Continuous spatial automaton: Difference between revisions

Revision as of 17:43, 7 February 2007

Revision as of 22:44, 22 February 2007

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[[Continuous spatial automata]], unlike [[cellular automata]], have a continuum of locations. The state of a ___location is a finite number of real numbers. Time is also continuous, and the state evolves according to differential equations. One important example is [[reaction-diffusion]] textures, differential equations proposed by [[Alan Turing]] to explain how chemical reactions could create the stripes on [[zebra]]s and spots on leopards. When these are approximated by CA, such CAs often yield similar patterns. MacLennan [http://www.cs.utk.edu/~mclennan/contin-comp.html] considers continuous spatial automata as a model of computation.

There are known examples of continuous spatial automata which exhibit propagating phenomena analogous to gliders in [[Conway's Game of Life]]. For example, take a [[2-sphere]], and attach a handle between two nearby points on the equator; because this manifold has [[Euler characteristic]] zero, we may choose a continuous nonvanishing vector field pointing through the handle, which in turns implies the existence of a [[Lorentz metric]] such that the equator is a closed [[timelike]] [[geodesic]]. An observer free falling along this geodesic falls toward and through the handle; in the observer's [[frame of reference]], the handle propagates toward the observer. This example generalizes to any [[Lorentzian manifold]] containing a closed timelike geodesic which passes through relatively flat region before passing through a relatively curved region. Because no [[closed timelike curve]] on a Lorentzian manifold is [[timelike homotopic]] to a point (where the manifold would not be locally causally well behaved), there is some [[timelike topological feature]] which prevents the curve from being deformed to a point. Because it has been conjectured that these might serve as a model of a photon, these are sometimes also called [[pseudo-photons]].