Discrete Dynamics Lab

The spiral glider-gun rule, v=3 k=7
Further details and results relating to the paper:
Wuensche,A., A.Adamatzky, (2006),
"On spiral glider-guns in hexagonal cellular automata: activator-inhibitor paradigm"
to appear in International Journal of Modern Physics C (preprint online here)

• Fundamental structures
  
• Spiral glider-guns
  
• Polymer gliders and eaters
  
• Snapshots of evolved systems
  
• Single mutations
  
• Pairwise collisions
  
• Memory
  
• Creating spiral-glider guns
  
• Mobile gliders-guns
  
• Building the spiral rule, thr rule-table, rule-matrix, and mutations
  

Fundamental structures in the spiral rule, which can be interpreted as a reaction-diffusion system (activator, inhibitor and substrate). Note also that the dynamics are symmetric — all phenomena can occur equally in any direction, or clockwise and anti-clockwise.
5 basic types of mobile gliders (G1-G5), all travelling NorthEast, in the direction of their "active" red heads.	2 types of static "eaters" (E1, E2), E2 has "memory" on its outer shell
S1: high-frequency spiral glider-gun shooting G1 gliders, frequency=6	S2: low-frequency spiral glider-gun shooting G2 gliders, frequency=21

S1 and S2 spiral glider-guns, 120x120, click to enlarge
S1: high-frequency spiral glider-gun	S2 low-frequency spiral glider-gun	high and low-frequency spiral glider-guns interacting

Polymer gliders and eaters forming chains, aggregations and oscillators of arbitrary length, can exist in many combinations.

Various types of small 2-headed gliders heading NE.

Examples of linked eaters.

Snapshots of evolved systems, with gliders, glider-guns and eaters, forming quasi-stable circuits, click to enlarge.
snapshot 88x88	snapshot 250x250

72 single mutations to the spiral rule 36 are quasi-neutral. Click to enlarge. The initial state is the 88x88 snapshot above.
index 0-5	0	1	2	3	4	5
index 6-11	6	7	8	9	10	11
index 12-17	12	13	14	15	16	17
index 18-23	18	19	20	21	22	23
index 24-29	24	25	16	27	28	29
index 30-35	30	31	32	33	34	35

Pairwise collisions between basic gliders - some examples

Mutual annihilation 2→0
3 steps				3 steps

Conservation + Destruction 2→1
2 steps				2 steps

One glider bouncing off another 2→2
180° bounce 6 steps

120° bounce 7 steps

Reproduction
2→4 6 steps

2→6 10 steps

2→8 13 steps

2→6 15 steps

Creating eater E1
eater E1 + 4 gliders 10 steps

Memory - Glider interacting with eater E2
Write

Read

Creating spiral-glider guns S1 and S2

S1 spiral glider-gun

S1 spiral glider-gun

S2 spiral glider-gun

mobile gliders-guns, some examples, with heading (moving head direction) indicated
heading NE	heading NE	heading NE	heading NE

