Three-dimensional cellular automaton: Difference between revisions

Latest revision as of 02:48, 4 November 2023

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:

Gliders of some sort must occur naturally from soups
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.

One way to allow birth on just a few neighbours without lightspeed expansion is to have birth on three cells as long as the three cells and the cell to be born are all orthogonally coplanar.

In general, the increase from 8 2D neighbours to 26 3D neighbours means there are vastly more possible rules.^[1] Bays developed several theorems to reduce the number of candidate rules. He found that rule B6/S567 produces behaviour similar to Life.^[2]^{[note 1]}

Gallery

3D CA in Golly^[3]

3D CA in Ready

3D CA in Visions of Chaos

Notes

↑ While ash looks similar, there is one major difference, and that's that soups settle much more quickly in the 3D rule than in Life.

References

↑ Carter Bays (2006). "A Note About the Discovery of Many New Rules for the Game of Three-Dimensional Life". Complex Systems.
↑ Carter Bays (1987). "Candidates for the Game of Life in Three Dimensions". Complex Systems.
↑ Andrew Trevorrow (May 15, 2018). 3D.lua (discussion thread) at the ConwayLife.com forums

Links

Forum threads

3D.lua (discussion thread) at the ConwayLife.com forums
3d and 4d cellular automata (discussion thread) at the ConwayLife.com forums
3D Geminoid challenge (discussion thread) at the ConwayLife.com forums
On three-dimensional cellular automata (discussion thread) at the ConwayLife.com forums
FCC3333 - a new 3D CA on an FCC grid (discussion thread) at the ConwayLife.com forums
3D Hensel/INT notation working group (discussion thread) at the ConwayLife.com forums

Other

WOLFRAM Demonstrations Project - 3D Totalistic Cellular Automata
Stephan Wolfram, A New Kind of Science, Chapter 5: Two Dimensions and Beyond
3D Cellular Automata on Softology's Blog
Wilensky, U. (1998). NetLogo Life 3D model. Center for Connected Learning and Computer-Based Modeling, Northwestern University, Evanston, IL.
Srivastava, Abhishek & Karunakaran, Deepak & Jagadeesan, Shailesh Kumar & Narayana, Vikram & Halakatti, Veeresh & Rao, Shrisha. (2015). Higher Dimensional Games of Life. 10.13140/RG.2.1.4679.3440.

[3] While ash looks similar, there is one major difference, and that's that soups settle much more quickly in the 3D rule than in Life.

[1] Carter Bays (2006). "A Note About the Discovery of Many New Rules for the Game of Three-Dimensional Life". Complex Systems.

[2] Carter Bays (1987). "Candidates for the Game of Life in Three Dimensions". Complex Systems.

[4] Andrew Trevorrow (May 15, 2018). 3D.lua (discussion thread) at the ConwayLife.com forums

[1]

[2]

[note 1]

[3]

@@ Line 2: / Line 2: @@
 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 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.
+Most commonly, the 3D space is thought of as being divided into [[Cubic honeycomb|a grid of cubic cells]]. 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==
@@ Line 11: / Line 11: @@
 # 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.<ref>{{cite web |url=https://wpmedia.wolfram.com/uploads/sites/13/2018/02/16-4-7.pdf |title=A Note About the Discovery of Many New Rules for the Game of Three-Dimensional Life |author=Carter Bays|date=2006|publisher=Complex Systems}}</ref> 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=Complex Systems}}</ref>
+One way to allow birth on just a few neighbours without lightspeed expansion is to have birth on three cells as long as the three cells and the cell to be born are all orthogonally coplanar.
+In general, the increase from 8 2D neighbours to 26 3D neighbours means there are vastly more possible rules.<ref>{{cite web |url=https://wpmedia.wolfram.com/uploads/sites/13/2018/02/16-4-7.pdf |title=A Note About the Discovery of Many New Rules for the Game of Three-Dimensional Life |author=Carter Bays|date=2006|publisher=Complex Systems}}</ref> Bays developed several theorems to reduce the number of candidate rules. He found that rule B6/S567 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=Complex Systems}}</ref><ref group=note>While ash looks similar, there is one major difference, and that's that soups settle much more quickly in the 3D rule than in Life.</ref>
+==Gallery==
+{|style="margin-left:auto; margin-right:auto; clear:both; text-align:center"
+|[[Image:3dgolly.png|300px]]{{br}}3D CA in [[Golly]]<ref>{{LinkForumThread|p=60043|format=ref|title=3D.lua|author=Andrew Trevorrow|date=May 15, 2018}}</ref>
+|-
+|[[Image:3dready.png|300px]]{{br}}3D CA in [[Ready]]
+|-
+|[[Image:3dvisionsofchaos.png|300px]]{{br}}3D CA in [[Visions of Chaos]]
+|}
 ==See also==
 * [[One-dimensional cellular automaton]]
+==Notes==
+<references group=note></references>
 ==References==
 <references></references>
-==External links==
+==Links==
-* [https://demonstrations.wolfram.com/3DTotalisticCellularAutomata/ WOLFRAM Demonstrations Project - 3D Totalistic Cellular Automata]
+; Forum threads
-* [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]
+* {{LinkForumThread|f=9|t=3404|title=3D.lua}}
-* [https://softologyblog.wordpress.com/2019/12/28/3d-cellular-automata-3/ Softology's Blog - 3D Cellular Automata]
+* {{LinkForumThread|f=11|t=5540|title=3d and 4d cellular automata}}
+* {{LinkForumThread|f=11|t=4182|title=3D Geminoid challenge}}
+* {{LinkForumThread|f=9|t=2289|title=On three-dimensional cellular automata}}
+* {{LinkForumThread|f=11|t=3757|title=FCC3333 - a new 3D CA on an FCC grid}}
+* {{LinkForumThread|f=7|t=6211|title=3D Hensel/INT notation working group}}
+; Other
+* WOLFRAM Demonstrations Project - [https://demonstrations.wolfram.com/3DTotalisticCellularAutomata/ 3D Totalistic Cellular Automata]
+* Stephan Wolfram, A New Kind of Science, [https://files.wolframcdn.com/pub/www.wolframscience.com/nks/nks-ch5.pdf Chapter 5]: Two Dimensions and Beyond
+* [https://softologyblog.wordpress.com/2019/12/28/3d-cellular-automata-3/ 3D Cellular Automata] on Softology's Blog
+* Wilensky, U. (1998). [http://ccl.northwestern.edu/netlogo/models/Life3D NetLogo Life 3D model]. Center for Connected Learning and Computer-Based Modeling, Northwestern University, Evanston, IL.
+* Srivastava, Abhishek & Karunakaran, Deepak & Jagadeesan, Shailesh Kumar & Narayana, Vikram & Halakatti, Veeresh & Rao, Shrisha. (2015). [https://www.researchgate.net/publication/281067727_Higher_Dimensional_Games_of_Life Higher Dimensional Games of Life]. 10.13140/RG.2.1.4679.3440.
 [[Category:Cellular automata]]
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