3-state inner-totalistic von Neumann (3ITVN) rules

[praosylen]

This is a somewhat interesting rulespace, and it's the smallest as-of-yet-unexplored (as far as I know) one that I could think of that can be simulated in Golly.

First, some clarifications:

• 3 cell states, von Neumann neighborhood.
• The state of a cell depends ONLY on the total number of cells of each state in its neighborhood (including the center cell in the totals) on the previous generation (e.g. if a state-2 cell with four state-1 neighbors dies, a state-1 cell with three state-1 and 1 state-2 neighbors must die as well).

There are exactly 3^21 such rules, ignoring rules equivalent under state transformations and the like. Rules are denoted by the following code:

Code: Select all

NNNNNN-NNNNN-NNNN-NNN-NN-N

Each N represents the state of each transition. The kth group of Ns represents transitions with k-1 state-2 cells. The kth N in a group represents a transition with k-1 state-1 cells. For example, the rulestring

Code: Select all

020200-00000-0000-100-00-0

represents the rule where a cell with one or three state-1 neighbors (including itself) and no state-2 neighbors becomes state 2, a cell with three state-2 neighbors and no state-1 neighbors becomes state 1, and all other cells die or remain state 0.

Here is a script to generate 3ITVN rules:

Code: Select all

import golly as g
size = map(int, "12222102333202332023202201")
table = """@RULE %s
@TABLE
n_states:3
neighborhood:vonNeumann
symmetries:permute
#00
00000%s
#10
01000%s
10000%s
#20
01100%s
11000%s
#30
01110%s
11100%s
#40
01111%s
11110%s
#50
11111%s
#01
02000%s
20000%s
#11
01200%s
12000%s
21000%s
#21
01120%s
11200%s
21100%s
#31
01112%s
11120%s
21110%s
#41
11112%s
21111%s
#02
02200%s
22000%s
#12
01220%s
12200%s
21200%s
#22
01122%s
11220%s
21120%s
#32
11122%s
21112%s
#03
22200%s
02220%s
#13
01222%s
12220%s
21220%s
#23
11222%s
21122%s
#04
02222%s
22220%s
#14
12222%s
21222%s
#05
22222%s"""
rule = g.getstring("Enter rule in format \"000000-00000-0000-000-00-0\"", "000000-00000-0000-000-00-0")
format = (rule,)
for i in xrange(len(rule)):
    format += (rule[i],)*size[i]
table %= format
ruledir = g.getdir("rules")
filename = ruledir + rule + ".rule"
with open(filename, 'w') as f:
    f.write(table)
g.setrule(rule)
g.show("Created rule in file: %s.rule" % rule)

Interesting rules appear few and far between. Here are four that may qualify:
000120-10000-2220-020-10-2
022112-02222-0101-000-01-1
000010-11210-2211-110-11-2
001200-10000-0202-120-01-1

The third one appears the most promising. Patterns:

Code: Select all

x = 45, y = 12, rule = 000010-11210-2211-110-11-2
.AB5.2A8.2A9.2B10.B.B$3A4.A2BA6.A2BA7.A2.A8.A.A.A$B2A15.2B8.4A10.A$
18.2A9.2A2$29.2A$28.4A$29.2B2$31.B$31.B$31.B!

Unfortunately, I have not found any stationary patterns so far.

EDIT: 5 more maybe-interesting rules:
022102-00201-0021-120-20-1
001211-10000-0000-110-12-2
020002-00020-2012-110-21-2
022010-00100-0002-111-11-2
001200-10221-0022-201-22-1

EDIT 2: 000100-10201-1220-210-11-1
021222-00001-0100-010-12-1

[praosylen]

Here's a rule that probably classifies as "very nearly exploding" and has a (rare) glider:

Code: Select all

x = 6, y = 6, rule = 022210-00000-0210-102-00-2
.2A$A2BA$A$3.A$2.2B.B$3.2B!

A number of odd-period oscillators, too, which seem to be rare in this rulespace.

