Truly Bilateral Life-Like rules

[BlinkerSpawn]

I was reflecting upon a non-totalistic rule I had developed that had the interesting property that two of the orthogonal ships (a 19c/48 even-bilateral and a 25c/50 odd-bilateral) evolved almost identically for 20 or so generations, and set about finding the rulespace for rules that have the property that similar odd- and even-bilateral patterns evolve identically.
To acheive this, a rule nust have either all or no members of each of the following "equivalency classes":
Birth conditions:
3e/3j, 3i/6i, 3q/3y, 4e/5y/4w, 4a/5i, 4n/5e, 6c/6k, 5a/7e/8
Survival conditions:
1e/3a, 2i/5i, 2c/2k, 3r/4i, 3i/4a, 3e/4r, 3y/4c/4q, 5e/5j, 5y/5q, 6c/7c, 6a/6i/7e
Also, to avoid B1 and preserve "true bilateral symmetry", the following birth conditions must be avoided:
2c, 4c, 2i, 4i, 2a, 2e
Whether birth/survival conditions not listed above are present in the rule or not should not change the property.
If somebody could find an interesting rule fitting these conditions (and check whether my hypothesis is correct) and post it here as an example, that would be nice.

[drc]

care to post a rule?
*must

B36i/S23-a4ciqy5i

[dvgrn]

also holy crap B36i/S23-a4ciqy5i is the most clusterfuck thing I've ever seen, unfortunately, it's explosive

The above line got reported for gratuitous ugliness:

I think that foul language is becoming an increasing problem on these forums. If I look back through very old posts, I don't find hardly any; the soonest I can find anything is in @Gustavo6046's posts; however recently, I can find many examples, mostly originating from the users @drc, @muzik, and @Gustavo6046. I don't know if use of foul language goes against any of the forum rules posted, but I would argue that there is (or at least used to be) a kind of unwritten rule about it.

Since the subject has come up, here are a few quick points:

• There certainly used to be an unwritten rule against profanity. Either that or it was just irrelevant. We're just talking about cellular automata around here -- who would need a law against foul language?
   
• The forum rules have recently been written down, so now they do say that
  "2b. Foul language/profanity is not acceptable here."
   
• In my personal opinion, one of the few things that's uglier and more annoying than misplaced profanity... is either moderators or non-moderators making a big deal out of it all the time.

In the long run the problem tends to solve itself, as each offensive poster notices that foul language doesn't improve communication, and really it just makes them look silly.

Apologies for the tangent. Now, back to our regularly scheduled programming (with any luck).

-------------------------------------
Obscenity is a crutch for crippled minds. -- Bonnie Abbzug, The Monkey Wrench Gang.
(Edward Abbey being Edward Abbey, of course, there's no shortage of profanity in The Monkey Wrench Gang)

[drc]

{insert sass here ew old me}

[BlinkerSpawn]

I was reflecting upon a non-totalistic rule I had developed that had the interesting property that two of the orthogonal ships (a 19c/48 even-bilateral and a 25c/50 odd-bilateral) evolved almost identically for 20 or so generations, and set about finding the rulespace for rules that have the property that similar odd- and even-bilateral patterns evolve identically.
To acheive this, a rule nust have either all or no members of each of the following "equivalency classes":
Birth conditions:
3e/3j, 3i/6i, 3q/3r, 4e/5r/4w, 4a/5i, 4v/5e, 6c/6k, 5a/7e/8
Survival conditions:
1e/3a, 2i/5i, 2c/2k, 3y/4i, 3i/4a, 3e/4y, 3r/4c/4q, 5e/5j, 5r/5q, 6c/7c, 6a/6i/7e
Also, to avoid B1 and preserve "true bilateral symmetry", the following birth conditions must be avoided:
2c, 4c, 2i, 4i, 2a, 2e
Whether birth/survival conditions not listed above are present in the rule or not should not change the property.
If somebody could find an interesting rule fitting these conditions (and check whether my hypothesis is correct) and post it here as an example, that would be nice.

Can't use the Hensel-to-.rule script on my tablet, but how's B36i/S2-i3eiq4ay?

[SuperSupermario24]

Can't use the Hensel-to-.rule script on my tablet, but how's B36i/S2-i3eiq4ay?

Incredibly boring.

Code: Select all

@RULE B36i_S2-i3eiq4ay

*** File autogenerated by saverule. ***


This is a two state, isotropic, non-totalistic rule on the Moore neighbourhood.
The notation used to define the rule was originally proposed by Alan Hensel.
See http://www.ibiblio.org/lifepatterns/neighbors2.html for details


@TABLE


n_states:2
neighborhood:Moore
symmetries:rotate4reflect

var a={0,1}
var b={0,1}
var c={0,1}
var d={0,1}
var e={0,1}
var f={0,1}
var g={0,1}
var h={0,1}

# Birth
0,1,1,1,0,0,0,0,0,1
0,1,1,0,1,0,0,0,0,1
0,1,1,0,0,1,0,0,0,1
0,1,1,0,0,0,1,0,0,1
0,1,1,0,0,0,0,1,0,1
0,1,1,0,0,0,0,0,1,1
0,1,0,1,0,1,0,0,0,1
0,1,0,1,0,0,1,0,0,1
0,1,0,0,1,0,1,0,0,1
0,0,1,0,1,0,1,0,0,1
0,1,1,0,1,1,1,0,1,1

