Environmental Engineering Reference
In-Depth Information
contexts. Here, it shows that a complicated dynamic process does not always need
to be based on complicated rules and mathematics.
Regular Random and Organic Forms
Many organic forms have components where certain aspects of its parts repeat the
overall shape. When a shape consists of parts of itself, it is called self-similar. Self-
similar structures can be highly complex, even though they can be captured in
relatively simple models involving random processes. Figure 1.4 was produced by
the following instruction: Select any arbitrary point in a plane as an initial value. Then
make a random decision between three alternative linear shifts. Keep repeating with
the obtained result.
Fig. 1.4 A self-similar, leaf-like shape was obtained by iterating a point in the plane. A random
decision was made between three transformations. (a) shows the transformations as specified by
the values given in Table 1.1 .(b) Iterations are started at 0.0 and repeated 420,000 times with each
point drawn. The overall figure yields an attractor: after sufficiently long time, the iteration of any
starting point would end up in the same figure. Many sponges, coral-, tree- or leaf-like forms can
be approximated in a comparable way
Table 1.1 Parameters for the three alternative shifts to produce a leaf-like shape. For each step
one of the transformations is chosen randomly
Transformation
First
Second
Third
Shrink co-ordinates along the x -axis
0.7
0.6
0.6
Shrink co-ordinates along the y -axis
0.9
0.7
0.7
Shift along the y -axis
0.2
-0.2
-0.2
Shift along the x -axis
-
-0.3
0.3
45
-45
Turn around the point [0, 0]
-
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