Nine Related Hexagonal Rules

[bprentice]

The rule trees

Code: Select all

@RULE H1
@TREE
num_states=3
num_neighbors=8
num_nodes=29
1 0 2 0
2 0 0 0
1 1 1 0
2 0 2 0
3 1 3 1
2 2 0 2
3 3 5 3
4 4 6 4
1 0 1 0
2 0 8 0
3 5 9 5
4 6 10 6
5 7 11 7
6 12 12 12
1 1 2 0
2 8 14 8
3 9 15 9
4 10 16 10
5 11 17 11
6 18 18 18
7 13 19 13
2 14 0 14
3 15 21 15
4 16 22 16
5 17 23 17
6 24 24 24
7 19 25 19
8 20 26 20
9 27 27 27
@COLORS
 0   0   0   0
 1 255   0   0
 2   0 255   0

Code: Select all

@RULE H2
@TREE
num_states=4
num_neighbors=8
num_nodes=27
1 0 2 3 0
1 0 1 3 0
2 0 1 0 0
1 1 2 3 0
2 1 3 1 1
3 2 4 2 2
2 3 3 3 3
3 4 6 4 4
4 5 7 5 5
2 3 0 3 3
3 6 9 6 6
4 7 10 7 7
5 8 11 8 8
6 12 12 12 12
2 0 3 0 0
3 9 14 9 9
4 10 15 10 10
5 11 16 11 11
6 17 17 17 17
7 13 18 13 13
3 14 9 14 14
4 15 20 15 15
5 16 21 16 16
6 22 22 22 22
7 18 23 18 18
8 19 24 19 19
9 25 25 25 25
@COLORS
 0   0   0   0
 1 255   0   0
 2   0 255   0
 3   0   0 255

Code: Select all

@RULE H3
@TREE
num_states=3
num_neighbors=8
num_nodes=27
1 0 2 0
2 0 0 0
1 1 1 0
2 0 2 0
3 1 3 1
2 2 0 2
3 3 5 3
4 4 6 4
3 5 1 5
4 6 8 6
5 7 9 7
6 10 10 10
1 0 1 0
2 0 12 0
3 1 13 1
4 8 14 8
5 9 15 9
6 16 16 16
7 11 17 11
2 12 0 12
3 13 19 13
4 14 20 14
5 15 21 15
6 22 22 22
7 17 23 17
8 18 24 18
9 25 25 25
@COLORS
 0   0   0   0
 1 255   0   0
 2   0 255   0

Code: Select all

@RULE H4
@TREE
num_states=3
num_neighbors=8
num_nodes=27
1 0 2 0
2 0 0 0
1 1 1 0
2 0 2 0
3 1 3 1
2 2 0 2
3 3 5 3
4 4 6 4
1 0 1 0
2 0 8 0
3 5 9 5
4 6 10 6
5 7 11 7
6 12 12 12
2 8 2 8
3 9 14 9
4 10 15 10
5 11 16 11
6 17 17 17
7 13 18 13
3 14 5 14
4 15 20 15
5 16 21 16
6 22 22 22
7 18 23 18
8 19 24 19
9 25 25 25
@COLORS
 0   0   0   0
 1 255   0   0
 2   0 255   0

Example patterns:

