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Cellular Automata have two visages. They are a scientific tool, and an artistic medium. I'll try to expose both sides of Cellular Automata on this page. Each month I'll put a new pattern here that made me thinking for a while, or enchanted (hypnotized?) me with its beauty, or both. Click on the thumbnail to see the enlarged picture of the pattern. Click on the pattern name to load it to MCell and run it. If you didn't install MCell, you can still enjoy the described rules in the MJCell Java applet and in the Cellular Automata gallery.
I hope that you will help me in keeping this page interesting. If the patterns you'd like to submit here can be loaded to MCell, please include their source version, plus optional bitmaps. You can also vote for any existing patterns from any CA libraries, if you want. It would be nice if you mentioned why those patterns attracted your attention.
|Brian Prentice, the author of the wonderful fractals
gallery and of the Snake rule
proposed a nice puzzle for the Star
Wars rule. He took the common in Star Wars still life objects that
support runners and constructed many different railroads. An example of such a
railroad is shown in the picture (note the train in the SW part of the
pattern.) The train is running up along the railroad. When it reaches the N
edge it starts moving down, and finally enters the "inner" part
of the figure at the S edge. Brian asks: is it possible to construct
crossings and switches so that the train's route can change? I suppose it
should be possible using Star Wars' guns.
It's very interesting to see other railroads created by Brian. They can be found at his site or in the updated patterns package for MCell 4.
|Billiard-ball machines (BBM) is a fascinating CA rule that at first seems hardly to be possible to be defined in
a classic range-1 Moore neighborhood. And indeed it's impossible - a very special
partitioning scheme has been defined for implementing this and several other
rules - a Margolus neighborhood. A good
discussion about various BBM rules can be found at Lotus
Billiard-ball machines rule has been defined by Edward Fredkin and was first published in 1987 in famous Cellular Automata Machines by Toffoli and Margolus.
Click on the picture to the left (or here) to see a short animation of the BBM rule. The animation shows 2 typical objects in the rule - balls being composed of at minimum two active cells and walls being made of at least four cells groupped in 2x2 blocks.
|The pattern of this month belongs to one of less explored CA
families - Larger than Life (LtLife). Rules
in the family check for neighbours in bigger ranges, up to 10 in MCell's
implementation. The larger distance results in exciting, very original
behaviour of cells.
One of noble representatives of the family, Bugs, was discovered by Kellie Evans while exploring LtLife phase space on the CAM8 cellular automaton machine. The rule works in R5 extended Moore neighbourhood; cells with the total count of live neighbours between 34 and 58 can survive; empty cells with 34 to 45 neighbours turn alive. It's a seriously endangered species in the crowded environment of the rule's characteristic dynamics. The rule features various bugs, often very amusing. Some of them (like Bosco) even are able to turn while traversing the space!
|WireWorld is one of
the oldest and well explored Cellular Automata rules. The automaton was
designed by Brian Silverman and
was included in his program PHANTOM FISH TANK in 1987. A. K. Dewdney publicized WireWorld in his "Computer Recreations" column (Scientific American, January, 1990).
Cells in WireWorld have one of four possible states: background (0), electron head (1), electron tail (2), and wire (3).
The rules for updating cells are:
|Jason Rampe has explored an interesting version of the
classic Rudy Rucker's Rug rule with 9 cells
neighbourhood and the increment of 3. It turns out that these parameters produce really
amazing results! Jason named this rule Digital
Inkblots. Most of the patterns go through stages of relatively small changes for a
while and then suddenly hit a chaotic stage that usually spreads out destroying the
previous stable state and creating a new stable (but more complicated) phase.
The visual effects depend very much to the color palette used. Best palettes seem to be shaded bands of colors, i.e. black=>white=>black=>white or black=>any color=>black=>any color etc. Shaded palettes without the black fade-out bands do not give as nice effects.
|Back to the roots...
David Eppstein (http://www.ics.uci.edu/~eppstein/) has just
found the first period 7 spaceship in Conway's Game of Life. The speed of the ship is 2c/7
- the spaceship travels 2 cells every 7 cycles. According to Stephen Silver, David's
discovery is the first new velocity for four years.
|Ebb&Flow is a rule
discovered by Michael Sweney. I remember when I first looked at this rule
I thought "Mhm... just another chaotic exploding rule." I couldn't be more
wrong! Michael soon sent me patterns designed for the rule that show that it has an
amazing tolerance for the creation of stable complex effects. Check the Michael's
Here is a very original oscillator - an example of how the plane can be tiled in this rule with blinkers. The timing of the blinkers is up to the designer with the constraint that neighboring orthogonal blinkers can not be in the same generation. In this example Michael created a "random" effect (sequential movement illusions get boring after awhile!)
Click on the picture to see the oscillator as an animated GIF.
|The pattern of this month illustrates how powerful and easy to implement
are hexagonal CA. The pattern shows 6 gliders sent in a loop using 6 reflectors. This
particular hexagonal rule, "Hexrule b2o",
was designed and explored by Paul Callahan.
