave the
pixel black as though it were in the set.**

**For locations where divergence is
indicated prior to hitting the limit, we color**

**the pixel depending on how many
iterations it took to indicate divergence. Under**

**this scheme points in or very near to the
set are black. The near boundary is**

**multicolored.**

**Points well outside the Mandelbrot set
are left black so the boundary is clearly**

**demarked. It is the boundary region of
this set which will be of most interest.**

**We will explore the Mandelbrot set
boundary in more detail later. In the next**

**display the first click on the Action
button will plot the outline of the**

**Mandelbrot set. For now just let the
program run until the general outline is**

**clear. Then click on Cut to stop the plot.
Then you can mark a point with the**

**mouse button and repeat the step by step
iterations as on the Iteration of z2+c**

**display. Use the outline of the set to
guide you in picking points. Iterating**

**points outside will give divergence.
Iterating points inside, convergence.**

**Points near the boundary - Who knows? It
took 230 iterations to determine that**

**the point picked in the illustration
above produced divergence. Run the**

**Iteration and the Mandelbrot Set display.**

**In the next display we will again paint
the boundary of the Mandelbrot set. This**

**time allow the program to complete the
portrait so you may see the intricate**

**detail. Those points in the boundary
closest to the set, based on escape time,**

**we color white. Farther out we use a
rotating multi-color scheme in which the**

**colors repeat as we get farther and
farther from the set. Finally if the point**

**is far enough from the set so that
divergence occurs quickly we leave the point**

**black. This coloring provides contrast
between regions. Run the Mandelbrot Set -**

**View 1, Full Scale display, or just look
at the image below.**

**
**

**You may have noticed that the central
feature of the Mandelbrot set is basically**

**a series of disks. The disks have
irregular borders and decrease in size as we**

**proceed along the negative real axis. And
the ratio of the diameter of one disk**

**to the next approaches a constant as we
look deeper and deeper into the series**

**of disks.**

**And the constant which the ratio
approaches is the Feigenbaum number,**

**approximately 4.669.**

**So what is going on here. The same number
keeps popping up in seemingly**

**unrelated places. The logistic function
is a quadratic, the sine function is**

**trigonometric, the Gaussian is
exponential and the Mandelbrot set is none of**

**these. The common thread in the
universality of the Feigenbaum number is**

**iteration. Is it the iterative process
itself that brings this order to chaos?**

**When we produce an object through
iteration like an attractor or a Mandelbrot**

**set, the whole is made of parts repeated.
The details on a small scale resemble**

**pieces in larger scale.**

**The notion of things looking the same on
different scales has profound**

**implications. Turbulent flow in a fluid
exhibits this characteristic. There are**

**eddies within eddies within eddies mixed
with smooth regions. Water at the point**

**of boiling, clouds, metals making the
transition to magnetism, many badly**

**behaved physical systems exhibit scaling.
To the mathematically sophisticated,**

**the geometric regularity shown by
Feigenbaum's number implies scaling phenomena.**

**The universality of it means that
difficult problems might be understood by**

**solving simple ones.**

**If you are intrigued by the line of
thought we have been exploring in the last**

**few paragraphs you will have to follow up
this course with additional study. We**

**are about at the limit of our exploration
in this direction. It was about**

**twenty-five years ago that the
foundations of this science was being put down.**

**There has been considerable building on
those foundations in the last two**

**decades and now there are structures upon
which even ordinary folks like us can**

**stand and perhaps see farther than before.
I urge you to go as high as your**

**curiosity can take you.**

**In the next few displays we offer
pictures of different parts of the Mandelbrot**

**set. The intention is to illustrate the
wealth of variety contained in the set**

**which is defined so simply. This may be
the extreme example of complexity**

**arising from simple origins. We got our
first look at this phenomena back in the**

**section on phase control maps when we
iterated the logistic function. In the**

**Mandelbrot set there is literally
infinite variety. No one has yet seen all the**

**detail available to you through this
program. In the research mode of this**

