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The Mandelbrot Set |
The Mandelbrot Set is a fractal and is said to be the most complex object
in mathematics, and yet, it has one of the simplest formulas.
There are many webpages about the Set, how it is produced, and the mathematics behind it.
Simply do a search on 'Mandelbrot Set' or 'Fractals' in your favorite search engine,
and you will find enough material to keep you busy for a long time.
There are lots of programs on the web you can use to generate fractals including the Mandelbrot Set on your own computer. This is a fascinating thing to do, and you can make beautiful pictures without knowing anything about the math. Again, the web contains many programs, many of which are free, which allow creation of these wonders. A good free program to get is named Chaos Pro. The Mandelbrot Set is a beautiful metaphor for the human mind because it contains infinity in a finite area and proves mathematically that this is possible! The boundary of the Set is literally infinitely long, and yet we can see the whole set on a finite plane on our computer screen. The Set is the area shown in the color black in the picture shown here. The colors around the black area of the set are produced by the arbitrary assignation of different colors to the areas outside of the set depending on the mathematical characteristics of the point under consideration. In this view, we cannot see the finer detail that would appear if we magnified a section near the boundary. Again, the colors are arbitrary and simply allow us to see the representation of the Set on the screen. If you were to start zooming in on the boundary of the Set, you would magnify the portion you chose to look into. The four pictures below should help to convey this. The best way to appreciate zooming in on the Set and to explore the infinite length of its boundary is to observe successively higher power zooms on your own computer using a fractal program. |
These four pictures are four frames from a full length movie of a zoom into the
Mandelbrot Set which was arbitrarily stopped at the last frame shown below.
Since the boundary of the Set is infinite in length, we can always zoom in
some more, and any end to a zoom is arbitrarily chosen.
The picture at the left shows the full Set and is the first frame in the movie.
The white box shows the portion of the Set we see in the next picture.
The interesting part of the Set is on and around the boundary, since this is
where the infinity of filagrees exist. This is where the points either pop
into or out of the set. As we descend deeper into the Set, the complexity of
the pictures never diminishes! Again, the colors are arbitrary and the Set is
color mapped to bring out the aspects we want to see in a particular portion
of the Set. |
This picture is a 6X magnification of the area shown in the first picture.
Notice how we are already seeing details of filagrees not visible in the
first picture. Again the box shows the approximate area shown in the next
picture.
Zooming into the set is like starting out in space looking at the earth and starting down towards the ground. As we get closer and closer, more and more detail is visible. If, once we reached the surface of the earth, we could continue into the atoms of the smallest objects, and then into the insides of one atom, and then into an electron, and then.... into infinitely small regions on our imagined zoom never stopping... zooming until the electron we entered was relatively as large as the universe and then.... further... infinity... amazing! This is the REALITY of the Mandelbrot Set! |
It is interesting how, as we zoom in, we see tiny little
shapes almost identical to the shape of the full set, though there are always
subtle differences depending on where in the set we are. This picture
is a 169X magnification of the original Set. If you look very closely at frame #2,
you can see the very tiny black dot in the center of the white box which in
this picture (frame #3) is clearly visible as one of the miniature near
duplicates of the full Set.Also illustrated here is a characteristic of all fractals called Self Similarity. Fractals have a tendency to resemble, or be similar to, themselves at all scales. Again, the white box contains the area of the next zoom, and much more, since the next zoom is a very large leap from this one! |
This is the last frame of the movie and it is a 1.85 x 10 ^ 19 magnification
of the Set! In English, this is 18.5 million trillion. One could do all sorts
of fun math. The diameter of an electron is approximately 10 ^ -15 meters
(10 to the minus fifteenth meters). An electron magnified as much as this
zoom into the set would be 1.85x10 ^ 19 * 10 ^ -15 = 1.85x10 ^ 4 (18,500) meters
in diameter. And, since the Set is infinite, you can easily see how we could
continue magnifying it until our electron was larger than the universe itself!
Notice that the detail and complexity of the picture are still as high as ever. Each exploration leads to a different and wonderful picture, and after a few zooms, you are no doubt the first person in the world to be looking at the particular little section of the Set you have found! Fractal geometry is the basis of how the actual universe works. Its nature is iterative and every state of a natural system depends upon the previous state being fed back into itself. It is uncomputable. We cannot calculate the future state of the system. We can only iterate our way to the future as it occurs. The Newtonian and LaPlacean clockworks universe is not real. The discovery and understanding of fractal geometry guarantees that we are not merely automatons. It provides the philosophical metaphor necessary to understand that we have free will. The processes of life and nature are not linear. They are non-linear and complex. They cannot be exactly imitated by a machine we can control. Control implies linearity. We are free. The universe is not computable. |