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Joshua
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Chicago,IL

Registered: Jun 2006
Posts: 10

Cellular Automata On Different Manifolds

Have CA's been defined on Curved or manifolds with holes or holes? If they have could you give me some papers that reference this?

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Old Post 07-10-2006 02:20 AM
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Todd Rowland
Wolfram Research
Maryland

Registered: Oct 2003
Posts: 113

I am not aware of much NKS-style work done on different underlying spaces.

There are a few examples of CA's on other tesselations in the flat plane, such as the hexagonal and triangular tesselations. See, e.g., Carter Bays article in Complex Systems. The main example of another flat surface besides the plane is the cylinder.

CA's on curved spaces are most easily defined on tesselations. There are a few technical issues that arise in representing the data, which may not seem insurmountable. But when one considers the problem of visually representing the dynamics, it is much more difficult than the 2D plane. Consider also that the 2D plane is difficult to study in a rigorous manner because some boring behaviors look interesting in higher dimensions, compared to the 1D case.

The last difficulty I want to mention is the motivation. One might be interested to investigate some geometrical or topological consequences. This is a hard problem, but potentially useful.

I think it might be more interesting to investigate the interplay between the equations that underly the tesselation and the dynamics of the cellular automata. For instance, one has the cylinder where y^n+1==y.

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Old Post 07-21-2006 07:26 PM
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