Code: Select all

x = 173, y = 73, rule = H1
37.5.A$
36.5.ABA$
36.4.AB3A$
35..3A2.3ABA$
35..2B4.ABA$
34..3A5.A$
34..2A11.A$
33..3A10.ABA$
33..2B10.AB3A$
32..3A11.3ABA$
32.16.ABA$
31.18.A2$
30.27.B2$
29..3A78.B.2A2.2A.B$
29..2B80.2A3.2A$
28..3A21.B55.2A8.2A$
28..2A31.B45.2A.B5.B.2A$
27..3A$
27..2B$
26..3A69.B.2A20.2A.B$
26.32.B41.2A21.2A$
25.72.2A26.2A$
25.41.A29.2A.B23.B.2A$
24.41.ABA84.B.2A2.2A.B$
24.40.AB3A84.2A3.2A$
23.42.3ABA23.A32.A23.2A8.2A$
23.43.ABA23.2A31.2A22.2A.B5.B.2A$
22.45.A24.A32.A$
22.50.A17.B2.B29.B2.B$
21.B3.B45.ABA17.A32.A15.B.2A20.2A.B$
21.49.AB3A15.2A31.2A16.2A21.2A$
20.51.3ABA14.A32.A14.2A26.2A$
20.52.ABA62.2A.B23.B.2A$
19.54.A2$
18.54.A$
18.52.ABA62.2A.B23.B.2A$
17.51.3ABA14.A32.A14.2A26.2A$
17.49.AB3A15.2A31.2A16.2A21.2A$
16.B3.B45.ABA17.A32.A15.B.2A20.2A.B$
16.50.A17.B2.B29.B2.B$
15.45.A24.A32.A$
15.43.ABA23.2A31.2A22.2A.B5.B.2A$
14.42.3ABA23.A32.A23.2A8.2A$
14.40.AB3A84.2A3.2A$
13.41.ABA84.B.2A2.2A.B$
13.41.A29.2A.B23.B.2A$
12.72.2A26.2A$
12.32.B41.2A21.2A$
11..3A69.B.2A20.2A.B$
11..2B$
10..3A$
10..2A31.B45.2A.B5.B.2A$
9..3A21.B55.2A8.2A$
9..2B80.2A3.2A$
8..3A78.B.2A2.2A.B2$
7.27.B2$
6.18.A$
6.16.ABA$
5..3A11.3ABA$
5..2B10.AB3A$
4..3A10.ABA$
4..2A11.A$
3..3A5.A$
3..2B4.ABA$
2..3A2.3ABA$
2.4.AB3A$
1.5.ABA$
1.5.A!

Code: Select all

x = 31, y = 21, rule = H2
11.2A$
10..AC9.C$
10.AC12.2A$
9..2A.C9.CB$
9..5A3.AC2.2BA$
8.3.A2CBA.2AC2BA$
8.3.AC.CB2.3A$
7.3.2A3C$
7.4.3A$
6.6.A2$
5.6.A$
5.4.3A$
4.3.2A3C$
4.3.AC.CB2.3A$
3.3.A2CBA.2AC2BA$
3..5A3.AC2.2BA$
2..2A.C9.CB$
2.AC12.2A$
1..AC9.C$
1.2A!

Code: Select all

x = 51, y = 62, rule = H3
31.3.BA$
30.4.2A$
30.B3.B$
29..A2BA$
29.2.A3$
27.11.3A$
27.11.2B$
26.11.3A$
26.11.2A$
25.10.3AB$
25.9.2A4$
23.9.ABA$
22.7.B.2ABA$
22.7.AB.A$
21.9.A3.A$
21.9.BA.2B$
20.9.AB$
20.10.AB.2A$
19.11.AB.2A8$
15.23.3A$
15.23.2B$
14.23.3A$
14.23.2A$
13.22.3AB$
13.21.2A4$
11.22.2A$
10.19.ABA.2A$
10.18.A2B.A$
9.19.A.2BA$
9.17.A.B.BA$
8.17.BAB$
8.19.BAB$
7.16.2B.A.A$
7.15.A2.BA$
6.13.B2.2A$
6.13.A2BA8.BA$
5.12.B2.2A10.B$
5.27.A$
4.26.AB$
4.25.A$
3.25.2A2.AB4.3A$
3.25.A2.B.B4.2B$
2.29.A5.3A4.A$
2.28.B11.2B2A$
1.26.A2BA8.A2BAB.B$
1.26.ABA9.AB.A.2A$
27.A11.A2.ABA!

Code: Select all

x = 38, y = 37, rule = H4
19.4.2A.B$
18.5.2A$
18..A5.B$
17..2A3.B$
17..A$
16.B2.B$
16..B14.B$
15.15.B$
15.16.2A$
14.15.B.2A7$
11..B19.A$
10.B2.B17.2A$
10..A19.A$
9..2A17.B2.B$
9..A19.B7$
5.5.2A.B$
5.5.2A$
4.8.B$
4.6.B14.B$
3.20.B2.B$
3.21.A$
2.17.B3.2A$
2.15.B5.A$
1.17.2A$
1.15.B.2A!

