The Beauty of CA
A bestseller by the name The
Beauty of Fractals by Heinz-Otto Peitgen, P. H. Richter,
Peter H. Richter depicts beauties of the Mandelbrot set. Here we present
different beauties with even more interesting features. The CA were
generated with three functions:
augment: [1, 0, 0 , 1, 20]
|delivery: [1, 1, 40 , 2]; change rule [1, 0, 1842, 1, 30]||0.56||0.91|
|delivery: [1, 1, 40 , 2]; change rule [1, 0, 1434, 1, 30]||-0.04||0.95|
|delivery: [1, 1, 60 , 2]; change rule [1, 0, 870, 1, 30]||1.21||0.87|
|delivery: [1, 1, 10 , 2]; change rule [1, 0, 624, 1, 30]||0.0||1.00|
|delivery: [1, 1, 10 , 2]; change rule [1, 0, 531, 1, 30]||0.0||1.00|
|delivery: [1, 1, 20 , 2]; change rule [1, 0, 528, 1, 20]||0.0||1.00|
|delivery: [1, 1, 20 , 2]; change rule [1, 0, 525, 1, 20]||0.0||1.00|
|delivery: [1, 1, 30 , 2]; change rule [1, 0, 3300, 1, 40]||2.38||0.75|
Delivery sets the CA boundaries. Augment initiates chaos between CA-0 states 1 – 20. Change rule activates a specified rule at given CA-0 states.
Bounded chaos with structure
Chaos is characterized by sensitivity
to initial conditions. All CA except the one generated with rule 3300,
are sensitive to their initial conditions, a periodic, and therefore
chaotic. Each displays a repertory of structures which will repeat
in various flavors and never be the same. They illustrate Heraclitus’
metaphor: “You never step twice into the same
river”, nevertheless you recognize it, and even give it a name.
The river is a bounded chaos and so are the CA. You can sort them by their repertory. Some favor triangles, and other columns. They display individuality and are unique. They maintain their individuality because they are WOB solutions. Like human beings, no two CA are the same.
Above all they are not fractals. Benoit Mandelbrot believed that Nature is a fractal. Superficially it may be so. Like the branching air-ways of the lung which seem to be fractal. In reality Life defies any geometry and so do these CA. Fractals are too simplistic for expressing the intricacy of life which is far more complex than the Mandelbrot set.
Actually fractals reduce the complexity of a system. We may distinguish between two kinds of complexity: One which can be mapped into a geometry, like fractals, and one which cannot. Since the present CA belong to the second category, they are more complex than the Mandelbrot set. (v. Complexity spectrum. From the present perspective, the Mandelbrot set is an isolated process. Let it interact with other processes and its fractal nature will vanish.
None of the above CA can be found in Wolfram’s book. His “A New kind of Science” is inhabited by non interacting CA which are less complex than the present. Two properties make the above CA so fascinating. They are controlled by a death mechanism, and interact.
Do you understand complexity?
augment: [j, j-1, While[p[j-1]
> set point] , start, end]
Argument: Augmented CA.
Argument: Augmenting CA.
Argument: Delivery condition.
Argument: The state of the augmenter at which augmentation starts.
Argument: The state of the augmenter at which augmentation ends.
p[j]: daily production.
change rule [j, j-1, rule, state[j-1, i], time]
Argument: CA whose rule changes.
Argument: Initiating CA.
Argument: The state at which the rule is implement.
Argument: time - how long the rule is implemented
delivery: [j, j-1, While[p[j-1] > set point], 2]
Argument: Activated CA.
Argument: Activating CA.
Argument: Delivery condition.
Argument: Delivery amount.
p[j]: daily production