**section you will be able to explore for a
lifetime. Run the Mandelbrot Set -**

**View 2 through - View 6. The only
difference in these views is the portion of**

**the complex plane at which we are looking.**

**Mandelbrot Set - View 2, Near the Axis**

**
**

**Mandelbrot Set - View 3, A River Delta**

**
**

**Mandelbrot Set - View 4, The Root of a
Grape Vine**

**
**

**Mandelbrot Set - View 5, Fireworks
Pinwheel**

**
**

**Mandelbrot Set - View 6, Turbulence**

**
**

**Next we will zoom in on a region of the
Mandelbrot set boundary. Multiple passes**

**allow you to get an early look at the
picture with details added as time goes**

**on. Points associated with a particular
number of iterations to divergence**

**(escape time) indicated by color, are
laid out in intricate patterns. The**

**patterns are infinitely detailed fractals.
The area to be expanded to full**

**screen is outlined by a white box. Run
the Mandelbrot Set - View 7 display to**

**see the location of the first zoom box.
Then run the Mandelbrot Set - View 8 for**

**the first zoom.**

**Mandelbrot Set - View 7, Full Scale with
Zoom Box**

**
**

**Mandelbrot Set - View 8, Once Zoomed**

**
**

**If we think of the set boundary as a
coastline, we have zoomed in here on a**

**peninsula, not quite all the way to the
point. If this were a coastline it would**

**have to be Maine. It is ragged in the
extreme. Next we will zoom in on the sort**

**of spiral structure enclosed in the box.
Run the Mandelbrot Set - View 9.**

**
**

**Mandelbrot Set - View 9, Twice Zoomed**

**
**

**We introduced fractals earlier in this
course but let us review the concept. The**

**whole Mandelbrot set is contained in the
complex plane such that a circle of**

**radius 2.5 centered at zero would
completely enclose it. The boundary though is**

**of infinite length. This is tough to
visualize without a program like this one**

**to actually plot the points in the set.
Lines of finite extent but infinite**

**length we know to be fractals. Fractals
involve an interesting concept in**

**addition to fractional dimension. That is
"self similarity". The boundary of**

**this set illustrates that.**

**Self similarity means that as you look at
a structure under magnification you**

**see tiny pieces of it which look like the
whole thing. With this program you**

**begin with the outline of the entire set.
In a later display you may set up a**

**window on the plot to examine any portion
of it in greater detail. By repeatedly**

**zooming in on the details you will see
that the boundary is indeed infinite and**

**that there are regions which look like
the outline of the entire set. The**

**Mandelbrot set also exhibits "symmetry".
Not the rigid symmetry of geometric**

**figures but an approximation of nature's
own.**

**Symmetry in this context refers to
sameness on both sides of some axis.**

**Generally mathematical functions which
exhibit symmetry are identically**

**symmetrical. The whole Mandelbrot set in
fact has this kind of symmetry about**

**the real axis. The set boundary however
on closer inspection has regions of**

**"almost symmetry" which are
reminiscent of the symmetry of trees and frogs. A**

**most unexpected and unmathematical thing
to find in a structure defined in only**

**a few lines of computer code. In fact
with a little imagination one may see**

**rivers, valleys, islands and mountains in
the set. Now run The Mandelbrot Set -**

**View 10 display.**

**Mandelbrot Set - View 10, Thrice Zoomed**

**
**

**In the last display you saw a copy of the
whole Mandelbrot set on a scale about**

**1:20000. Just to clarify a point, the
scale refers to linear dimensions. The**

**area of this display in the complex plane
is about 1/400,000,000 of the full**

**scale display. If you are willing to wait
for the picture to develop, you have a**

**very powerful microscope here. Soon you
will be able to control the**

**magnification by repeated zooming to
smaller and smaller scales. There is a**

**limit inherent in the way personal
computers deal with floating point decimal**

**numbers, in how far we can take this
magnification.**

**The number of significant digits in
numbers handled by PCs is limited to about**

**15. The computer has trouble telling the
difference between 100000.0000000001**

**and 100000.0. When the finest details of
the plot we are making approaches this**

**scale, there is uncertainty about exactly
which pixels represent the actual**

**value. Since several pixels may all
represent the same value, they all are lit.**

**This gives the picture a blocky
appearance. Ultimately every pixel on the screen**

**might be lit for a single point, reducing
the resolution of the picture to zero.**