Brian Prentice

[SuperSupermario24]

That H2 pattern is hypnotic o_o

[bprentice]

Five more rule trees:

Code: Select all

@RULE H5
@TREE
num_states=5
num_neighbors=8
num_nodes=28
1 0 2 3 4 0
2 0 0 0 0 0
1 1 1 3 4 0
2 0 2 0 0 0
3 1 3 1 1 1
2 2 0 2 2 2
3 3 5 3 3 3
4 4 6 4 4 4
1 0 1 3 4 0
2 0 8 0 0 0
3 5 9 5 5 5
4 6 10 6 6 6
5 7 11 7 7 7
6 12 12 12 12 12
2 8 8 8 8 8
3 9 14 9 9 9
4 10 15 10 10 10
5 11 16 11 11 11
6 17 17 17 17 17
7 13 18 13 13 13
2 8 0 8 8 8
3 14 20 14 14 14
4 15 21 15 15 15
5 16 22 16 16 16
6 23 23 23 23 23
7 18 24 18 18 18
8 19 25 19 19 19
9 26 26 26 26 26
@COLORS
 0   0   0   0
 1 255   0   0
 2   0 255   0
 3   0   0 255
 4 255 255   0

Code: Select all

@RULE H6
@TREE
num_states=5
num_neighbors=8
num_nodes=29
1 0 2 3 4 0
2 0 0 0 0 0
1 1 1 3 4 0
2 0 2 0 0 0
3 1 3 1 1 1
1 0 2 3 1 0
2 2 5 2 2 2
3 3 6 3 3 3
4 4 7 4 4 4
1 0 1 3 4 0
2 5 9 5 5 5
3 6 10 6 6 6
4 7 11 7 7 7
5 8 12 8 8 8
6 13 13 13 13 13
2 9 9 9 9 9
3 10 15 10 10 10
4 11 16 11 11 11
5 12 17 12 12 12
6 18 18 18 18 18
7 14 19 14 14 14
2 9 0 9 9 9
3 15 21 15 15 15
4 16 22 16 16 16
5 17 23 17 17 17
6 24 24 24 24 24
7 19 25 19 19 19
8 20 26 20 20 20
9 27 27 27 27 27
@COLORS
 0   0   0   0
 1 255   0   0
 2   0 255   0
 3   0   0 255
 4 255 255   0

Code: Select all

@RULE H7
@TREE
num_states=5
num_neighbors=8
num_nodes=30
1 0 2 3 4 0
1 0 2 3 2 0
2 0 1 0 0 0
1 1 1 3 4 0
2 1 3 1 1 1
3 2 4 2 2 2
1 0 2 3 1 0
2 3 6 3 3 3
3 4 7 4 4 4
4 5 8 5 5 5
1 0 1 3 4 0
2 6 10 6 6 6
3 7 11 7 7 7
4 8 12 8 8 8
5 9 13 9 9 9
6 14 14 14 14 14
2 10 10 10 10 10
3 11 16 11 11 11
4 12 17 12 12 12
5 13 18 13 13 13
6 19 19 19 19 19
7 15 20 15 15 15
2 10 0 10 10 10
3 16 22 16 16 16
4 17 23 17 17 17
5 18 24 18 18 18
6 25 25 25 25 25
7 20 26 20 20 20
8 21 27 21 21 21
9 28 28 28 28 28
@COLORS
 0   0   0   0
 1 255   0   0
 2   0 255   0
 3   0   0 255
 4 255 255   0

Code: Select all

@RULE H8
@TREE
num_states=5
num_neighbors=8
num_nodes=29
1 0 2 3 4 0
2 0 0 0 0 0
1 1 1 3 4 0
2 0 2 0 0 0
3 1 3 1 1 1
1 0 2 1 1 0
2 2 5 2 2 2
3 3 6 3 3 3
4 4 7 4 4 4
1 0 1 3 4 0
2 5 9 5 5 5
3 6 10 6 6 6
4 7 11 7 7 7
5 8 12 8 8 8
6 13 13 13 13 13
2 9 9 9 9 9
3 10 15 10 10 10
4 11 16 11 11 11
5 12 17 12 12 12
6 18 18 18 18 18
7 14 19 14 14 14
2 9 0 9 9 9
3 15 21 15 15 15
4 16 22 16 16 16
5 17 23 17 17 17
6 24 24 24 24 24
7 19 25 19 19 19
8 20 26 20 20 20
9 27 27 27 27 27
@COLORS
 0   0   0   0
 1 255   0   0
 2   0 255   0
 3   0   0 255
 4 255 255   0

