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The beehive rule
Further details and results relating to the paper:
Wuensche,A.,"Glider dynamics in 3-value hexagonal cellular automata: the beehive rule",
Int. Journ. of Unconventional Computing, Vol.1, No.4, 2005, 375-398.
Preprint available HERE

 Beehive rule in 3d 3d CA v=3 k=6 n=40x40x10 kcode= 0022000220022001122200021210 (v3k6x1.vco in DDLab) panel 1. panel 2. panel 3. panel 4. panel 5. panel 6. panels 1,2 and 3 started as random initial states - panels 3,4 5,6 are consecutive time-steps

 The beehive rule-table (kcode) showing all 56 possible mutations, and neighborhoods controling the basic glider 0022000220022001122200021210 (rule v3k6x1.vco in DDLab) \begin{verbatim} kcode = 0022000220022001122200021210 kcode index / totals: 2s+1s+0s=k=6 / / kcode basic / / / glider / / / mutations --------- / 2_1_0 / 2___1___0 background-> 0: 0 0 6 -> 0 o c - head+-> 1: 0 1 5 -> 1 0 - 0 2: 0 2 4 -> 2 - Sg cg 3: 0 3 3 -> 1 -+ G - G out4 4: 0 4 2 -> 2 -+ - G G out3 5: 0 5 1 -> 0 -+ G G - out1 6: 0 6 0 -> 0 -+ G G - side2-> 7: 1 0 5 -> 0 c c - side1-> 8: 1 1 4 -> 2 - c c side1+ 9: 1 2 3 -> 2 - cg G 10: 1 3 2 -> 2 -+ - G G out2 11: 1 4 1 -> 1 -+ G - G tail 12: 1 5 0 -> 1 -+ G - G head-> 13: 2 0 4 -> 0 c c - 14: 2 1 3 -> 0 Gs c - 15: 2 2 2 -> 2 - gc gc 16: 2 3 1 -> 2 -+ - G G 17: 2 4 0 -> 0 -+ G G - 18: 3 0 3 -> 0 g c - 19: 3 1 2 -> 2 - c cg 20: 3 2 1 -> 2 - cg Gd 21: 3 3 0 -> 0 -+ G G - 22: 4 0 2 -> 0 G c - center-> 23: 4 1 1 -> 0 g cg - 24: 4 2 0 -> 2 - cg G 25: 5 0 1 -> 2 - cg G 26: 5 1 0 -> 0 g gc - 27: 6 0 0 -> 0 G Gd - key to mutations: quasi-neutral G=25/56, wildcards -+ 10/28 G/g=gliders, G=same/similar dynamics, g=weak/different, S=spirals, d=dense, s=sparse, c=chaos, o=order, 0=all 0s  >

 21 types of glider collisions collision diagrams  no type no before after head-on odd: 4 2->0 3 6 0 2->2 1 2 2 head-on odd: 4 2->0 3 6 0 2->2 1 2 2 oblige head-on: 8 2->0 3 6 0 2->1 2 4 2 2->4 1 2 4 2->5 1 2 5 2->6 1 2 6 oblique tail-on: 5 2->0 1 2 0 2->1 2 4 2 2->2 1 2 2 2->6 1 2 6 ----------------------------------- totals: 21 21 42 31 gliders type no before after self-destruction: 2->0 10 20 0 one-survivor:.... 2->1 4 8 4 conservation:.... 2->2 3 6 6 self-reproduction: 2->4 1 2 4 2->5 1 2 5 2->6 2 4 12 ------------ totals 21 42 31  oblique 60 degree head-on collisions - 8 types oblique 120 degree tail-on collisions - 5 types 180 degree 0dd head-on collisions - 4 types 180 even head-on collisions - 4 types

 8 60degree (head-on oblique) collisions 1a 2->5 2a 2->1 3a 2->1 4a 2->4 5a 2->6 6a 2->0 7a 2->0 8a 2->0

 5 120degree (oblique tail-on) collisions 1b tail 2->1 2b tail 2->1 3b tail 2->2 note bounce 4b 2->6 continues as 5a 5b 2->0

 4 180degree (head-on) collisions, odd 1h-odd 2->2 2h-odd 2->0 3h-odd 2->0 4h-odd 2->0

 4 180 (head-on) collisions, even 1h-even 2->2 2h-even 2->0 3h-even 2->0 4h-even 2->0

 Exploding red cell makes 6 new gliders single red->6

 polymer-like gliders made up from sub-units p=1 .. p=2 .. .. .. p=2 .. p=4 .. p=4 p=4

 glider-guns (puffer-trains) with various periods shooting 1 to 4 glider streams 1a streams=1 period=4 1b streams=1 period=4 1c streams=1 period=4 2a streams=2 period=4 2b streams=2 period=4 2c streams=2 period=4 2d streams=2 period=4 3a streams=3 period=4 3b streams=3 period=4 3c streams=3 period=4 3d streams=3 period=4 3e streams=3 period=4 4a streams=4 period=8

 Static glider-gun: period=13, multiple streams in 6 directions (found behind puffer-train) Puffer-train: moving west, absorbing boundary conditions, 222x122 click to enlarge

 2 interesting mutations index 23: 4 1 1 -> 0 changed to 2 index 2: 0 2 4 -> 2 changed to 1
 56 single mutations to the beehive rule 28/56 are quasi-neutral, click to enlarge index 0-6 0 1 2 3 4 5 6 index 7-13 7 8 9 10 11 12 13 index 14-20 14 15 16 17 18 19 20 index 21-27 21 22 23 24 25 26 27

 Classifying rule-space to find v=3 k=6 complex rules

 Examples of other complex-rules kcode=2200021000222201110201212210 0=10 2=11 1=7 kcode=0222200220000200100201102110 0=14 2=9 1=5 kcode=0220002120022202120200112110 0=11 2=11 1=6 kcode= 0200001120100200002200120110 0=15 2=6 1=6 kcode= 0200202022222200012100002100 0=14 2=9 1=3 kcode=0122120102200122010000102000 0=14 2=8 1=6