GliderGuns in 3d Cellular Automata
 Discrete Dynamics Lab 
k=6 Beehiverule period=6   k=7 Spiralrule period=8 

Beehiverule
3d glidergun v=3 k=6 ijh=40x10x40  click panels to enlarge kcode= 0022000220022001122200021210=(hex)0a0282816a0264 (v3k6x1.vco in DDLab) "Glider dynamics in 3value hexagonal cellular automata: the beehive rule", Int.J.Unconv.Comp, Vol.1, No.4, 2005, 375398. (PREPRINT)  
     

Spiralrule
3d glider gun v=3 k=7 ijh=40x10x40  click panels to enlarge kcode=000200120021220221200222122022221210=(hex)020609a2982a68aa64 (v3k7w1.vco in DDLab) "On spiral gliderguns in hexagonal cellular automata: activatorinhibitor paradigm" Int.J.Mod.Phys C, Vol. 17, No. 7,10091026, (PREPRINT)  
        
In these examples, imagine looking down into a shallow box 40 x 40, and 10 cells deep. Periodic boundaries have been suppressed so that the gliders fall out of the space so an not to interfere with the generating core.
The 3d gliderguns are periodic structures fixed in one location, which spontaneously emerged in the dynamics from random initial states. The CA rules are identical to the ktotalistic rules  beehive and spiral, but apply to a 3d (nearest neighbour) cubic neighbourhood. Because ktotalistic rules are symmetric  independent of the location of values in the neighbourhood  the gliderguns exist in each chirality and in all possible orientations. Many other structures also emerge  gliders, compoundgliders, and mobile gliderguns, but as yet no static structures analagous to 2d eaters in the 2d spiral rule. I discovered the 3d glidergun in the beehiverule in 2004, and in the spiralrule in June 2009. The best way to search for these structures is to set a small random 3d block in DDLab, 2 or 3 cells in diameter, and observe the result. The core of the glidergun can then be isolated (and saved) with other DDLab methods.