Monday, March 19, 2007
Cellular Automata and Dimensions
Wolfram characterizes the class of cellular automata pictured here (of which this is but a sample) as "one-dimensional". He further characterizes any cellular automaton rendered on a two-dimensional grid as being two-dimensional.
But is this actually right?
The problem is this. Rendering the model in a single dimension does not make it one-dimensional any more than rendering it in two dimensions (note that this, in fact, is what Wolfram's model does as pictured here) makes it two-dimensional.
The state of every cell (black or white) in a Wolfram-style automaton depends upon the state of exactly three other cells. If the term "dimension" is to have any meaning in this field, nascent though it may be, it must have some correlation to the realities of the underlying model.
Each cell in this picture is of the following form:
Each cell has three ancestors. Wolfram conveniently deems the Middle Ancestor to be "Cell on a previous time step," but there are certainly plenty of applications for cellular automata where time is not even a consideration (at least not in the way physicists think it is).
It turns out that Wolframs simple class of cellular automata could just as easily be rendered on a 3-d grid like the Game of life (The three dimensions being height, width, and time). They could even be rendered as a (possibly infinite) 3-d solid.
So why settle for sloppy definitions of dimension?
Well, it's quicker, for one. It's obvious, for two. But it's still wrong.
The picture we started with should be called a "two-dimensional projection of a three-dimensional cellular automaton." This would, at the very least, give the cellular automaton with two neighbors first-class citizenship (come to the talk or ask me afterward why this is important for the field). ¶ 1:55 PM
Wolfram characterizes the class of cellular automata pictured here (of which this is but a sample) as "one-dimensional". He further characterizes any cellular automaton rendered on a two-dimensional grid as being two-dimensional.
But is this actually right?
The problem is this. Rendering the model in a single dimension does not make it one-dimensional any more than rendering it in two dimensions (note that this, in fact, is what Wolfram's model does as pictured here) makes it two-dimensional.
The state of every cell (black or white) in a Wolfram-style automaton depends upon the state of exactly three other cells. If the term "dimension" is to have any meaning in this field, nascent though it may be, it must have some correlation to the realities of the underlying model.
Each cell in this picture is of the following form:
Each cell has three ancestors. Wolfram conveniently deems the Middle Ancestor to be "Cell on a previous time step," but there are certainly plenty of applications for cellular automata where time is not even a consideration (at least not in the way physicists think it is).It turns out that Wolframs simple class of cellular automata could just as easily be rendered on a 3-d grid like the Game of life (The three dimensions being height, width, and time). They could even be rendered as a (possibly infinite) 3-d solid.
So why settle for sloppy definitions of dimension?
Well, it's quicker, for one. It's obvious, for two. But it's still wrong.
The picture we started with should be called a "two-dimensional projection of a three-dimensional cellular automaton." This would, at the very least, give the cellular automaton with two neighbors first-class citizenship (come to the talk or ask me afterward why this is important for the field). ¶ 1:55 PM
