B34/S03456

[checkman]

This post also connects with the "useless discoveries" thread elsewhere in this forum. By looking at random fields and this rule, I found an oscillator that everyone has probably seen:

x = 4, y = 2, rule = B34/S03456
bo$2obo!

but then I noticed that you can put two of them together, to make:

x = 7, y = 2, rule = B34/S03456
boboo$2o2bobo!

but unfortunately, it also has period 2.

There's also a very interesting p16:

x = 4, y = 2, rule = B34/S03456
2o$b3o!

If you look at it closely, you see that the original pattern is rotated 90 degrees every 4 iterations. Unfortunately it doesn't move, so it's not a spaceship.

[checkman]

The reason I started looking at this CA is because of Eppstein's "most wanted" page; https://www.ics.uci.edu/~eppstein/ca/wanted.html

It lists several CAs which he thinks have gliders/space ships, but which he has been unable to find any. He says that it is "one of the rules studied by Wolfram. An expanding octagonal region filled with stable junk. Has many small p2 and p4 oscillators."

This CA has a lot of patterns which I have dubbed "tar". These consist of irregularly shaped blobs whose boundary is two cells thick, and they expand to infinity. Close to the boundary -- within 5-10 cells of it -- there is quite a bit of action, but it eventually settles down into a fixed pattern or a p2 further in.

Other oscillators that I have discovered are:

* a p10, which in one generation is a line of 5 live cells. (5o!) After 5 iterations, the row of cells has rotated 90 degrees.

* a p2 "rotary saw": x = 4, y = 4, bo$b3o$3o$2bo! This actually turns up quite a bit in random fields. If you remove one of the cells in the middle, the saw still works.

* p2: Y tetromino: x = 3, y = 3, 2bo$2o$bo!

* p2: T pentomino: x = 3, y = 3, 3o$bo$bo!

* a couple more p2s: bo$2o$2b2o$3o$2bo! and bobo$bobo$2ob2o$2bo!

I also read through Donald Knuth's rough draft for his 6A fascicle, and decided to look for some still lifes (p1). Based on some guesswork, I found the following by hand:

x = 5, y = 5, 2ob2o$5o$bobo$5o$2ob2o! [20 cells]

x = 7, y = 7, b2ob2o$ob3obo$2obob2o$b5o$2obob2o$ob3obo$b2ob2o! [33 cells, "the cloverleaf"]

x = 9, y = 9, 3b2ob2o$3b5o$4ob4o$3obob3o$bob3obo$3obob3o$4ob4o$2b5o$2b2ob2o! [53 cells]

The first and the third look like fortresses (from above) to me. The symmetry is intentional, of course.

Questions that might be worth investigating are:

(1) Are there any odd-period oscillators, other than still lifes?

(2) Are there any phoenixes? (Phoenices?)

[dvgrn]

...but then I noticed that you can put two of them together, to make:

x = 7, y = 2, rule = B34/S03456
boboo$2o2bobo!

but unfortunately, it also has period 2.

I seem to be missing some key detail here. When I copy this into Golly, it appears to be period 4, as does any linear extension of it:

Code: Select all

x = 13, y = 2, rule = B34/S03456
obo2bo2bo2bo$2b2ob2ob2ob2o!

There's also a very interesting p16:

x = 4, y = 2, rule = B34/S03456
2o$b3o!

This one turns into a rotary saw after a few ticks.

Everything in the next post works as advertised, except for the 53-cell fortress -- the first two lines should only be inset by two cells, not three. Here's the whole collection, cleaned up for pasting into Golly:

Code: Select all

x = 46, y = 20, rule = B34/S03456
32bo$5o7bo8b3o7b2o9bobo$12b2o8bo10b2o7bobo$11bo10bo8b3o7b2ob2o$33bo9bo
7$20b2ob2o11b2ob2o$3b2ob2o11bob3obo10b5o$3b5o11b2obob2o8b4ob4o$4bobo
13b5o9b3obob3o$3b5o11b2obob2o9bob3obo$3b2ob2o11bob3obo8b3obob3o$20b2ob
2o9b4ob4o$36b5o$36b2ob2o!

Just out of curiosity: what program are you using to investigate these patterns? You seem to be generating your RLE by hand, and it's non-standard -- there's no rule in these new patterns, so Golly pastes it in as B3/S23 (and then of course it blows up). And the extra comma at the end of the header instead of a newline makes Golly ignore the entire pattern.