EDIT: My favorite rule so far:

Code: Select all

x = 267, y = 176, rule = 000211-11012-2101-001-20-1
B.B2.2A.B.A2.A2BAB3.2B2A.A.2B2.B.2A.2A.B.B.B2.A3B2A.B.B2.ABA2B.A2BA2B
.B3A.B2.2B2.2AB2A.3A2B2A.B3A.2A.4ABA2BA2B.2A.A2.4BA2.ABA.B3ABA.2BAB.
2A2.B2.2B3.B.B3.A.B2A.BAB.ABA.B4ABA.ABA.A.B2A.B.2ABAB.AB.ABA.B2.4B.B.
2A2B2.2B.3B.B.BAB.B2.BA.2B2A.A.3B.3BAB.2AB$B5.A.2A2.2A2BAB2.2A.A.2A2B
.B3A2B2.A.2A.2BA4.B.2A3B2.B2.4ABABAB.2A2.A2B2.BA.2BAB.3A.A.A.AB.A2BAB
A.BA.B2ABA2B.A.B.3B2ABA.B.2A.B4A.ABA2.B.B.2BAB.B.2A3.A.A4.B.2B3AB2.B
2A2.B4.B3.A2.A3.B.2B3.4B.2BA3B.AB.ABAB.B.A3BA2.2BAB4A.A.AB.2BAB2A7.B
2A$2AB.ABABA2.2AB.A.5AB2A.2A3.3B.AB.B2.B.B2A.B3A3BA2BA.ABA.A2.A2B2A2.
B3.BAB.B.B.2BAB.ABA2.2BA.A3.A3.2A2.AB2AB2A.4ABA4BA.B2A2B.3A.3BA.2B2.A
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B.BA$B.3A.B2A2.A.A3.AB2A.3B.AB3ABA3.B.B3A3B2.B2.2BAB.B2.AB.4A.BABABA.
3BAB2A.B.AB2A2.2BA.B4A.B.A2B2AB.B2.2B3.A.A3B2.B.3B.BA.B.2ABA.2B3.2A.B
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2.B3AB2AB.2A2B.AB3.2B.4B.2B2.B.A3.2BA.AB2.BA2B2A3.AB.3A.BA.B.A2B.2A.
2A2BA3.A.BA3.B2A.2A3.A.AB.2AB2AB3ABAB$2A2B2.7A2.2B2.3B.ABA.B2A.BA2.A
2.A.AB4ABA2.B.B2A.B.ABA.A.B.3B.A.2A.AB4AB.2B2A2.ABAB2AB2A.B.A.BA3.A3B
.A.2A.2AB.B.B.BA.4B2.3A4.2A.B.2A.BA2.BA2BA.BA3B.B.2A.2B.2B3.A2B2A.B5.
3BA.A2.BA.B.BA.2BA.2A3.BABA2.BA2.B2A2B.BA.2A3.BA2B.4AB3AB.A2.AB.B2.B$
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2.2B2A2B.B2AB2A2B6A.BA.AB.B.B2.A4BAB2A.B2A2BAB3.2A$2A.B.2B.2A.A2.A.B.
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A2B.B.2ABA2BA2B2A2.BA.BAB.B.2A2.B2.B.ABAB.B!

EDIT 2: 000212-11012-2101-200-10-1 has some nice things too, including c/2 and c/11 ships and a 2-cell p10.

[praosylen]

A nice rule with a common puffer:

