FCC3333 - a new 3D CA on an FCC grid

[Tim Hutton]

FCC is the 'face-centred cubic' lattice, also known as the Rhombic Dodecahedral Honeycomb

Brian D. Eubanks recently (Dec 2018) posted some results of a rule on the FCC. He calls it '3333' and it is very simple:

each cell is alive in the next generation if (and only if) it has three live neighbors in the previous generation.

Here are some animations of the rule in action:

50% random initialization dies down, leaving one p2 blinker:


50% random initialization dies down, leaving a p2 glider:


I've put a Ready file here: https://github.com/GollyGang/ready/tree/gh-pages/Patterns/CellularAutomata/FCC3333 to encourage exploration.

Brian says there's a 'Christmas tree' oscillator, but I haven't managed to recreate it yet. He says he hasn't yet found a glider gun. Twitter discussion

[eubanksb]

I'm not familiar with Ready, I had homegrown my simulator, I'm going to look at ready tonight and figure out how to enter the data for the 126 cycle oscillator.

[wildmyron]

The p126 oscillator in Golly's 3D.lua pattern format:

Code: Select all

3D version=1 size=50 pos=19,14,19 gen=267
x=12 y=23 z=12 rule=3D3/3H
11bo$$10bo$$9bo$$8bo$$7bo$$6bo$$5bo$$4bo$$3bo$$bbo$$bo$$o/10boo$$9b
oo$$8boo$$7boo$$6boo$$5boo$$4boo$$3boo$$bboo$$boo$$oo/9b3o$$8b3o$$7b
3o$$6b3o$$5b3o$$4b3o$$3b3o$$bb3o$$b3o$$3o/8b4o$$7b4o$$6b4o$$5b4o$$4b
4o$$3b4o$$bb4o$$b4o$$4o/7b5o$$6b5o$$5b5o$$4b5o$$3b5o$$bb5o$$b5o$$5o
/6b6o$$5b6o$$4b6o$$3b6o$$bb6o$$b6o$$6o/5b7o$$4b7o$$3b7o$$bb7o$$b7o$$
7o/4b8o$$3b8o$$bb8o$$b8o$$8o/3b9o$$bb9o$$b9o$$9o/bb10o$$b10o$$10o/b
11o$$11o/12o!

The two gliders in this rule presented by Carter Bays in his 1987 paper are shown in the 3D.lua thread.

[Andrew]

Thanks Arie -- you just beat me to it by a few minutes!

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

[Andrew]

Adding or removing base layers from the p126 Christmas tree pattern usually results in oscillators with different periods (or occasionally patterns that die). The periods are typically rather small, but this one has a period of 1,022:

Code: Select all

3D version=1 size=60 pos=24,18,24
# p1022 oscillator
x=18 y=35 z=18 rule=3D3/3H
17bo$$16bo$$15bo$$14bo$$13bo$$12bo$$11bo$$10bo$$9bo$$8bo$$7bo$$6bo$$
5bo$$4bo$$3bo$$bbo$$bo$$o/16boo$$15boo$$14boo$$13boo$$12boo$$11boo$$
10boo$$9boo$$8boo$$7boo$$6boo$$5boo$$4boo$$3boo$$bboo$$boo$$oo/15b3o
$$14b3o$$13b3o$$12b3o$$11b3o$$10b3o$$9b3o$$8b3o$$7b3o$$6b3o$$5b3o$$
4b3o$$3b3o$$bb3o$$b3o$$3o/14b4o$$13b4o$$12b4o$$11b4o$$10b4o$$9b4o$$
8b4o$$7b4o$$6b4o$$5b4o$$4b4o$$3b4o$$bb4o$$b4o$$4o/13b5o$$12b5o$$11b
5o$$10b5o$$9b5o$$8b5o$$7b5o$$6b5o$$5b5o$$4b5o$$3b5o$$bb5o$$b5o$$5o/
12b6o$$11b6o$$10b6o$$9b6o$$8b6o$$7b6o$$6b6o$$5b6o$$4b6o$$3b6o$$bb6o
$$b6o$$6o/11b7o$$10b7o$$9b7o$$8b7o$$7b7o$$6b7o$$5b7o$$4b7o$$3b7o$$bb
7o$$b7o$$7o/10b8o$$9b8o$$8b8o$$7b8o$$6b8o$$5b8o$$4b8o$$3b8o$$bb8o$$
b8o$$8o/9b9o$$8b9o$$7b9o$$6b9o$$5b9o$$4b9o$$3b9o$$bb9o$$b9o$$9o/8b10o
$$7b10o$$6b10o$$5b10o$$4b10o$$3b10o$$bb10o$$b10o$$10o/7b11o$$6b11o$$
5b11o$$4b11o$$3b11o$$bb11o$$b11o$$11o/6b12o$$5b12o$$4b12o$$3b12o$$bb
12o$$b12o$$12o/5b13o$$4b13o$$3b13o$$bb13o$$b13o$$13o/4b14o$$3b14o$$
bb14o$$b14o$$14o/3b15o$$bb15o$$b15o$$15o/bb16o$$b16o$$16o/b17o$$17o
/18o!