# Survival
1,1,1,0,0,0,0,0,0,1
1,1,0,1,0,0,0,0,0,1
1,1,0,0,1,0,0,0,0,1
1,0,1,0,1,0,0,0,0,1
1,0,1,0,0,0,1,0,0,1
1,1,1,0,0,0,1,0,0,1
1,1,1,0,0,0,0,0,1,1
1,1,0,1,0,1,0,0,0,1
1,1,1,1,1,0,0,0,0,1
1,1,1,0,1,0,1,0,0,1

# Death
1,a,b,c,d,e,f,g,h,0


@COLORS
1 255 255 255



@ICONS

circles

[gameoflifeboy]

There are two 4C/8 spaceships in this rule. One is just like the other except with even symmetry instead of odd.

Code: Select all

x = 2, y = 14, rule = B36i_S2-i3eiq4ay
o$2o$2o$2o$o4$o$2o$2o$2o$2o$o!

EDIT: Yes, I know that if one works in this rule the other would too.

EDIT 2: It's actually how I found the bigger spaceship.

[BlinkerSpawn]

There are two 4C/8 spaceships in this rule. One is just like the other except with even symmetry instead of odd.

Code: Select all

x = 2, y = 14, rule = B36i_S2-i3eiq4ay
o$2o$2o$2o$o4$o$2o$2o$2o$2o$o!

The odd-even equivalency exists in every rule obeying the rules in the first post. I simply tossed that one together as an example.

[LaundryPizza03]

After reviewing it, I have corrected the criteria in the lead.

To verify this, I looked through all possible combinations of the following pair of templates, with matching-colored cells representing cells that must have the same state:

Code: Select all

x = 9, y = 3, rule = LifeSuper
DLD2.D2LD$HTH2.H2TH$JBJ2.J2BJ!

There are 40 distinct combinations under vertical reflection, all of which are shown in the table below.

Code: Select all

x = 22, y = 98, rule = LifeSuper
3B2.4B4.3B2.4B$BDB2.B2DB4.BCB2.B2CB$3B2.4B4.3B2.4B3$3B2.4B4.3B2.4B$BD
B2.B2DB4.BCB2.B2CB$BAB2.B2AB4.BAB2.B2AB3$3B2.4B4.3B2.4B$BDB2.B2DB4.BC
B2.B2CB$ABA2.A2BA4.ABA2.A2BA3$3B2.4B4.3B2.4B$BDB2.B2DB4.BCB2.B2CB$3A
2.4A4.3A2.4A3$3B2.4B4.3B2.4B$ADA2.A2DA4.ACA2.A2CA$3B2.4B4.3B2.4B3$3B
2.4B4.3B2.4B$ADA2.A2DA4.ACA2.A2CA$BAB2.B2AB4.BAB2.B2AB3$3B2.4B4.3B2.
4B$ADA2.A2DA4.ACA2.A2CA$ABA2.A2BA4.ABA2.A2BA3$3B2.4B4.3B2.4B$ADA2.A2D
A4.ACA2.A2CA$3A2.4A4.3A2.4A3$ABA2.A2BA4.ABA2.A2BA$BDB2.B2DB4.BCB2.B2C
B$BAB2.B2AB4.BAB2.B2AB3$ABA2.A2BA4.ABA2.A2BA$BDB2.B2DB4.BCB2.B2CB$ABA
2.A2BA4.ABA2.A2BA3$ABA2.A2BA4.ABA2.A2BA$BDB2.B2DB4.BCB2.B2CB$3A2.4A4.
3A2.4A3$ABA2.A2BA4.ABA2.A2BA$ADA2.A2DA4.ACA2.A2CA$BAB2.B2AB4.BAB2.B2A
B3$ABA2.A2BA4.ABA2.A2BA$ADA2.A2DA4.ACA2.A2CA$ABA2.A2BA4.ABA2.A2BA3$AB
A2.A2BA4.ABA2.A2BA$ADA2.A2DA4.ACA2.A2CA$3A2.4A4.3A2.4A3$BAB2.B2AB4.BA
B2.B2AB$BDB2.B2DB4.BCB2.B2CB$BAB2.B2AB4.BAB2.B2AB3$BAB2.B2AB4.BAB2.B
2AB$BDB2.B2DB4.BCB2.B2CB$3A2.4A4.3A2.4A3$BAB2.B2AB4.BAB2.B2AB$ADA2.A
2DA4.ACA2.A2CA$BAB2.B2AB4.BAB2.B2AB3$BAB2.B2AB4.BAB2.B2AB$ADA2.A2DA4.
ACA2.A2CA$3A2.4A4.3A2.4A3$3A2.4A4.3A2.4A$BDB2.B2DB4.BCB2.B2CB$3A2.4A
4.3A2.4A3$3A2.4A4.3A2.4A$ADA2.A2DA4.ACA2.A2CA$3A2.4A4.3A2.4A!