Code: Select all

x = 50, y = 50, rule = 022101-00120-0120-202-01-2
2.B.AB.B.B.A2.2A2B.AB2ABA.BABAB2.B.B.A2.ABA3BA2BA$A.2ABA.2B.ABA.4AB.A
4.3B2A.A.3A2.3BA.B.4A$4.2B.B.ABABA2.2A.3A.AB.BAB.2A.A7.4AB.BAB$A2BA.A
.A3BA4.AB.B.2BA4.B2.B2A.BA2.A.2A.A.A.BA$3B.2A5B.ABA.A.A.3B.A2B.BA.6A
2.2A2.AB.2A.B$2.BA3BA.2A3B.AB.4A2.B2A.AB.2B.B.B.AB8A2B$.A2B2.B.3ABA2.
BA.2AB2ABA2B2A.AB3A2.AB.4B2ABA.A$B2.B.A.A.A4.B2.2A7B.2A.B.B2AB2.2B3A
3.B2A$4.2B.B2.AB3AB.A2B2A.5B.B.AB2.B.B4A2B3A.A$B2A.A.B.2A4.A2.AB.3B.B
2.BA2B.2BA2B.3A.B4AB$B.ABA.A2B4.AB4.3B.2BA.3B.B.A.3AB4.B2A2B.B$2B3.AB
3A.A2BA3.B3ABA.A2.2AB2AB.BAB2.2A2B2.4A$2A3B.B.A2BA.B3ABA.2A2.2BAB2A2.
B5AB2.B2.A.2AB$A.2A3.B.2B.B.2B.A2B2.B2AB2.A2BABA3B4A2.B.B.2AB$3A.BAB
5.3BA.B.2BA.2B2A2BA3B.2A2B2A.2B.B.2ABA$2.2A.B.A2B2A.A.B.2A2.B.2A2BAB
2.5B.2A.BA.2A.3B$2A4.B2A.A3B2A.A.BA2.2B2.B.ABA.B.2A3.3A2.B.A.B$BAB3.
2B.ABA2B3ABA.2BA.B2.ABA2.A.2B2.3AB.3B2A.B$B.B.BA2.2BA3.AB4.BA.2A.BA2B
2.A.A3B2A.A2.A2B2AB$B2.3A2B.BA2.A.B.B.A.A2.BA2BA.BABABABA2.A.A.2A.2AB
$2B.A3B.A4.A2.A2B4AB.5BA.B.A6B2.BA.3B$3A.B3.2B.B2ABA3B.AB.A2B.BA4B4.
2AB2ABA2B.3A$A2B.A.2B.A2BA.2B2A2.A4BA3.B.A2.B5AB.AB3.B.A$B.A.5A.2A2.A
.BA.2BABABA2.B.A3BA2B2A2B3.2BA.2B$4AB.ABAB.B.BA3B3.A.BA.B.2B.A2.B2AB.
BA.2BA.2A.B$.2B.B2.B.A2B3.A.BABABA3B6A.2B.2BABAB2.2B3A$AB2.2B2.A2.A.B
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A.B$BA2BA2BA2.3BAB.BA.A.2B.ABABAB.A.B.B3A.A.2B.AB2AB$5.2B.3B5ABA2B3.B
.BA.B.BABA.A2.B.BA.2A.2A.B$BAB.AB.A.B2A.B.B.2B4A.A.BAB.2AB3AB.A2.B.2B
.B.BA$BA.2ABABA.B.ABA3.AB5.A3.ABA.2B.B2.A6B.AB$A.B3.A.A2.2B4.AB.BABA
6BAB.A.A2.BA2B6.A$A.BAB.2A.2BAB.B.A.2A2.BA.2A2.BA.BA5.3B.2A.A2B$.2B2A
2.AB.B.B2A.B2.A.2A.2A2B4.B.A2.AB.B2.2A.A$.A.B.A.B.B.2A.B.B.BABA3.2A.A
.3B.A.A.BA.BA2.AB3A$4.AB4A3BA2B.AB2A2.A.ABA3BABA.A3B.AB.3BA2.B$A.B2AB
A2.ABA2.B2A.A.BA.BABA.2B.B2.A.B4.B.5ABA$BAB.A.A.BAB.3B2.A.B5A3B.A3BA
2.BA.A2.A4.BA$BA2.A.2B.B2A2BA.B3ABAB.BABA.2AB2.A2BA.BA2B.2A3.B$2A.2B
2A.B.2A2B.B.2A3.B2.B3.4AB2AB.B2A3.B.A$B.BA2.B.AB2A.2BA2B.B.B2A2.AB.A.
2A.A.B2A.3A2BA2.2B$2B.A2B2A2.B.B2AB3.A.ABA3.2B2ABA.2A3.B2A.AB2A2.A$AB
A3.B2A.2AB2.B2.2BABA.2B.B3A.A.A3.ABA2.3A.ABA$2.B.2A4BA2.B2.ABA2BA3.2A
.2A3B.B.BA2.A.B2.ABA2B$3AB.A2B.2A.5B2A.B.B.4B.B.A4B.B2A2BA2B3.AB$B.A.
2B.A.ABA.B.BA4.3B.2BA.A.2B4AB.2A.AB.A2.2B$B.ABA.B.B3.3B4A.4A.BABA.AB
3A.B2.2B2ABAB2A.A$B3.BA2B.AB2.2A.A3.3AB.B2A2B2.A.A2.2BA2B4A.3A$2B3A2.
BA.BA2.B.AB.2ABA2.BA2BA.B2AB.4B4.BA3BA!