heading NE

heading East

Building the spiral rule
The lookup table v=3 cell-states (cell-values) [2, 1, 0] are represented by cell-colors: black, red, white in the examples above. A cell updates its state depending on the totals of [2s, 1s, 0s] in its neighborhood, size k=7 — the cell and its nearest neighbors on a lattice with hexagonal tiling. There are S = (vk - 1)! / (k!(v -1)!) possibilities, so 36 [2, 1, 0] sub-rules (and their outputs) set out in DDLab's format below. The output string defines the rule (also represented in hexadecimal). 35 0 - rule-table index | | 2:766555444433333222222111111100000000 1:010210321043210543210654321076543210 0:001012012301234012345012345601234567 |||||||||||||||||||||||||||||||||||| 000200120021220221200222122022221210 = 020609a2982a68aa64 in hexadecimal The filename of the spiral rule in DDLab is v3k7w1.vco	k7 hex neighbourhood
The consequences of mutating the spiral glider-gun rule's lookup table Each lookup table output (subrule) has two possible mutations, 72 in all, This table indicates the consequences of each mutation, and which subrules are quasi-neutral (+,-), and which are responsible for gliders (>). This relates to the neighborhood frequency histogram, the frequency of sub-rules in the evolving CA. A snapshot of each mutation is shown below. rule-table index / totals: 2s+1s+0s=k=7 / / rule-table / / / mutations neighborhood frequency histogram / 2_1_0 / 2___1___0 0: 0 0 7 : 0 > o c . |||||||||||||||||||||||||||||| 85.7% => 1: 0 1 6 : 1 > o . o |||||||||||| 0.7% 2: 0 2 5 : 2 . g g || 3: 0 3 4 : 1 + G . G | 4: 0 4 3 : 2 - . G G : 5: 0 5 2 : 2 - . G G : 6: 0 6 1 : 2 + . G G : 7: 0 7 0 : 2 + . G G : 8: 1 0 6 : 0 > c c . |||||||||||||||||||||||||||||| 4.5% => 9: 1 1 5 : 2 > . c c |||||||||||||||||||||||| 1.4% 10: 1 2 4 : 2 . g g |||| 11: 1 3 3 : 1 + G . G | 12: 1 4 2 : 2 + . G G : 13: 1 5 1 : 2 + . G G : 14: 1 6 0 : 2 + . G G || 15: 2 0 5 : 0 > c c . |||||||||||||||||||||||||||||| 2.3% => 16: 2 1 4 : 0 > o c . ||||||||||||||||||||||| 1.3% 17: 2 2 3 : 2 . c g |||| 18: 2 3 2 : 1 - G . G || 19: 2 4 1 : 2 + . G G | 20: 2 5 0 : 2 + . G G || 21: 3 0 4 : 0 > g c . ||||||||||||| 0.8% 22: 3 1 3 : 2 > . c g |||||||||||||||||||||||||| 1.6% 23: 3 2 2 : 2 . g g |||||| 24: 3 3 1 : 1 g . g |||| 25: 3 4 0 : 2 - . G G | 26: 4 0 3 : 0 g G . ||| 27: 4 1 2 : 0 > G c . |||||||||| 28: 4 2 1 : 2 . cg g || 29: 4 3 0 : 1 g . g |||||||| 30: 5 0 2 : 0 G c . || 31: 5 1 1 : 0 > G c . ||| 32: 5 2 0 : 2 - . G G || 33: 6 0 1 : 0 G c . || 34: 6 1 0 : 0 G cg . || 35: 7 0 0 : 0 - G G . | Key to mutations G:quasi-neutral, g:related but different glider dynamics emerge c:chaotic, cg:tends to chaos but with gliders present o:order. 36 mutations and 15 sub-rules are wildcards, making no impact, or minimal impact on the dynamics. Sub-rules are marked as: +strong wildcards (9), –weak wildcards (6), and >glider-critical (10) The rule as a matrix The rule can be represented more simply as an ij matrix as shown below, were i represents the number of 2s, and j the number of 1s. The number of 0s follows: 7-(i+j). The output of each sub-rule can be read off each ij. The matrix on the right shows: +:strong wildcards, and –:weak wildcards. the spiral glider-gun rule as a k=7 matrix j=1s wildcards 0 1 2 3 4 5 6 7 -weak +strong ............... 0: 0 1 2 1 2 2 2 2 0 1 2 + - - + + 1: 0 2 2 1 2 2 2 0 2 2 + + + + 2: 0 0 2 1 2 2 0 0 2 - + + i=2s 3: 0 2 2 1 2 0 2 2 1 - 4: 0 0 2 1 0 0 2 1 5: 0 0 2 0 0 - 6: 0 0 0 0 7: 0 -	neighbourhoods frequency histogram: snaphot from DDLab
The rule as a k=6 outer-totalistic rule The same rule can be expressed as a k=6 outer-totalistic rule, which consists of three centre-rules, one for each state (or value [2, 1, 0]) of the central cell, c. Each centre-rule has 28 sub-rules, which are shown in the DDLab format, and tabular and matrix form below, indicating strong (+), and weak (-) wildcards. 28 0 - rule-table index | | | 2:6554443333222221111110000000 v| 1:0102103210432105432106543210 | 0:0010120123012340123450123456 hexadecimal filename in DDLab |||||||||||||||||||||||||||| -------------- ---------- | 0:0002001220212002212202221210 - 020689829a2a64 w1otc0.vco c| 1:0201202122221202221222222121 - 2189aa62a6aa99 w1otc1.vco | 2:0002001200212202212002221220 - 020609a2982a68 w1otc2.vco 3 k=6 outer-totalistic rules rule table index / totals: 2s+1s+0s=k=6 3 k=6 outer-totalistic matrices / / rule tables / / ---------------- j=1s wildcards / 2_1_0 C=2 C=1 C=0 0 1 2 3 4 5 6 -weak +strong 0: 0 0 6 : 0 1 0 ............. 1: 0 1 5 : 2 2 1 c=2 0: 0 2 2 1 2 2 2 0 2 2 + - - + 2: 0 2 4 : 2 1+ 2 1: 0 0 2 1 2 2 0 0 2 - + + 3: 0 3 3 : 1+ 2- 1+ 2: 0 2 2 1 2 0 2 2 1 - 4: 0 4 2 : 2- 2- 2- i=2s 3: 0 0 2 1 0 0 2 1 5: 0 5 1 : 2- 2+ 2- 4: 0 0 2 0 0 - 6: 0 6 0 : 2+ 2+ 2+ 5: 0 0 0 0 7: 1 0 5 : 0 2 0 6: 0 0 8: 1 1 4 : 0 2 2 9: 1 2 3 : 2 1+ 2 10: 1 3 2 : 1- 2+ 1+ 11: 1 4 1 : 2+ 2+ 2+ c=1 0: 1 2 1 2 2 2 2 1 2 + - - + + 12: 1 5 0 : 2+ 2+ 2+ 1: 2 2 1 2 2 2 2 2 + + + + 13: 2 0 4 : 0 0 0 2: 0 2 1 2 2 0 2 - + + 14: 2 1 3 : 2 2 0 i=2s 3: 2 2 1 2 2 2 1 - 15: 2 2 2 : 2 1- 2 4: 0 2 1 0 2 1 16: 2 3 1 : 1 2+ 1- 5: 0 2 0 - 17: 2 4 0 : 2- 2+ 2+ 6: 0 0 18: 3 0 3 : 0 2 0 19: 3 1 2 : 0 2 2 20: 3 2 1 : 2 1 2 21: 3 3 0 : 1 2- 1 c=0 0: 0 1 2 1 2 2 2 0 1 2 + - - + 22: 4 0 2 : 0 0 0 1: 0 2 2 1 2 2 0 2 2 + + + 23: 4 1 1 : 0 2 0 2: 0 0 2 1 2 0 0 2 - + 24: 4 2 0 : 2- 1 2 i=2s 3: 0 2 2 1 0 2 2 1 25: 5 0 1 : 0 0 0 4: 0 0 2 0 0 2 26: 5 1 0 : 0 2- 0 5: 0 0 0 0 27: 6 0 0 : 0- 0 0 6: 0 0	k6 outer-totalistic hex neighbourhood