Two Alternating Rules

[Extrementhusiast]

Here are two alternating rules that are very similar, and yet very different:
First rule:

Code: Select all

n_states:3
neighborhood:Moore
symmetries:permute

var a={0,1}
var b={0,2}
var c={1,2}

# Birth
0,1,b,b,b,b,b,b,b,2
0,1,1,1,b,b,b,b,b,2

# Change
1,b,b,b,b,b,b,b,b,2
1,b,b,b,b,b,b,b,1,2
1,b,b,b,b,b,b,1,1,2
1,b,b,b,b,b,1,1,1,2
1,b,b,b,b,1,1,1,1,2
1,b,b,b,1,1,1,1,1,2
1,b,b,1,1,1,1,1,1,2
1,b,1,1,1,1,1,1,1,2
1,1,1,1,1,1,1,1,1,2

# Death
2,a,a,a,a,a,a,a,a,0
2,2,a,a,a,a,a,a,a,0
2,2,2,a,a,a,a,a,a,0
2,2,2,2,a,a,a,a,a,0
2,2,2,2,2,a,a,a,a,1
2,2,2,2,2,2,a,a,a,0
2,2,2,2,2,2,2,a,a,0
2,2,2,2,2,2,2,2,a,0
2,2,2,2,2,2,2,2,2,0

Second rule in next post.

[Extrementhusiast]

Second rule:

Code: Select all

n_states:3
neighborhood:Moore
symmetries:permute

var a={0,1}
var b={0,2}
var c={1,2}

# Birth
0,1,b,b,b,b,b,b,b,2
0,1,1,b,b,b,b,b,b,2

# Change
1,b,b,b,b,b,b,b,b,2
1,b,b,b,b,b,b,b,1,2
1,b,b,b,b,b,b,1,1,2
1,b,b,b,b,b,1,1,1,2
1,b,b,b,b,1,1,1,1,2
1,b,b,b,1,1,1,1,1,2
1,b,b,1,1,1,1,1,1,2
1,b,1,1,1,1,1,1,1,2
1,1,1,1,1,1,1,1,1,2

# Death
2,a,a,a,a,a,a,a,a,0
2,2,a,a,a,a,a,a,a,0
2,2,2,a,a,a,a,a,a,0
2,2,2,2,a,a,a,a,a,0
2,2,2,2,2,a,a,a,a,1
2,2,2,2,2,2,a,a,a,0
2,2,2,2,2,2,2,a,a,0
2,2,2,2,2,2,2,2,a,0
2,2,2,2,2,2,2,2,2,0

I still haven't gotten all of the bugs out in terms of interactions between the two colors, but I'm just looking at single-color patterns, so it shouldn't be too much of a problem.

[Wojowu]

I played a while with these rules, and I think second one is better (I named it alternating2).
I found some patterns:

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x = 125, y = 111, rule = alternating2
54.A$53.A3$50.A$54.A2$53.A70.A$52.A70.A$120.A$117.A$114.A$111.A$108.A
$105.A$102.A$99.A$96.A$93.A$90.A$87.A16$61.A3$62.A9$51.A$52.A$47.A$
50.A4$41.A3$42.A11$29.A$30.A$25.A$28.A5$18.A3$19.A5$40.A2$42.A3$41.A
3$4.A35.A$5.A33.A$A$3.A4$28.A$27.A$25.A5$8.A3$9.A!

there is an extensible puffer, rake, two spaceships, extensible oscillator based on replicator.
Here is also breeder:

Code: Select all

x = 12, y = 4, rule = alternating2
.A$A$11.A$10.A!

I think this breeder is not repeating pattern in middle (there always appears something new) but I don't have any proof

[ebcube]

Alternating2 seems to have a few interesting properties:

- Both the basic oscillator from this rule and the oscillator generated by the long puffer found by Wojowu, as well as most of the body of the long puffer itself (its behaviour is certainly defined by this) seem to be a part of a wider family of oscillators and possibly spaceships.

- This family of oscillators is an emulation of the simpler CA with Margolus neighbourhood exhibited by the rule 2x2 (B36S125) on blocks of size 2. This emulation happens on a oblique, non-orthogonal grid with a slope of (3, 1).

Now, to the explanation (it has cheap webcam photos of fast drawings!)

