03 Cellular automata

What are cellular automata?

  • boolean network
  • spatial pattern of nodes
  • local interactions, only nearest neighbors in rules
  • all nodes have same rules
\([X0][X1][X2][X3]...\)

Simplest case: 1-dimension

  • every node has 3 neighbors (left, center, right)
  • circular boundary conditions
  • example rules
\(000 \rightarrow 0\)
\(001 \rightarrow 1\)
\(010 \rightarrow 0\)
\(011 \rightarrow 1\)
\(100 \rightarrow 1\)
\(101 \rightarrow 0\)
\(110 \rightarrow 1\)
\(111 \rightarrow 0\)
Example
By applying the rules the next time step is calculated. For instance for
\([0][0][0][1][0]\)
\(\downarrow\)
\([0][0][1][0][1]\)

Such simple rules can result in complex patterns.

linear map

Time evolution of cellular automaton with 101 states and the rules from above. The code is provided in the examples.

Insights: local interactions can lead to global patterns → Pattern formation

2D automata

The neighborhoods can be extended to 2 dimension.
Famous example: Conway’s Game of life