Talk:Elementary cellular automaton

rule 22
What is rule 22? What is rule 210? I can't find them anywhere... Sugestion: make a simple table with current pattern/new state for center cell for all the rules. — Preceding unsigned comment added by 187.65.203.8 (talk) 19:36, 11 July 2012 (UTC)


 * See, for example, On Patterns and Dynamics of Rule 22 Cellular Automaton. It will be good to add a section on Rule 22 in this article. Lbeaumont (talk) 14:29, 4 November 2023 (UTC)

Algorithm example
I have implemented the Game of Life in the past and read this entire article but I still don't understand how an elementary cellular automaton works. Say we've got the Rule 110. If we start with just "1", then based on what rule does this expand to what string? Could someone explain this for a few more steps well? &mdash;ZeroOne ( talk / @ ) 20:47, 27 January 2011 (UTC)
 * It was difficult for me to understand at first as well. Essentially, rule 110 corresponds to the following transition rule:

If one starts with a single 1 cell, or 0001000, the array can be divided into the following substrings of length 3:

000, 001, 010, 100, 000

Each substring is then "looked up" in the transition table, resulting in the following:

01100

This is then added below the first array:

0001000

0011000

Note the added zeroes on either side; these are added since the zeroes continue infinitely on either side, and 000 in rule 110 produce 0. You can therefore think of it as "expanding" by one cell on either side at every time step, if the zeroes are trimmed:

1

110

11100

etc.

Sorry if this explanation isn't clear. I might make a diagram to add to the page which will hopefully make this more clear. Inverse Hypercube 20:44, 12 June 2011 (UTC)

ZeroOne: This is in fact simpler than Life. This is more-or-less the single dimensional equivalent of Life. So in Life, you look at a single cell, which will "live/breed/die" according to the EIGHT cells around it. You repeat that process for every cell. Yes? Here, you look at a single cell, and what happens to it depends on the TWO neighbours - one either side.

The other difference is that in Life, there is only one "useful" rule that determines what happens to a cell, the rule that you alread know, which Conway determined by trial and error. Here, all rules are potentially "useful".

Maybe an example might help. You might start with the simplest case, all cells empty except for one critter: ........................................O..................................................................

The next generation might be: .......................................O.O.................................................................

or it might be: .......................................OOO.................................................................

Depending on what rule you use. Does that help? 124.184.13.246 (talk) 11:46, 31 August 2012 (UTC)

depending on what Rule you use.

Images of all rules
The images of all rules sections showing the histories of a single bit under all rules has a couple issues. First, the size of images is much too large at 400 by 400 and makes the page difficult to navigate; 100 by 100 as in the random initial conditions section allows for a gallery view. Second, including every rule is too much detail since many rules either die out, are copies or mirror images of others, or in general aren't very interesting. Readers looking for a higher level of detail can find images in the references and external links section. A few representative rules should be sufficient of an encyclopedic treatment.--RDBury (talk) 06:50, 26 February 2012 (UTC)
 * I would agree (although I put them in the article in the first place). Note it is not all the rules, but 88 non-duplicate ones. I wouldn't object to someone using a few examples for each class. Inverse Hypercube  18:40, 26 February 2012 (UTC)

Discrepancy in orbit description
The second section states that there are 88 rules that are inequivalent under the transformations of mirroring and complementing, which effectively means that there are 88 distinct orbits in the action of the group generated by those transformations on the rules. This group is isomorphic to the Klein-four group, and the number of rules fixed by each transformation are given in the section as 64 by mirroring, 16 by complementing, and 8 by mirroring and then complementing (or vice versa; the group is abelian). Adding that there are 256 rules fixed by the identity, Burnside's lemma tells us that there should be (256 + 64 + 16 + 8)/4 = 86 distinct orbits, not 88. If my reasoning is correct, then either one of the fixed point counts is wrong, or the total count of orbits is wrong.

I believe that the error is in the count of the fixed points of mirrored-complementing, which I will denote by T. Mirrored-complementing simply applies the permutation (18)(24)(36)(57) to the indices and then complements the entries, which means that the fixed points are exactly the rules where precisely one of the entries in each pair of transposed indices is 1. This gives us four choices between two indices each to make, for a total of 16 rules. We then have (256 + 64 + 16 + 16)/4 = 88 distinct orbits, which agrees with the current given value.

Could someone please check my reasoning and correct the article if necessary? Thanks. — Preceding unsigned comment added by 50.34.254.170 (talk) 23:38, 12 May 2013 (UTC)

Going ternary
Replacing bits with trits gives 3^27 = 7625597484987 CA rules instead of just 256. That makes sense. But.. where is information? Where are researches? Any article links? I've found nothing.. and made ternary 1D CA generator. Check it out at http://xakepp35.bitbucket.org/base27/celluar/ However it has flaws: 27 trits can not fit in 32 bits, only 20 lower trits are working. some simple things are easy to figure out: for instance, my defaults is rule#590601, which is ternary version of rule #110. But maybe someone knows some interesting ternary rule numbers? Any info appreciated! Xakepp35 (talk) 02:33, 14 October 2013 (UTC)

Rule 110 and it's mirror
I do not understand how rule 110 (binary 01101110) reflects to 124 (binary 01111100). Because if we turn it's binary representation around vertical line we obtain 118 (binary 01110110). I am unable to find anywhere an explanation, please enlighten me. — Preceding unsigned comment added by Bleak sphex (talk • contribs) 09:34, 2 May 2022 (UTC)

How many rules?
The article states there are 256 rules total, of which 88 are not equivalent under some combination of mirror and complement transformations. It also states that there are 64 rules that are equivalent under a mirror transformation, 16 rules equivalent under a complement transformation and a further 16 equivalent under a combination of the two.

256 - 64 - 16 - 16 = 160

160 is not equal to 88.

Why is there a discrepancy? Signsold (talk) 10:29, 17 June 2023 (UTC)