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Subsections
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.
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.
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.
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.
Next: 2.2 Neighborhood size
Up: 2. Choices in the
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Jörg R. Weimar