It is symmetric (not totalistic) with 2 states. Note that hexagonal rules do not
necessarily have to be played on a hexagonal lattice. A simple way to emulate a hexagonal
grid is to use the Moore neighborhood omitting two opposite diagonal corners, what is
especially easy to do in MCell in Weighted Life rules
"Hexrule b2o" has many properties associated with Life: gliders, reflectors, oscillators, still life, duplicators, reflectors, etc. It's also missing some features, for example it's not chaotic enough - it dies down too quickly.
Paul Callahan designed a number of interesting patterns for the rule. I've converted them to MCell format and made them available here for downloading. I encourage you also to read the original Paul's essay on the rule.
|"Balloons" rule is a
very interesting example of aggregation. It was discovered by Brian
Silverman using his "Phantom Fishtank" program. Balloons is implemented in MCell
as a rules table.
Balloons is driven by Silverman's Brain rule. If enough firing Brain cells are together, they turn on a permanent firing cell. These permanent firing cells serve as seeds around which more turned-on cells agglutinate. If a turned on cell is entirely surrounded, it changes state, so that one soon gets the effect of cells with membranes. As a final fillip, if there is too much excitement at a cell's membrane, the membrane bursts and the cell goes over to a "dead" state which can slowly be nibbled away by the ever active Brain rule.
The rule plus its description was copied from CelLab by Rudy Rucker & John Walker.
|This month's pattern shows one of very surprising features of Cellular
Automata - the capability of replicating initial configurations.
Replicators occur commonly in the CA space. They are known in both von Neumann and Moore
Perhaps the best known replicator is the "Fredkin" (13579) rule in the "Vote" family found by Edward Fredkin. It's a totalistic rule. After 32 steps every starting pattern is replicated 9 times. The patterns keep self-replicating and soon the whole universe is covered with stamps of the initial pattern. There exist also replicators in other families of rules. The "Replicator" rule (1357/1357) in the "Life" rules family creates 8 copies of initial configurations. "Fredkin2" and "Fredkin3" rules are implementations of the Fredkin rule in the von Neumann neighbourhood. Both create 5 copies of initial configurations, but Fredkin3 is a unique rule: it uses 3 states, and it's period is 27.
|The pattern of this month combines both beauty and engineering
capabilities. It illustrates the Janko Gravner's "Flakes" rule, also known as "Life
without Death" (LwoD). The rule, when started from simple groups of
cells usually produces beautiful growing flakes (try for example various filled circles
with radius > 20 cells). Often ladder-like formations of cells emerge and travel to
David Griffeath and Cris Moore noticed that the ladders can be used to construct a universal computer. In fact they proved that thanks to possibilities of turning, stopping and rephasing ladders the rule is P-complete. The updated MCell 2.20 library of patterns will contain illustrations of all basic operations on ladders.
Yin Yang Fire
|This month's pattern illustrates the Yin Yang Fire
Cellular Automaton, discovered by Jack Ruijs.
Yin Yang Fire (YYF) is a new cellular automaton that exhibits very complex chaotic behaviour and even shows beautiful fractal-like structures. The whole automaton is a very dynamical system with appearing and disappearing structures in a seemingly ever-changing fashion.
The algorithm behind the Yin Yang Fire rule is very simple and can be described in one clear sentence, however the algorithm will not be revealed until more research has been done on it's significance.
For more details about the rule refer to the Yin Yang Fire home page. Note that the page contains also a YYF mpeg animation that was shown in MTV's program Top Selection on June 10th 1999.
Guns of all periods.mcl
|The relatively new "Star Wars"
rule has exceptional patterns-creating capabilities, what was clearly shown by Stephen Silver. The pattern of
this month shows how simple is the construction of a universal gun with any period > 5
(the construction is simple, not the process of designing it...). Note that it's also easy
to create irregular guns! Here is the original recipe from Stephen:
"To construct a gun of period n > 5 do the following:
Does any other automaton exist, where a universal gun can be created with such a small number of cells?
|This pattern was created using the "Fireworks" rule by John
Elliott. "Fireworks" is one of the most beautiful rules I've seen so far. It
produces interesting results from both random and prepared initial states. Even several
cells scattered over the lattice will produce long-running ravishing pictures. John
discovered the rule using his own Webside CA Java applet.
Load the included Cross.mcl pattern and watch it running for at least several minutes. You'll soon understand why I prize the rule so high. I suggest that you use the "MCell Standard" color palette.
Day & Night.lif
|"Day & Night"
is one of the most surprising rules in the Life family. Discovered in April 1997 by Nathan
Thompson, it has one unique property: this property is that the rule is symmetric with
respect to ON and OFF cells. That is, if you take an arbitrary object in this universe,
and invert it so that all ON cells become OFF cells, and vice versa, then the inverted
object has the same evolution as the original object, except for the inversion.
The properties of the rule were in-depth explored by David Bell.
|Webmaster: Mirek Wójtowicz
Last update: 5 Dec 2000