Three-dimensional cellular automaton: Difference between revisions
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A '''three-dimensional cellular automaton''' operates in three-dimensional space. Like one- and two-dimensional [[cellular automata]], 3D automata may operate in different [[neighbourhood]]s, be [[totalistic]] or [[non-totalistic]], [[isotropic]] or [[non-isotropic]]. | A '''three-dimensional cellular automaton''' operates in three-dimensional space. Like one- and two-dimensional [[cellular automata]], 3D automata may operate in different [[neighbourhood]]s, be [[totalistic]] or [[non-totalistic]], [[isotropic]] or [[non-isotropic]]. | ||
Most commonly, the 3D space is thought of as being divided into a grid of | Most commonly, the 3D space is thought of as being divided into a grid of cubic [[cell]]s. For a 3D version of the [[Moore neighbourhood]], each cell is at the center of a {{times|3|3|3}} neighbourhood, giving it 26 neighbouring cells it touches. For a 3D version of the [[von Neumann neighbourhood]], a cell has 6 neighbours with which it shares a face. | ||
==3D Game of Life== | ==3D Game of Life== | ||
In 1987, [[Carter Bays]] wrote a paper analyzing what it meant to project the [[Game of Life]] into | In 1987, [[Carter Bays]] wrote a paper analyzing what it meant to project the [[Game of Life]] into a 3D [[universe]] with cubic cells in which a cell has 26 neighbours instead of the 8 neighbours in 2D. Bays proposed two criteria for such a [[rule]] having Life-analogous behaviour: | ||
#[[Glider]]s of some sort must occur naturally from [[soup]]s | # [[Glider]]s of some sort must occur naturally from [[soup]]s | ||
#Soups must exhibit bounded growth | # Soups must exhibit bounded growth | ||
Simply applying the [[B3/S23]] rule in 3D space does not meet criterion 2. Having birth on 4 neighbours or fewer results in lightspeed expansion similar to B2 rules in 2D. | Simply applying the [[B3/S23]] rule in 3D space does not meet criterion 2. Having birth on 4 neighbours or fewer results in lightspeed expansion similar to B2 rules in 2D. | ||
In general, the increase from 8 2D neighbours to 26 3D neighbours means there are vastly more possible rules. Bays developed several theorems to reduce the number of candidate rules. He found that rule B6/S57 produces Life | In general, the increase from 8 2D neighbours to 26 3D neighbours means there are vastly more possible rules. Bays developed several theorems to reduce the number of candidate rules. He found that rule B6/S57 produces behaviour similar to Life.<ref>{{cite web |url=https://content.wolfram.com/uploads/sites/13/2018/02/01-3-1.pdf |title=Candidates for the Game of Life in Three Dimensions |author=Carter Bays|date=1987|publisher=Department of Computer Science, University of South Carolina}}</ref> | ||
==See also== | ==See also== | ||
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==External links== | ==External links== | ||
* [https://demonstrations.wolfram.com/3DTotalisticCellularAutomata/ WOLFRAM Demonstrations Project-3D Totalistic Cellular Automata] | * [https://demonstrations.wolfram.com/3DTotalisticCellularAutomata/ WOLFRAM Demonstrations Project - 3D Totalistic Cellular Automata] | ||
* [https://files.wolframcdn.com/pub/www.wolframscience.com/nks/nks-ch5.pdf Stephan Wolfram, A New Kind of Science, Chapter 5: Two Dimensions and Beyond] | * [https://files.wolframcdn.com/pub/www.wolframscience.com/nks/nks-ch5.pdf Stephan Wolfram, A New Kind of Science, Chapter 5: Two Dimensions and Beyond] | ||
[[Category:Cellular automata]] | [[Category:Cellular automata]] | ||
Revision as of 17:49, 29 September 2022
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A three-dimensional cellular automaton operates in three-dimensional space. Like one- and two-dimensional cellular automata, 3D automata may operate in different neighbourhoods, be totalistic or non-totalistic, isotropic or non-isotropic.
Most commonly, the 3D space is thought of as being divided into a grid of cubic cells. For a 3D version of the Moore neighbourhood, each cell is at the center of a 3 × 3 × 3 neighbourhood, giving it 26 neighbouring cells it touches. For a 3D version of the von Neumann neighbourhood, a cell has 6 neighbours with which it shares a face.
3D Game of Life
In 1987, Carter Bays wrote a paper analyzing what it meant to project the Game of Life into a 3D universe with cubic cells in which a cell has 26 neighbours instead of the 8 neighbours in 2D. Bays proposed two criteria for such a rule having Life-analogous behaviour:
Simply applying the B3/S23 rule in 3D space does not meet criterion 2. Having birth on 4 neighbours or fewer results in lightspeed expansion similar to B2 rules in 2D.
In general, the increase from 8 2D neighbours to 26 3D neighbours means there are vastly more possible rules. Bays developed several theorems to reduce the number of candidate rules. He found that rule B6/S57 produces behaviour similar to Life.[1]
See also
References
- ↑ Carter Bays (1987). "Candidates for the Game of Life in Three Dimensions". Department of Computer Science, University of South Carolina.