A multi-n-dimensional cellular automata m:n-CAk is a generalization of a cellular automata defined as follows:

Its representation is:



  • m is the number of layers in the automata.
  • n is the number of dimensions of the layers.
  • k is the number of main layers (1 by default).

With this notation, a two-dimensional cellular automata is represented as 1:2-CA. A transition to an m:2-CA cellular automata is defined like a two-dimensional cellular automata, but the main-layer cell state is a combination of the data contained in the same position of the m-1 secondary layers.

GIS data can be represented in different layers of the m:n-CA. Vector data (2DLayers) and raster data (3DLayers) are suitable. All layers must be georeferenced.

An m:n-CA implements a set of rules that determine its transitions. Some new features are included:

  1. Layers that modify their cell state are called main layers. The maximum number of main layers is m. The number of main layers is represented by k (m>=k). A given automata may have more than one main layer.
  2. The combination function Ψ allows state calculation in a main layer, which depends on the state in all other layers of the automata.
  3. Data need not be in raster format since all layers are georeferenced. Thus, vector data may be used in cellular automata. The Ψ function determines the cell state independently of the structure of the layer data.

Obviously, a 1:n-CA (a cellular automata with just one raster layer) is the same as an n-dimensional cellular automata.

Some media related with the project. 

In this video a simple evolution rule for a m:n-ACk over Z2 is shown.   
In this video a simple evolution rule for a m:n-ACk over R2 is shown. Here we are not using any simplification rule to reduce the number of points added to the surface.

The computer program (implemented in C++) that are represented in these videos can be downloaded here (windows installer).

Some publications

If you have a question about the publications please feel free to ask me anything.


GIS and Simulation system integration in a Virtual Reality Environment.


5-7th April 2006. University of East Anglia (Norwich)  (U.K.)

I3M 2005

Simplifying GIS data use inside discrete event simulation model through m:n-CAk cellular automaton


Marseille, France, October 20-22 2005



Vectorial data use in a m:n-CAk cellular automaton


5-7th April 2006. Nottingham (U.K.)


Pau Fonseca i Casas
Department of Statistics and Operations Research

Universitat Politècnica de Catalunya - BarcelonaTECH
North Campus - C5218 Room
Barcelona, 08034, SPAIN

Tel. (+34) 93 4017035
Fax. (+34) 93 4015855