[Andrew]

... I'm going to look at ready tonight and figure out how to enter the data for the 126 cycle oscillator.

I think you'll find this to be a rather frustrating experience. Ready's editing capabilities are quite primitive. Much easier to use the 3D.lua script included in the latest version (3.2) of Golly. I recommend opening Preferences > Keyboard and assigning the "3" key to open 3D.lua. The script has a number of useful features:

* Full editing functions (cut, copy, paste, etc).

* Unlimited undo/redo for all editing and generating changes.

* It can run other Lua scripts, so you can create complicated patterns programatically, run searches, etc.

* Customizable keyboard shortcuts (see 3D.lua's help for how to create a startup script).

[77topaz]

Adding or removing base layers from the p126 Christmas tree pattern usually results in oscillators with different periods (or occasionally patterns that die). The periods are typically rather small, but this one has a period of 1,022:
[RLE]

Hmm, the periods 126 and 1022 being of the form 2^N - 2 suggests that these oscillators use a XOR mechanism somehow? (Compare A160657 (B36/S125's 2x2 oscillators).)

[Andrew]

Hmm, the periods 126 and 1022 being of the form 2^N - 2 suggests that these oscillators use a XOR mechanism somehow? (Compare A160657 (B36/S125's 2x2 oscillators).)

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

Code: Select all

L   result
1   dies
2   p2
3   dies
4   p6
5   p4
6   p14
7   dies
8   p14
9   p12
10  p62
11  p8
12  p126
13  p28
14  p30
15  dies
16  p30
17  p28
18  p1022
19  p24
20  p126

Note that for even L the periods are 2, 6, 14, 14, 62, 126, 30, 30, 1022, 126 which matches A160657! Clearly can't be a coincidence but I'm struggling to see how tree-shaped patterns in a 3D rule can be related to rectangular patterns in a 2D rule.

[77topaz]

Interesting. If I look at the trees when they're oscillating, I can at times see resemblances to the behaviour of the 2x2 rectangular oscillators, but it's more difficult to visualise/comprehend (Blinkerspawn or Calcyman, who've also investigated these kinds of oscillators, could probably describe it better than I).

The odd-L terms also match with an XOR-related sequence in the OEIS, A268754, which contains A160657 as a subsequence.

[wildmyron]

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.

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):

Code: Select all

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:

Code: Select all

#C [[ SQUARECELLS ]]
x = 7, y = 17, rule = B2/S3H
bobo$o$bo2bo$3bobo$2bo2bo$2bo$4bobo4$2bo$2bobo$2o$3bo2bo$bo3bo$4b2o$3b
o!

Hmm, the periods 126 and 1022 being of the form 2^N - 2 suggests that these oscillators use a XOR mechanism somehow? (Compare A160657 (B36/S125's 2x2 oscillators).)

