Straight lines, squares, triangles and circles
are fundamental geometrical objects that we are all familiar
with. Graphic designers often put together a number of them
to create pretty patterns. Mathematicians on the other hand
are more ambitious. They have learned to compose extraordinarily
complex but yet highly regular patterns using infinitely many
basic objects each of infinitesimal sizes. The following figures
shows two famous examples: the Sierpinski gasket and the Koch
curve. In the 1970's, mathematician Benoit Mandelbrot named
these and many other related geometrical objects 'fractals'.
(Sierpinski gasket )
(Koch curve )
Let us first take a closer look at the Sierpinski
gasket. Its construction requires defining a generator, also
called the level 1 object shown below. The objects at subsequent
levels are then obtained by replacing all solid triangles
in the previous level by downsized copies of the generator.
To do the drawing by hand, we will find it very tiring after
going up a few levels. However, in the very imaginative minds
of mathematicians, we can easily go to infinite level and
only then we obtain the Sierpinski gasket.
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(level 1 - generator) |
(level 2) |
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(level 3)
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(level 4) |
The picture on this web page in fact is incapable
of showing clearly the finest details of the Sierpinski gasket.
Luckily, we can watch the animation below instead. It shows
what happens if we keep zooming into the lower left corner.
Note that the same Sierpinski gasket reappears again and again.
Smaller copies of itself are in fact embedded everywhere inside
itself! This is called self-similarity and is an essential
property of all types of fractals. However, do not try to
wait for the basic infinitesimal triangles to show up in the
movie. It will take infinitely long time to attain an infinite
magnification factor to see them!
Sierpinski
gasket animation
A filled triangle is two-dimensional, while
a wire even after being bent into the shape of a triangular
frame is still a one-dimensional object. One the other hand,
the Sierpinski gasket is less solid than a filled triangle
but much more bulky than a triangular frame. It is for this
reason that mathematicians invented a generalization of the
concept of dimension, which Mandelbrot called the fractal
dimension. Very loosely speaking, it illustrates the degree
of "closeness" to normal non-fractal objects with
dimension one, two, three, etc. They found that the Sierpinski
gasket is best described as having a dimension
log 3/log 2 = 1.585! Similarly, the Koch curve can be constructed
by repeatedly replacing line segments by its generator as
shown below. It has a fractal dimension log 4/log 3 = 1.262.
Why do scientists study fractals? This is because
fractals are not only found in laboratories but are all around
us in nature. Famous examples include coastline, mountains,
river networks, clouds, blood vessels, broccoli, fern leafs,
etc.
The area of Hong Kong Island is 78 square km,
but how long is the coastline? If you try to look for an answer
from your geography textbook, you may be disappointed. In
fact, coastlines are fractals very much like the Koch curve.
Remember that you can measure the length of a curve by trying
to best coincide it with a string and then measuring the length
of the string. Try it on a circle. Indeed, I am not interested
at the actual length. More importantly, note that the thickness
of the string does not matter as long as it is much thinner
than the diameter of the circle. Now, try to measure the length
of the Koch curve using strings with different thickness.
This time, you may be surprised to find that the thinner the
string is, the longer the Koch curve appears to be! Actually,
the Koch curve is infinitely long. To show it, you need an
infinitely thin string, or better still you need mathematics
instead. Coastlines are similar fractal curves with infinite
length. Hong Kong Island is perhaps a bit too small and too
artificial for an in-depth analysis. The coast of Britain
for example was found to be a fractal with dimension 1.25.
It is not difficult to build our own fractals
which can be found in nature. Physicists Tom Witten and Len
Sander invented a beautiful fractal called diffusion limited
aggregation in 1981. You can create examples of it simply
by pressing a button using the simulation below. It gives
a different pattern every time you run it. One of its original
motivation was to simulate how tiny dust particles gather
to form larger dust balls. Therefore, in the simulation, you
observe small particles keep wandering randomly around until
they happen to be caught and stick to the growing dust ball.
Later on, scientist found that it better resembles mineral
deposits inside rocks, branched corals, bacterial colonies
and so on. It is a fractal with dimension 1.72.
( java applet)
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