Cellular Automata

Подпись: John von NeumannStanislaw Ulam [220] worked with John von Neumann and became inspired by von Neumann’s concept of cellular automata. In this concept, the space, either an arbitrary dimensional abstract space or – for botanical simulation – the usual two – or three-dimensional space, is divided entirely into cells of the same dimension.

Aside from the division into cubes, triangles, and tetrahedrons, as well as other primitives can be used. Each of the cells generated in such a way has a fixed number of predefined neighbors. Each cell is assigned a state chosen from a finite set. In an iterative procedure all cells change their states according to the same rule, which describes the next state of a cell as a function of its previ­ous state, and the states of its neighbors. For example, all cells at the start are assigned the state “invisible”. During the simulation process, more and more cells are made visible. This also permits the simulation of growth procedures.

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Figure 4.1

Discrete branching pattern by Ulam:

(a) simple pattern with illustration of

different generations;

(b) more complex branching pattern


Подпись: (a)(b)

A simple example uses square cells (a grid). Starting from a single cell, a growth rule activates all adjacent cells, with the exception of those that already have two or more already activated cells. Figure 4.1a shows a corresponding pattern. A modification of the growth rule allows for the first branching struc-

ture to develop (see Fig. 4.1b). Hereby those cells are excluded from the activa – Section 4.2

tion that are adjacent to a cell that already was activated in the same iteration. A First Continuous Model

Later Ulam extended the procedure to other cell types, and in this way was able

to generate a number of different branching patterns.

Extensions of this concept of branching structures were later implemented by Meinhardt [138], who analyzed the formation of net-like structures, and by Greene (see Sect. 4.12), who used three-dimensional models for the growth of climbers.