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:

Code: Select all

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

Code: Select all

-- 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.

[dvgrn]

Interesting. If I look at the trees when they're oscillating, I can at times see resemblances to the behaviour of the 2x2 rectangular oscillators, but it's more difficult to visualise/comprehend (Blinkerspawn or Calcyman, who've also investigated these kinds of oscillators, could probably describe it better than I).

The odd-L terms also match with an XOR-related sequence in the OEIS, A268754, which contains A160657 as a subsequence.

For anyone who hasn't seen the various forum discussions, a wide variety of 1D and 2D rules produce isomorphic XOR behavior. The 2x2 rectangular oscillators show behavior along their edges that matches the one-dimensional pattern made by a single initial ON cell in Wolfram Rule 90. Here's the 126-tick oscillator in W90, with two duplicated rows at the top and bottom:

Code: Select all

x = 25, y = 128, rule = W90:T26,0
obobobobobo3bobobobobobo$11bobo$10bo3bo$9bobobobo$8bo7bo$7bobo5bobo$6b
o3bo3bo3bo$5bobobobobobobobo$4bo15bo$3bobo13bobo$2bo3bo11bo3bo$bobobob
o9bobobobo$o7bo7bo7bo$bo5bobo5bobo5bo$obo3bo3bo3bo3bo3bobo$3bobobobobo
bobobobobo$2bo19bo$bobo17bobo$o3bo15bo3bo$bobobo13bobobo$o5bo11bo5bo$b
o3bobo9bobo3bo$obobo3bo7bo3bobobo$5bobobo5bobobo$4bo5bo3bo5bo$3bobo3bo
bobobo3bobo$2bo3bobo7bobo3bo$bobobo3bo5bo3bobobo$o5bobobo3bobobo5bo$bo
3bo5bobo5bo3bo$obobobo3bo3bo3bobobobo$7bobobobobobo$6bo11bo$5bobo9bobo
$4bo3bo7bo3bo$3bobobobo5bobobobo$2bo7bo3bo7bo$bobo5bobobobo5bobo$o3bo
3bo7bo3bo3bo$bobobobobo5bobobobobo$o9bo3bo9bo$bo7bobobobo7bo$obo5bo7bo
5bobo$3bo3bobo5bobo3bo$2bobobo3bo3bo3bobobo$bo5bobobobobobo5bo$obo3bo
11bo3bobo$3bobobo9bobobo$2bo5bo7bo5bo$bobo3bobo5bobo3bobo$o3bobo3bo3bo
3bobo3bo$bobo3bobobobobobo3bobo$o3bobo11bobo3bo$bobo3bo9bo3bobo$o3bobo
bo7bobobo3bo$bobo5bo5bo5bobo$o3bo3bobo3bobo3bo3bo$bobobobo3bobo3bobobo
bo$o7bobo3bobo7bo$bo5bo3bobo3bo5bo$obo3bobobo3bobobo3bobo$3bobo5bobo5b
obo$2bo3bo3bo3bo3bo3bo$bobobobobobobobobobobobo$o23bo$bo21bo$obo19bobo
$3bo17bo$2bobo15bobo$bo3bo13bo3bo$obobobo11bobobobo$7bo9bo$6bobo7bobo$
5bo3bo5bo3bo$4bobobobo3bobobobo$3bo7bobo7bo$2bobo5bo3bo5bobo$bo3bo3bob
obobo3bo3bo$obobobobo7bobobobobo$9bo5bo$8bobo3bobo$7bo3bobo3bo$6bobobo
3bobobo$5bo5bobo5bo$4bobo3bo3bo3bobo$3bo3bobobobobobo3bo$2bobobo11bobo
bo$bo5bo9bo5bo$obo3bobo7bobo3bobo$3bobo3bo5bo3bobo$2bo3bobobo3bobobo3b
o$bobobo5bobo5bobobo$o5bo3bo3bo3bo5bo$bo3bobobobobobobobo3bo$obobo15bo
bobo$5bo13bo$4bobo11bobo$3bo3bo9bo3bo$2bobobobo7bobobobo$bo7bo5bo7bo$o
bo5bobo3bobo5bobo$3bo3bo3bobo3bo3bo$2bobobobobo3bobobobobo$bo9bobo9bo$
obo7bo3bo7bobo$3bo5bobobobo5bo$2bobo3bo7bo3bobo$bo3bobobo5bobobo3bo$ob
obo5bo3bo5bobobo$5bo3bobobobo3bo$4bobobo7bobobo$3bo5bo5bo5bo$2bobo3bob
o3bobo3bobo$bo3bobo3bobo3bobo3bo$obobo3bobo3bobo3bobobo$5bobo3bobo3bob
o$4bo3bobo3bobo3bo$3bobobo3bobo3bobobo$2bo5bobo3bobo5bo$bobo3bo3bobo3b
o3bobo$o3bobobobo3bobobobo3bo$bobo7bobo7bobo$o3bo5bo3bo5bo3bo$bobobo3b
obobobo3bobobo$o5bobo7bobo5bo$bo3bo3bo5bo3bo3bo$obobobobobo3bobobobobo
bo$11bobo!