A Margolus rule is defined by a series of transitions such as that {A,B,C,D} -> E. When simulating Margolus, this is thought of as if E were a "middle block" between A, B, C and D; this is also how 2x2 emulates it:

Code: Select all

AABB             
AABB   ---\   EE 
CCDD   ---/   EE 
CCDD             

In Alternating2, this is done as:

Code: Select all

  ····D
  ·C···
  ··E··
  ···B·
  A····

http://cl.ly/413n473Z2T1y271K2z2K/Foto_ ... _22_34.jpg

On the picture, the circles are the positions of E on the odd generation of the Margolus, which is any 2n+1 generation for 2x2 and any 4n+2 generation for Alternating2, while the squares are in the original positions of A, B, C or D (generations 2n on 2x2 or 4n on Alternating2)

This is interesting not only because of the strange method of emulation, but because every known block-based oscillator on 2x2 of period n translates into a oscillator of period 2n on Alternating2. Because of this, oscillators of period 2 * A160657(n) are already "known" on Alternating2.

Finally, a few examples of this behavior:

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x = 16, y = 46, rule = alternating2
15.A3$14.A3$13.A3$12.A3$11.A3$10.A3$9.A3$8.A3$7.A3$6.A3$5.A3$4.A3$3.A
3$2.A3$.A3$A!

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x = 61, y = 61, rule = alternating2
60.A$57.A$54.A$51.A7.A$48.A7.A$45.A7.A$42.A7.A7.A$39.A7.A7.A$36.A7.A
7.A$33.A7.A7.A7.A$30.A7.A7.A7.A$27.A7.A7.A7.A$24.A7.A7.A7.A7.A$21.A7.
A7.A7.A7.A$18.A7.A7.A7.A7.A$15.A7.A7.A7.A7.A7.A$20.A7.A7.A7.A7.A$17.A
7.A7.A7.A7.A$14.A7.A7.A7.A7.A7.A$19.A7.A7.A7.A7.A$16.A7.A7.A7.A7.A$
13.A7.A7.A7.A7.A7.A$18.A7.A7.A7.A7.A$15.A7.A7.A7.A7.A$12.A7.A7.A7.A7.
A7.A$17.A7.A7.A7.A7.A$14.A7.A7.A7.A7.A$11.A7.A7.A7.A7.A7.A$16.A7.A7.A
7.A7.A$13.A7.A7.A7.A7.A$10.A7.A7.A7.A7.A7.A$15.A7.A7.A7.A7.A$12.A7.A
7.A7.A7.A$9.A7.A7.A7.A7.A7.A$14.A7.A7.A7.A7.A$11.A7.A7.A7.A7.A$8.A7.A
7.A7.A7.A7.A$13.A7.A7.A7.A7.A$10.A7.A7.A7.A7.A$7.A7.A7.A7.A7.A7.A$12.
A7.A7.A7.A7.A$9.A7.A7.A7.A7.A$6.A7.A7.A7.A7.A7.A$11.A7.A7.A7.A7.A$8.A
7.A7.A7.A7.A$5.A7.A7.A7.A7.A7.A$10.A7.A7.A7.A7.A$7.A7.A7.A7.A7.A$4.A
7.A7.A7.A7.A$9.A7.A7.A7.A$6.A7.A7.A7.A$3.A7.A7.A7.A$8.A7.A7.A$5.A7.A
7.A$2.A7.A7.A$7.A7.A$4.A7.A$.A7.A$6.A$3.A$A!

[flipper77]

Second rule:

Code: Select all

n_states:3
neighborhood:Moore
symmetries:permute

var a={0,1}
var b={0,2}
var c={1,2}

# Birth
0,1,b,b,b,b,b,b,b,2
0,1,1,b,b,b,b,b,b,2

# Change
1,b,b,b,b,b,b,b,b,2
1,b,b,b,b,b,b,b,1,2
1,b,b,b,b,b,b,1,1,2
1,b,b,b,b,b,1,1,1,2
1,b,b,b,b,1,1,1,1,2
1,b,b,b,1,1,1,1,1,2
1,b,b,1,1,1,1,1,1,2
1,b,1,1,1,1,1,1,1,2
1,1,1,1,1,1,1,1,1,2

# Death
2,a,a,a,a,a,a,a,a,0
2,2,a,a,a,a,a,a,a,0
2,2,2,a,a,a,a,a,a,0
2,2,2,2,a,a,a,a,a,0
2,2,2,2,2,a,a,a,a,1
2,2,2,2,2,2,a,a,a,0
2,2,2,2,2,2,2,a,a,0
2,2,2,2,2,2,2,2,a,0
2,2,2,2,2,2,2,2,2,0

I don't know if this is a mistake, but shouldn't the 5th transition under #death be this

Code: Select all

2,2,2,2,2,a,a,a,a,0

instead of

Code: Select all

2,2,2,2,2,a,a,a,a,1

I'm just pointing this out. I might be right or wrong.

