by
Erik Oosterwal
"Now as I beheld the living creatures, behold one wheel upon the Earth by the living creatures, with his four faces. The appearance of the wheels and their work was like unto the color of a beryl: and they four had one likeness: and their appearance and their work was as it were a wheel in the middle of a wheel." - Ezekiel 1:15,16 KJV
In 1977 and 1982, Benoit Mandelbrot authored the books Fractals: Form,
Chance, and Dimension and The Fractal Geometry of Nature, spawning
a new science called Fractal Mathematics.
Fractals are shapes or objects that are made up of parts identical or similar
to the object itself.
| Let's take a look at some regular fractals: This one is called Koch's snowflake. It is made by starting with a regular triangle, then replacing each line segment with a line, two parts of a regular triangle, and another line ( _/\_ ). | |
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Another example is Heighway's dragon. Here, each line segment is replaced with a right angle. The interesting thing about this fractal is that is exists in nature - Take a piece of paper, hold on to one end, and repeatedly fold the other end to the origin. Unfold the paper, allowing each crease to form a 90° angle, and you will see Heighway's dragon. |
Fractals appear everywhere in nature - huge cumulonimbus clouds (the big
puffy ones) can be looked at as clumps of smaller clouds. Each of these can
in turn be looked at as clumps of even smaller clouds, continually breaking
these clouds down until you get to the individual drops of water vapor that
make of the clouds - and even these drops can be broken down further until
you get to the individual molecules. At each step, the new part resembles
the original cloud.
Trees are another good example of fractals - The whole tree looks somewhat
like a lollipop. You can take one of the branches from the tree, and what
do you know? Another tree! Take a twig off of this branch, and again we have
another tree! Even the leaves resemble the original tree.
This building is also a fractal - the building itself is basically a block
(with other blocks attached (wings, additions)). When you walk into the building
you become aware that the inside of the building is made up of large rooms.
The rooms are subdivided into cubicles, the cubicles are subdivided into
sections (cupboards, drawers).
I am priviledged to have two young children who point out all the new things
in life that I have forgotten about long ago. "Hey Dad!", they say. "What's
that?" Or, "Hey Dad! How come...?" Then I see the world through their eyes
- a new way of seeing a toad, or a renewed interest in looking at the stars,
or planes. My challenge to you today is to look at the shapes of things around
you and see them in a new way - take time to fold a piece of paper and see
what comes out of it. Sit up and notice the wheel inside the other wheel.