From top to bottom, then left to right, these configurations encode the equivalence rules:

Code: Select all

1. B0=B0
2. B1e=B2a
3. B2c=B1c
4. B3i=B3i
5. B2i=B1e
6. B3e=B3j
7. B4i=B2a
8. B5i=B4a
9. B3y=B3q
10. B4c=B2c
11. B5e=B4n
12. B5y=B4w
13. B6i=B3i
14. B7e=B5a
15. B2i=B4i
16. B4t=B5r
17. B4e=B5y
18. B6c=B6k
19. B6i=B6i
20. B8=B7e
21. S0=S1e
22. S1e=S3a
23. S2c=S2k
24. S3i=S4a
25. S2i=S2i
26. S3e=S4r
27. S4i=S3r
28. S5i=S5i
29. S3y=S4q
30. S4c=S3y
31. S5e=S5j
32. S5y=S5q
33. S6i=S4t
34. S7e=S6a
35. S2i=S5i
36. S4t=S6a
37. S4e=S6c
38. S6c=S7c
39. S6i=S7e
40. S8=S8

When you assemble these rules into a graph, you get this:

D8_2 transition equivalence classes.png (28.06 KiB) Viewed 667 times

There are 56 involved transitions, which are grouped into 24 equivalence classes:

• B4i=B2a=B2i=B1e
• S6a=S7e=S4t=S6i
• S4c=S3y=S4q
• B1c=B2c=B4c
• S3a=S1e=S0
• S7c=S6c=S4e
• B5y=B4w=B4e
• B8=B7e=B5a
• B5e=B4n
• B4a=B5i
• B4t=B5r
• S5y=S5q
• S5e=S5j
• B6i=B3i
• B6c=B6k
• S4i=S3r
• B3j=B3e
• B3y=B3q
• S2k=S2c
• S4a=S3i
• S3e=S4r
• S5i=S2i
• B0
• S8

The studious observer may notice that these classes form 12 pairs that are black-white reversals of each other. This is not a coincidence, since the template does not pay attention to which cell is black and which cell is white. When we add in the 46 irrelevant transitions, we find that there are exactly 2^70 D8_2-symmetric rules, of which 2^35 are self-complementary since the 46 omitted transitions also form 23 complementary pairs.

And here's an example of a D8_2-symmetric rule constructed using these rules:

Code: Select all

x = 110, y = 20, rule = B3-ckr6i/S01e2cik3-ckq4aciqr5i
2b4o3bo2bobo2bo3bo2bobo2bo3b4o35b4o3bo2bobo2bo4bo2bobo2bo3b4o$bo2b4o3b
obobo3bo3bobobo3b4o2bo33bo2b4o3bobobo3b2o3bobobo3b4o2bo$2b3obobo3b4ob
5ob4o3bobob3o35b3obobo3b4ob6ob4o3bobob3o$bobo3b3ob2ob3o5b3ob2ob3o3bobo
33bobo3b3ob2ob3o6b3ob2ob3o3bobo$5ob3ob4o2bo5bo2b4ob3ob5o31b5ob3ob4o2bo
6bo2b4ob3ob5o$bob2ob3obobob4obob4obobob3ob2obo33bob2ob3obobob4ob2ob4ob
obob3ob2obo$obob2o2b6o4bobo4b6o2b2obobo31bobob2o2b6o4bo2bo4b6o2b2obobo
$o2bo4b3ob2o4b3o4b2ob3o4bo2bo31bo2bo4b3ob2o4b4o4b2ob3o4bo2bo$2o2bobo3b
obo2bo3bo3bo2bobo3bobo2b2o31b2o2bobo3bobo2bo3b2o3bo2bobo3bobo2b2o$2b2o
b3obo2b2o2b7o2b2o2bob3ob2o35b2ob3obo2b2o2b8o2b2o2bob3ob2o$o2bobobo4b2o
bobo3bobob2o4bobobo2bo31bo2bobobo4b2obobo4bobob2o4bobobo2bo$o4bo2b2ob
5o7b5ob2o2bo4bo31bo4bo2b2ob5o8b5ob2o2bo4bo$ob3o2b3o4b4obob4o4b3o2b3obo
31bob3o2b3o4b4ob2ob4o4b3o2b3obo$obo2bobob3ob4obobob4ob3obobo2bobo31bob
o2bobob3ob4obo2bob4ob3obobo2bobo$bob2o10b9o10b2obo33bob2o10b10o10b2obo
$ob2o2bo2b2o2bob2obobob2obo2b2o2bo2b2obo31bob2o2bo2b2o2bob2obo2bob2obo
2b2o2bo2b2obo$3o4bobobob13obobobo4b3o31b3o4bobobob14obobobo4b3o$b2o2b
2obo2b2o2bo2bobo2bo2b2o2bob2o2b2o33b2o2b2obo2b2o2bo2bo2bo2bo2b2o2bob2o
2b2o$2b3o2b2o5b3ob3ob3o5b2o2b3o35b3o2b2o5b3ob4ob3o5b2o2b3o$2b2ob5o4bo
3bobo3bo4b5ob2o35b2ob5o4bo3bo2bo3bo4b5ob2o!