I've tried opening two copies of Golly and running the 126-tick oscillator in 3D.lua, using the W90 picture as a map for what should happen next. There's definitely a mapping in there somewhere -- for example, after 63 ticks the 3D oscillator becomes its mirror image, whereas the W90 pattern returns to its original configuration but with the left and right halves swapped.

I have the feeling that I'm just trying to map the first phase of the 1D oscillator to the wrong phase of the 3D oscillator, and if I just found the correct column (or something) through the 3D oscillator, the corresponding phase of the 1D oscillator could be read off directly, same as the edges of the XOR rectangles in the various 2D cases.

[Andrew]

... in my defense the rule and its gliders are mentioned in the earlier paper.

I'd forgot about that -- my apologies!

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.

Of course! I keep getting tripped up by that. I'll edit my post to remove the spurious claim.

... I didn't verify L=36 due to the time required.

I can confirm the resulting period is 174,762.

[BlinkerSpawn]

There's a noticeable spatial-parity thing going on within the oscillators which lends credence to the hypothesis that these are indeed some sort of Triangolus oscillator.
That being said, I'm having difficulty probing much further because it's kind of hard to visualize what's going on even in Golly. Is there a way to view individual layers or something?
fcc-grid.lua would be really helpful right about now but that's probably too much to ask

[wildmyron]

... I'm having difficulty probing much further because it's kind of hard to visualize what's going on even in Golly. Is there a way to view individual layers or something?

If you change to Select or Edit mode there will be one layer which is prominently visible. Use '.' and ',' to move the layer and I think it's 'a' to change the visible layer orientation. (Sorry, on mobile atm, can't check key bindings.) This won't help with visualising the "diagonal" layers in the FCC lattice, but it might help a bit.

[Andrew]

Is there a way to view individual layers or something?

Some useful tips:

Hit the M/D/S keys to switch to Move/Draw/Select mode. The latter 2 modes show a blue active plane.

Hit shift-A to cycle the active plane thru 3 orthogonal orientations. (Diagonal orientations might be nice to have -- I'll think about that.)

Hit the period/comma keys to move the active plane one cell at a time, or shift-click anywhere in the plane and drag it to the desired position.

The Draw/Select modes let you draw/select cells only in the active plane, but it's also possible (and often more convenient) to do some editing in Move mode:

Alt-click in a live cell and drag the cursor to delete that cell and all live cells behind it.
Shift-click in a live cell and drag the cursor to select that cell and all live cells behind it.

Hit H to see 3D.lua's help which includes a table of all the keyboard shortcuts and many other tips.

[BlinkerSpawn]

Sure enough, there it is!

1PwFINr.png (10.47 KiB) Viewed 237 times