Here's one that acts a lot like a mix between Brian's Brain and cb2:

Code: Select all

x = 50, y = 50, rule = 000211-10022-0100-111-12-0
.A.AB.A6.B2.BAB.2B5.A2.AB.A4.B2ABA.3B3A$AB.4A3.BA3.A2.2ABA2.B2.B.A3B.
B3AB.A.A.3B.2B$2A.3BAB.A.A.BABAB2.ABAB2A2.B.BA.B.A.A.B.ABA2BA.AB$A2.
2A2BA.3A.ABABA.B.A2B3.ABA2BABA3B2.ABA.A2B2.B$.BABABA.AB.2B2A.ABABA2.B
.A4.A3BA.A4BA.2A.2AB$.2AB5ABA4.B2.2A.BAB2.2A.BAB.B.BA.A2B.B4.B2A$2.A
3.B4A2.AB2AB3A2B2A2.2AB.A2.BA2B.B2.2AB.A.BA$2.BA2B.A2.B.B2.BA.2A.A.AB
.3A2BA.A.ABAB.B.BA2BAB.B$.A2.B2.B3.B2A2BA2.AB2A2.A4.A2BA3B2.2ABA.2B.B
A$B2A2.2BA.2B.B2A.2B.2AB.ABA2.A3.A2.2AB.B2.B2.A.3A$3.A2.3AB.B2AB.A3.A
B.A.2B2AB2.A2B2.2B2.2A3.2AB$BA.ABA2BA2B.2A.2B.A3BA.2BA.A2B2.B3A.A4BA.
2BA.B$A2BA.A2B2.2A.BABA2.BA2.2B2A3B.B.A.B2AB3.A5BA$.A.2AB.A3.A2B.AB2A
2.ABA4B3.BA.B.B.AB2A2BAB.BAB$BA.A.BA.A.B2A2B.A.2AB2.AB.B2A2.A2BAB2A2B
ABA.2A2.2B$BA.A.B2.BAB4.2A2.A.B.B.B2ABA.B.B.A.2AB4.2AB.AB$3A3.B.2A2.
2B2.A.2A.B2.B.B2.B3AB2A.BA2.A.A2BA3B$2B.AB2.A.2A4.B.B2AB3.2A2.AB4AB.
2A5B2ABA.AB$2ABA.A.B4A2.2B.A5BAB.2A6B3.4A.3BA2B2A$A2B.ABA.2BAB.A.ABAB
AB2.5B2.B3.B.A2.3A.B.A2B.A$A.A.A4B4.A.2BAB2AB3.2A3B2.B.3A.A.B.AB.BAB
2A$.A.AB.2ABA2.2A2.B3A2.AB.4B2.A.2AB.AB.2BAB3A$2A2B.A.3B.3A2.A.A2B2A.
3BAB3.BA.B.A.BA2.3A.3A$A.2A.B2A2.2A.B.B.B2AB4.ABAB2.AB.BAB2.A2BA2B3.B
$A2.ABA.B.AB2.A.B.3A2.2A4.B.A.A.B2A.A2.ABA.A.2B$2.6A3B.B.B.A.2B4.3A.
2B3.B.4B2.2BA.B.3B$A.ABA2B3.AB.A3.B.AB2A.3A.2A.ABA.B.3BA.BABABA.A$2A
5BA2.B4.BA.B.B.A3B.2AB.AB4.BA.3BAB.2B.B$.AB.ABA2B3.2B2A.AB.B2.A.A2.A
3.B2A.AB.A3B.B.A.AB$AB.4B.2BAB5.BA2BA.2B.BA4B.3A2.2A2B2A5.A$2.2BA2.2B
4A.A2BA.BAB2A3BA.ABA.3A.B2A.B.A.3B$2AB.B2.B.2B3.3BAB.B2.B5.B2A.B2A2.A
2.B.3A2BA$2.2A2.2B2.A2.4B4A.B.2A.B.B.A2BABA.AB3.ABABA2B$AB.2A.A.BAB.B
.A.ABA2.AB.AB3A3B.B.B2.A.BA.2A2B.BA$2A.2A2.BA2B2A.2A.5AB.2ABAB3A.A2.A
B.B.ABA2.A2B$2B.2A2BA2B3.B2.2B2A2B3A2.A.A.BA.A3.B2.B3A.B2.B$2BA.A.A.
2A.2ABA.B3A2BAB.ABA.AB.BAB.A2.2BAB.B2.B2A$.B3.ABA6.2ABABA.AB2A2.A2.A
2BA2.B2A.B.B.4B2A$BA.B4.A2.B.4A3.2A.BA.A3BA2BA.ABAB3A.BA3B$2A.A.A.A3B
4A.AB.2A2.2AB.2A6BA5BA.2BA3B$.4A.3A3BA.A.B.AB.AB.3A2.2A.AB.AB2.A.BAB.
B2ABA$A2.B.2A.A.A.AB.B.A2.AB2.2B.B2A.A.B.BA4BA.ABA2B.A$2A2B3ABAB.A.BA
3B2A.A2B.A2B.2A2B.A2.BA.B2.2ABA3B$4A.2AB2.A2BAB.BA.A.B2A4.3B4.B.AB.BA
2B.A2.2A$BA3B3A.A3.2B.2A.3A2B3.AB2A.B.2A3.B.2A.3A.2A$2.2BA.BA2.A.AB4.
ABA3.3BAB2ABABA.B2A4B.A2B.2B$BAB.A.ABA2B.2AB.2B.BA.B3.2B6A2B.2A.3B2A
2.BA$.B.B4.2A4BA.A.B2AB2.A2.2B.B2.2B.4BA.BAB3.A$2B2.2A5B.B3A.A2.4BAB
2A3BAB.B2ABA.B.2B2.AB$.B2.3B.B2.B.BA2.AB.B.A.BAB.B.2A.A.BA.2B.BA.A.B.
2A!