EDIT:
I believe that this was done on purpose because the results are many times interesting this way instead of the other.

[Extrementhusiast]

The first one of ebcube's oscillators also works in the other rule, appropriately named "alternating1":

Code: Select all

x = 16, y = 46, rule = alternating1
15.A3$14.A3$13.A3$12.A3$11.A3$10.A3$9.A3$8.A3$7.A3$6.A3$5.A3$4.A3$3.A
3$2.A3$.A3$A!

What patterns I have so far:

Code: Select all

x = 188, y = 70, rule = alternating1
29.B.B7.B.3B$29.B.B.3B2.B2.B.B$29.3B.B4.B2.3B$31.B.3B2.B2.B.B$31.B5.B
3.3B2$173.3B.3B$89.3B.3B20.B.B.B.B22.B.3B23.B3.B.B$15.B.B46.B.3B20.B
3.B22.B.B.B.B22.B3.B23.3B.B.B$15.B.B46.B.B.B20.3B.3B20.3B.3B22.B.3B
23.B.B.B.B$15.3B46.B.3B22.B.B.B22.B3.B22.B.B25.3B.3B$17.B46.B.B.B20.
3B.3B22.B3.B22.B.3B$17.B46.B.3B5$3B$2.B$3B$B35.A$3B32.A3$35.A28.A31.A
90.A$20.A42.A31.A$17.A80.A$97.A88.A$68.A$67.A56.A$90.A34.A59.A$.2A86.
A$92.A57.A$91.A92.A2$149.A$183.A$116.A$117.A30.A$182.A2$147.A$181.A3$
180.A3$179.A3$178.A3$177.A3$176.A3$175.A3$174.A3$173.A3$172.A!

I am still looking for a gun in the second rule.

[flipper77]

Here's a group of oscillators I found:

Code: Select all

x = 54, y = 10, rule = alternating1
52.A$53.A$27.A$28.A$4.A$5.A3$A20.A22.A$.A20.A22.A!

EDIT:
Apparently Extrementhusiast already beat me to it

[Tropylium]

If I'm reading this right, "alternating1" is B13~S4, and "alternating2" is B12~S4.

I'm not seeing a whole lot in common between them actually, other than that they both share the "skew Margolus" family of oscillators. The first is one of a number of rules which mostly stabilize quickly, but where a domino or duoplet leads to a large explosion that dies out… the second is almost entirely dominated by the small replicator.

Wojowu's breeder actually turns out to be a chaotic explosion in the long run BTW; the central SW and NE shoots start sprouting tertiary arms around 2k generations, which eventually overtake the regular NW and SE growth.

For another example of "2-cell explosion" rules, B1345~S4 (which also supports the Margolus patterns) has a fairly spectacular explosion of 194 generations which can make at least a large p64 oscillator.

Code: Select all

n_states:3
neighborhood:Moore
symmetries:permute

var a={0,1,2}
var b={a}
var c={a}
var d={a}
var e={a}
var f={a}
var g={a}
var h={a}

# birth
 0,1,0,0,0,0,0,0,0,2
 0,1,1,1,0,0,0,0,0,2
 0,1,1,1,1,0,0,0,0,2
 0,1,1,1,1,1,0,0,0,2

# phase change
1,a,b,c,d,e,f,g,h,2

# survival
 2,2,2,2,2,0,0,0,0,1

# death otherwise
2,a,b,c,d,e,f,g,h,0

Code: Select all

x = 46, y = 2, rule = alternlife_13454
.A11.A30.A$A13.A30.A!

[Wojowu]

I have found new rake which shoots more complicated spaceships:

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x = 6, y = 6, rule = alternating2
5.A$2.A.A$.A2$.A$A!

And I've found Sierpiński triangle generator!

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x = 9, y = 5, rule = alternating2
A$.A4.3B$.A3.B.2B$5.2B.B$5.3B!

I found both new patterns when searching for high-density patterns

Edit:
I also wanted to post a high density pattern:

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x = 11, y = 10, rule = alternating2
4.A$5.A$5.A$2B5.3B$4B3.B2.B$B2.B3.4B$.3B5.2B$5.A$5.A$6.A!

This is densest pattern I've found with density of 0.1499 in gen 5432. I think there are denser patterns, but it is best I found

I've also found much smaller breeder:

Code: Select all

x = 6, y = 2, rule = alternating2
AB3.A$B3.A!