Continuous spatial automaton
Encyclopedia
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.
It is an important open question whether pseudo-photons can be created in an Einstein vacuum space-time, in the same way that a glider gun in Conway's Game of Life fires off a series of gliders. If so, it is argued that pseudo-photons can be created and destroyed only in multiples of two, as a result of energy-momentum conservation.
Alan Turing
Alan Mathison Turing, OBE, FRS , was an English mathematician, logician, cryptanalyst, and computer scientist. He was highly influential in the development of computer science, providing a formalisation of the concepts of "algorithm" and "computation" with the Turing machine, which played a...
to explain how chemical reactions could create the stripes on zebra
Zebra
Zebras are several species of African equids united by their distinctive black and white stripes. Their stripes come in different patterns unique to each individual. They are generally social animals that live in small harems to large herds...
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
Conway's Game of Life
The Game of Life, also known simply as Life, is a cellular automaton devised by the British mathematician John Horton Conway in 1970....
. For example, take a 2-sphere, and attach a handle between two nearby points on the equator; because this manifold has Euler characteristic
Euler characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent...
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
Geodesic
In mathematics, a geodesic is a generalization of the notion of a "straight line" to "curved spaces". In the presence of a Riemannian metric, geodesics are defined to be the shortest path between points in the space...
. An observer free falling along this geodesic falls toward and through the handle; in the observer's frame of reference
Frame of reference
A frame of reference in physics, may refer to a coordinate system or set of axes within which to measure the position, orientation, and other properties of objects in it, or it may refer to an observational reference frame tied to the state of motion of an observer.It may also refer to both an...
, 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
Closed timelike curve
In mathematical physics, a closed timelike curve is a worldline in a Lorentzian manifold, of a material particle in spacetime that is "closed," returning to its starting point...
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.
It is an important open question whether pseudo-photons can be created in an Einstein vacuum space-time, in the same way that a glider gun in Conway's Game of Life fires off a series of gliders. If so, it is argued that pseudo-photons can be created and destroyed only in multiples of two, as a result of energy-momentum conservation.