[toroidalet]

Sierpinski builder:

Code: Select all

x = 12, y = 5, rule = 000211-11012-2101-001-20-1
.A.B$A2.3B5AB$A4.4A.AB$A2.3B5AB$.A.B!

[praosylen]

Sierpinski builder:

Code: Select all

x = 12, y = 5, rule = 000211-11012-2101-001-20-1
.A.B$A2.3B5AB$A4.4A.AB$A2.3B5AB$.A.B!

A growing spaceship backrake that eventually degenerates into a normal backrake:

Code: Select all

x = 12, y = 23, rule = 000211-11012-2101-001-20-1
.A.B$A2.3B5AB$A4.4A.AB$A2.3B5AB$.A.B14$.A.B$A2.3B5AB$A4.4A.AB$A2.3B5A
B$.A.B!

EDIT: Another nice rule:

Code: Select all

x = 146, y = 16, rule = 021022-00021-2011-100-01-1
40.2A2$2B6.2B30.2B$29.B$28.2B110.A$30.B$8.B7.B9.B$7.2B6.2B8.2B$9.B7.B
9.B117.A$17.A2B$16.B5$141.A!

Puffers can be made from combinations of the c/3 diagonal ships.

[Naszvadi]

This is a somewhat interesting rulespace, and it's the smallest as-of-yet-unexplored (as far as I know) one that I could think of that can be simulated in Golly.

First, some clarifications:

• 3 cell states, von Neumann neighborhood.
• The state of a cell depends ONLY on the total number of cells of each state in its neighborhood (including the center cell in the totals) on the previous generation (e.g. if a state-2 cell with four state-1 neighbors dies, a state-1 cell with three state-1 and 1 state-2 neighbors must die as well).

