Think Complexity: Complexity Science and Computational Modeling by Allen Downey

Author:Allen Downey [Downey, Allen]
Language: eng
Format: epub
Publisher: O'Reilly Media
Published: 2018-07-10T23:00:00+00:00

Fractals

To understand fractals, we have to start with dimensions.

For simple geometric objects, dimension is defined in terms of scaling behavior. For example, if the side of a square has length l, its area is l2. The exponent, 2, indicates that a square is two-dimensional. Similarly, if the side of a cube has length l, its volume is l3, which indicates that a cube is three-dimensional.

More generally, we can estimate the dimension of an object by measuring some kind of size (like area or volume) as a function of some kind of linear measure (like the length of a side).

As an example, I’ll estimate the dimension of a 1-D cellular automaton by measuring its area (total number of “on” cells) as a function of the number of rows.

Figure 7-6 shows three 1-D CAs like the ones we saw in “Wolfram’s Experiment”. Rule 20 (left) generates a set of cells that seems like a line, so we expect it to be one-dimensional. Rule 50 (center) produces something like a triangle, so we expect it to be 2-D. Rule 18 (right) also produces something like a triangle, but the density is not uniform, so its scaling behavior is not obvious.

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