Andrew wrote:wildmyron wrote:The two gliders in this rule presented by Carter Bays in his

1987 paper ...

The correct paper and link for the above rule is:

"Patterns for Simple Cellular Automata in a Universe of Dense-Packed Spheres"

http://www.complex-systems.com/pdf/01-5-1.pdf

That is a much more comprehensive discussion of the rule, though in my defense the rule and its gliders are mentioned in the earlier paper.

Andrew wrote:I used [3D.lua's] Random Pattern command for a few minutes and discovered this previously unknown diagonal glider (the Bays paper mentions only 2 known gliders, both orthogonal):

`3D version=1 size=40 pos=19,19,19`

# diagonal c/2 spaceship

x=3 y=3 z=2 rule=3D3/3H

bbo$bbo$oo/boo$o!

This has to be the same as the other glider, just traveling in the direction of one of the "diagonal" neighbours when considered from the PoV of the cubic lattice used to simulate the FCC lattice.

I'm sure it's the same effect as why these two orientations of a c/5 glider in B2/S3H appear to be different orthogonal and diagonal gliders when viewed on a square grid:

`#C [[ SQUAREDISPLAY ]]`

x = 7, y = 17, rule = B2/S3H

bobo$o$bo2bo$3bobo$2bo2bo$2bo$4bobo4$2bo$2bobo$2o$3bo2bo$bo3bo$4b2o$3b

o!

Andrew wrote:Could well be so! Here's a table of the resulting patterns for each tree with L layers:

Andrew: Ha, you beat me this time. Continuing the Christmas Tree sequence:

`L result`

21 p124

22 p4094

23 p16

24 p2046

25 p252

26 p1022

27 p56

28 p32766

29 p60

30 p62

31 dies

32 p62

33 p60

34 p8190

35 p56

36 unknown

37 p2044

38 p8190

39 p48

40 p2046

This continues to match A160657 for even L, though I didn't verify L=36 due to the time required.

It's worth noting that some of the trees (in particular with odd L) evolve into oscillators with the given period - they don't all have the triangular layer pattern as a phase of the resulting oscillator. This may or may not help with understanding where this behaviour comes from.

For anyone else interested, here's a script to generate these tree patterns

`-- Christmas tree for 3D.lua`

local size = 12

local offset = size//2

ClearCells()

for len = size, 1, -1 do

local dy = size - 2*len

for x = 0, len-1 do

for z = 0, x do

SetCell(x-offset, dy, len-z-offset)

end

end

end

For larger trees the grid size will need to be increased above the default.