Code: Select all

@RULE H9
@TREE
num_states=5
num_neighbors=8
num_nodes=30
1 0 2 3 4 0
2 0 0 0 0 0
1 1 1 3 4 0
2 0 2 0 0 0
3 1 3 1 1 1
1 0 2 1 1 0
2 2 5 2 2 2
3 3 6 3 3 3
4 4 7 4 4 4
1 0 1 3 4 0
2 5 9 5 5 5
3 6 10 6 6 6
4 7 11 7 7 7
5 8 12 8 8 8
6 13 13 13 13 13
1 0 1 1 4 0
2 9 15 9 9 9
3 10 16 10 10 10
4 11 17 11 11 11
5 12 18 12 12 12
6 19 19 19 19 19
7 14 20 14 14 14
2 15 0 15 15 15
3 16 22 16 16 16
4 17 23 17 17 17
5 18 24 18 18 18
6 25 25 25 25 25
7 20 26 20 20 20
8 21 27 21 21 21
9 28 28 28 28 28
@COLORS
 0   0   0   0
 1 255   0   0
 2   0 255   0
 3   0   0 255
 4 255 255   0

Example patterns:

Code: Select all

x = 8, y = 2, rule = H5
1.2A$
.A!

Code: Select all

x = 10, y = 4, rule = H6
2.A.A$
1.4A$
1..A$
.2A!

Code: Select all

x = 13, y = 6, rule = H7
3..D$
3.DCBC$
2.D2C2A$
2..B2ABA$
1.2.CABA$
1.4.A!

Code: Select all

x = 14, y = 6, rule = H8
3.5.A$
3.2.2ABA$
2.2.B2ABA$
2.2AB2A$
1.3ABA$
1.2A!

Code: Select all

x = 17, y = 8, rule = H9
4.2.BA2.AB$
4..CA.A.AC$
3.2.D4AD$
3..7A$
2..CA4.AC$
2.4BD4B$
1..2AC2.C2A$
1..BC3.CB!

Brian Prentice

[bprentice]

Each of these rules is specified by a rule table. To obtain the next state of a cell, the table columns are indexed by the sum of the six nearest neighbors in state one and the table rows are indexed by the current state of the cell. The tables are shown below in Java code. Note that the tables define either a Generations rule or a slightly modified Generations rule. None of these rules is suitable for engineering interesting constructions. The intent of this exploration was to determine how small changes in a rule table affect the the rule's behavior. The example patterns above show a three barrel gun changing to a six barrel gun and then to a two barrel gun. A shuttle oscillator can be seen changing to an oscillating ship. Also, subtle changes in the small ships can be observed.

Here are the rule tables:

H1

final static int ruleTable[][] =
{
{0,0,1,0,0,1,0},
{2,2,1,2,1,2,2},
{0,0,0,0,0,0,0}
};

H2

final static int ruleTable[][] =
{
{0,0,1,1,0,1,0},
{2,1,2,2,2,2,2},
{3,3,3,3,3,3,3},
{0,0,0,0,0,0,0}
};

H3

final static int ruleTable[][] =
{
{0,0,1,0,0,0,0},
{2,2,1,2,2,1,2},
{0,0,0,0,0,0,0}
};

H4

final static int ruleTable[][] =
{
{0,0,1,0,0,1,0},
{2,2,1,2,1,1,2},
{0,0,0,0,0,0,0}
};

H5

final static int ruleTable[][] =
{
{0,0,1,0,0,0,0},
{2,2,1,2,1,1,2},
{3,3,3,3,3,3,3},
{4,4,4,4,4,4,4},
{0,0,0,0,0,0,0}
};

H6

final static int ruleTable[][] =
{
{0,0,1,0,0,0,0},
{2,2,1,2,1,1,2},
{3,3,3,3,3,3,3},
{4,4,4,1,4,4,4},
{0,0,0,0,0,0,0}
};

H7

final static int ruleTable[][] =
{
{0,0,1,0,0,0,0},
{2,2,1,2,1,1,2},
{3,3,3,3,3,3,3},
{4,2,4,1,4,4,4},
{0,0,0,0,0,0,0}
};

H8

final static int ruleTable[][] =
{
{0,0,1,0,0,0,0},
{2,2,1,2,1,1,2},
{3,3,3,1,3,3,3},
{4,4,4,1,4,4,4},
{0,0,0,0,0,0,0}
};

H9

final static int ruleTable[][] =
{
{0,0,1,0,0,0,0},
{2,2,1,2,1,1,2},
{3,3,3,1,3,1,3},
{4,4,4,1,4,4,4},
{0,0,0,0,0,0,0}
};

Brian Prentice