The New Science of Cities by Michael Batty

Author:Michael Batty [Batty, Michael]
Language: eng
Format: epub
ISBN: 9780262019521
Publisher: The MIT Press
Published: 2013-11-01T04:00:00+00:00

Figure 8.1

Constructing a space-filling curve: The Koch snowflake curve. (a) Successive displacement of the central section of a line at ever-finer scales. (b) Application of the displacement rule to the lines defining a triangle shape called a Koch Island.

If we now call the initiator line L(1), the length of the line L(n) at any iteration of the recursion is

L(n) = nL(1), (8.1)

where is the composite scaling ratio, in this case 4/3. Clearly, we can vary the scaling ratio. If it approaches 2, then this means the line fills the two-dimensional space. This construction is a recursion of the same rule at different scales, and it generates a pattern which is self-similar in that the motif—the triangular displacement—occurs at every scale and is in a sense the hallmark of the entire construction. The structure grown from the bottom up produces a shape that is a fractal, a regular geometry composed of noncontinuous, sometimes statistically irregular parts repeated on successive scales, which is indicative of the same processes being applied over and over again. The process can be viewed as a hierarchy that is clearly present in the pattern itself, but in terms of the recursive process can be abstracted into the usual treelike diagram shown in figure 8.2.

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