272 A Programmer’s Guide to the Mind
It was discovered in the 1960s that, with the help of computers, it was
possible to use simple mathematical equations to produce geometrical
shapes of incredible beauty and complexity. These visual masterpieces
were called fractal curves. The best known one is the Mandelbrot Set. Its
basic shape is two unequal circles, sitting side by side, bordered by a „lace‟
of color. The amazing thing about this image is that further complexity and
beauty can be uncovered by zooming in on any of the colored areas. Each
time the computer is instructed to blow up one of the „lacy‟ regions, this
lace expands into another scene of beauty. If the computer has sufficient
accuracy, and one zooms in on the right areas, then it is possible to expand
a tiny region into a full picture dozens of times or more before reaching the
end.
The Mandelbrot Set and other fractal curves are examples of order
combined with complexity. Using the order of a simple equation, it is
possible to create a seemingly infinite complexity of visual curves.A When
the mind looks at these visual curves, Teacher strategy detects the
underlying order which ties together all of the complex lines and shapes.
The result is that the picture looks beautiful. Notice how, again, we find a
blending between two aspects of Teacher thought—in this case, beauty and
mathematics.
A tree is an example of a fractal object which occurs in everyday life. It
consists of a trunk with branches. Each branch has several twigs, and each
twig has many leaves. The leaves themselves have a main stem with
several branching veins. Notice how at each scale the same general shape
of trunk and branches can be seen. This creates order within complexity,
because the same overall feature—trunk and branches, is continually
repeated at different places and at varying levels of detail. There is order
because the same pattern recurs, but there is also complexity since each
individual reoccurrence is slightly different from the others. The result is
good Teacher emotion, because Teacher strategy loves to discover order
within complexity.B
A The equation for the Mandelbrot Set is Z
new^ = Zold
(^2) + C. It is hard to think
of a much simpler equation. The only hassle is that Z is a complex number,
with both a real and an imaginary component. An imaginary number
includes i, the square root of -1. Since no real number multiplied by itself
equals -1, it must be imagined. Hence, imaginary numbers. If you do not
understand this, then go talk to a physicist or electrical engineer. Except,
make sure that when you talk to the engineer you use j and not i.
B If a tree contains so much order within complexity, why do we not
experience „gobs‟ of Teacher emotion every time we look at one?
Sometimes we do. However, usually boredom and emotional insensitivity
conspire to make us pass by without noticing what we are really seeing.
We will discover later that Teacher thought in any case tends to be