2.1 Lattice geometry

The first choice is the selection of a specific lattice geometry. The definition of cellular automata requires the lattice to be regular. We consider the different possibilities for one, two, and three dimensions in turn.

2.1.1 One dimension

For one-dimensional automata, there is only one possibility: a linear array of cells. A one-dimensional realization of our example automaton is shown in Figure 2.1. In this case only simple traveling waves are observed.

The presentation of 1-D CA is usually as a linear array of cells. By including a history of the CA, i.e., displaying a time-space diagram for a number of subsequent steps, more information can be conveyed. Another possibility for periodic CA is to display the cells in a circle to emphasize the periodic conditions.

Applet 2.1: Example automaton in one dimension.
Applet 2.1a: One dimensional automaton using circle representation.

2.1.2 Two dimensions

In two dimensions there are three regular lattices, namely triangular, square, and hexagonal lattices. Figures 2.2 to 2.4 shows the evolution of a spiral wave in each of the three geometries.

Applet 2.2: Spiral wave in the example automaton in two dimensions on a triangular lattice.

Applet 2.3: Spiral wave in the example automaton in two dimensions on a square lattice.

Applet 2.4: Spiral wave in the example automaton in two dimensions on a hexagonal lattice.
The main characteristics of the three lattices are:

• Triangular lattice: Advantage: small number of nearest neighbors (three), which can be useful in some cases. Disadvantage: More difficult to represent and visualize, since it must be mapped to square arrays and display pixels.
• Square lattice: Advantage: simple representation using square arrays and simple visualization. Disadvantage: In some cases the square lattice has insufficient isotropy.
• Hexagonal lattice: Advantage: The hexagonal lattice has the lowest anisotropy of all regular two-dimensional lattices. Often this lower anisotropy makes simulations appear more natural, and in some cases it is absolutely necessary to model the phenomena correctly (for example in lattice gas models for fluid flow [6] ). Disadvantage: More difficult to represent and visualize, since it must be mapped to a square lattice.

2.1.3 Three dimensions

In three dimensions there are many possible regular lattices. We will only consider the cubic lattice, since it is easiest to handle and represent. There are no regular lattices in three dimensions that have sufficient symmetry to correctly simulate hydrodynamics in a particle representation [6].

One difficulty with three-dimensional cellular automata is the graphical representation (on two-dimensional paper or a computer screen) of the state. Figure 2.5 shows a representation where only two of the three states are indicated by boxes.

Applet 2.5: Representation of a three-dimensional automaton. Cells in state 0 are transparent on the left and filled on the right. To show part of the inside states, one octant of the cube is clipped on the right.