There are exactly 3^21 such rules, ignoring rules equivalent under state transformations and the like. Rules are denoted by the following code:

Code: Select all

NNNNNN-NNNNN-NNNN-NNN-NN-N

Each N represents the state of each transition. The kth group of Ns represents transitions with k-1 state-2 cells. The kth N in a group represents a transition with k-1 state-1 cells. For example, the rulestring

Code: Select all

020200-00000-0000-100-00-0

represents the rule where a cell with one or three state-1 neighbors (including itself) and no state-2 neighbors becomes state 2, a cell with three state-2 neighbors and no state-1 neighbors becomes state 1, and all other cells die or remain state 0.

Here is a script to generate 3ITVN rules:

Code: Select all

import golly as g
size = map(int, "12222102333202332023202201")
table = """@RULE %s
@TABLE
n_states:3
neighborhood:vonNeumann
symmetries:permute
#00
00000%s
#10
01000%s
10000%s
#20
01100%s
11000%s
#30
01110%s
11100%s
#40
01111%s
11110%s
#50
11111%s
#01
02000%s
20000%s
#11
01200%s
12000%s
21000%s
#21
01120%s
11200%s
21100%s
#31
01112%s
11120%s
21110%s
#41
11112%s
21111%s
#02
02200%s
22000%s
#12
01220%s
12200%s
21200%s
#22
01122%s
11220%s
21120%s
#32
11122%s
21112%s
#03
22200%s
02220%s
#13
01222%s
12220%s
21220%s
#23
11222%s
21122%s
#04
02222%s
22220%s
#14
12222%s
21222%s
#05
22222%s"""
rule = g.getstring("Enter rule in format \"000000-00000-0000-000-00-0\"", "000000-00000-0000-000-00-0")
format = (rule,)
for i in xrange(len(rule)):
    format += (rule[i],)*size[i]
table %= format
ruledir = g.getdir("rules")
filename = ruledir + rule + ".rule"
with open(filename, 'w') as f:
    f.write(table)
g.setrule(rule)
g.show("Created rule in file: %s.rule" % rule)

Interesting rules appear few and far between. Here are four that may qualify:
000120-10000-2220-020-10-2
022112-02222-0101-000-01-1
000010-11210-2211-110-11-2
001200-10000-0202-120-01-1

The third one appears the most promising. Patterns:

Code: Select all

x = 45, y = 12, rule = 000010-11210-2211-110-11-2
.AB5.2A8.2A9.2B10.B.B$3A4.A2BA6.A2BA7.A2.A8.A.A.A$B2A15.2B8.4A10.A$
18.2A9.2A2$29.2A$28.4A$29.2B2$31.B$31.B$31.B!

Unfortunately, I have not found any stationary patterns so far.

EDIT: 5 more maybe-interesting rules:
022102-00201-0021-120-20-1
001211-10000-0000-110-12-2
020002-00020-2012-110-21-2
022010-00100-0002-111-11-2
001200-10221-0022-201-22-1

EDIT 2: 000100-10201-1220-210-11-1
021222-00001-0100-010-12-1

The subset of this ruleset with 3^20 elements would be the most simple (minimal number of cell states), totalistic and "symmetric" set in E^2 {4,4} that might have a universal rule that also contains "0 0000 0" transition, which is necessary criteria declared by Neumann. B0*/*V two-state rules are blinking rules trivially violating this; life-like rules have asymmetric neighbourhood graphs - not edge-transitive. However all B0*/*V rules have 3 state equivalents, but not necessarily inner-totalistic.

[Naszvadi]

This is a somewhat interesting rulespace, and it's the smallest as-of-yet-unexplored (as far as I know) one that I could think of that can be simulated in Golly.

Please check this out:

Code: Select all

x = 4, y = 4, rule = 020222-00000-0000-100-00-1
A$A.A$.2A$3.A!

../forums/viewtopic.php?f=11&t=2233&p=93327#p93327

Gun when? Gun + circuitry + universal constructor = article phd etc world domination