I don't mean to be too Golly-centric -- many other CA editors would have similar problems -- but probably more people would look at these patterns if they were easier to copy and paste. A simple Ctrl+C in Golly will copy a selected pattern as standard RLE.

Code: Select all

Surrounding RLE with [code] and 

also helps...[/code]

[checkman]

...but then I noticed that you can put two of them together, to make:

x = 7, y = 2, rule = B34/S03456
boboo$2o2bobo!

but unfortunately, it also has period 2.

I seem to be missing some key detail here. When I copy this into Golly, it appears to be period 4, as does any linear extension of it:

I must not have been counting correctly that day. Yes, it's still a p4.

There's also a very interesting p16:

x = 4, y = 2, rule = B34/S03456
2o$b3o!

This one turns into a rotary saw after a few ticks.

So it does. I must have been working with another rule set when I found that one.

Just out of curiosity: what program are you using to investigate these patterns? You seem to be generating your RLE by hand, and it's non-standard -- there's no rule in these new patterns, so Golly pastes it in as B3/S23 (and then of course it blows up). And the extra comma at the end of the header instead of a newline makes Golly ignore the entire pattern.

Yes, I'm doing my RLEs by hand (and eyeball). I've modified http://www.quesucede.com/public/gameoflife/ so that it actually runs with other rule sets. One of these days, I'll get serious and download one of the "real" programs.

Code: Select all

Surrounding RLE with [code] and 

also helps...[/code]

Thanks for the tip; I'm new here.

Last night, I also did some brute-force searches for still lifes. An isolated cell is a still life, as well as a 2x2 block; I call these (and any non-interfering combination of them) trivial. There are no nontrivial still lifes that fit inside a 4x4 grid, and the only one that fits in the 5x5 grid is the fortress that I posted last night. The only one that fits inside a 6x6 grid and which takes up 6 rows is

Code: Select all

x = 5, y = 6, rule = B34/S03456
2ob2o$5o$2bo$2bo$5o$2ob2o!

oo.oo
ooooo
..o..
..o..
ooooo
oo.oo

which looks like a Van Der Graaf in the traditional B3/S23.

There are probably 100 or so that fit inside a 7x7 grid and which take up 7 rows; I did some crude symmetry checking and got 308, although better symmetry-checking would cut that number in half, or more. Here's an example from each "equivalence class":

Code: Select all

oo.o.oo
ooooooo
.oo.oo.
oo.o.oo
.oo..o.
ooooooo
oo.o.oo (four-cornered fortress)

oo.o.oo
ooooooo
.oo.oo.
oo.o.oo
oooo.o.
..ooooo
..oo.oo (five-cornered fortress)

oo.o.oo
ooooooo
.oo.oo.
oo.o.oo
o.oo.o.
.o.oooo
..oo.oo (damaged fortress)

oo.o.oo
ooooooo
..o..o.
..ooooo
...o.o.
..ooooo
..oo.oo (trapezoidal fortress)

.oo.oo.
o.ooo.o
oooo.oo
.oo.oo.
oo.oooo
o.ooo.o
.oo.oo. (a la cloverleaf)

oo.oo.
oooo.o
.o..oo
ooooo.
.o..oo
oooo.o
oo.oo. (B)

oo.oo
ooooo
.o.o.
ooooo
.o.o.
ooooo
oo.oo

..oo..
..oo..
oooooo
oo..oo
.oooo.
oo..oo
oo..oo (A (Mountain) -- "A Mountain" is a mountain near the ASU campus)

.....oo
.....oo
...ooo.
..o.ooo
..oo.o.
ooooooo
oo.o.oo (three-cornered fortress)

For many of the fortresses, and the Bs, you can change the middle 9 or 15 cells and get more still lifes. The boundaries of these still lifes also show what the boundaries of larger still lifes will look like; once you get inside of the boundary (i.e., once you've looked at what the top two rows look like), case analysis explodes.

I don't think I'll be looking at 8x8s until I work with a "real" program --- these were found by a C program that had N^2 nested for loops, and which only took a few seconds to run; I don't expect anything new, though.

[velcrorex]

Here's a few oscillators I've found of periods, 3,4,5,6, and 7.

Code: Select all

x = 33, y = 90, rule = B34/S03456
21booboo$20bob3obo$4boobooboo7bobobobobo$4b8o6bobooboboobo$5boboobo6bo
b9obo$4b3obb3o5boobb5obboo$4b3obb3o6b11o$5boboobo6boobb5obboo$4b8o5bob
9obo$4boobooboo6bobooboboobo$19bobobobobo$20bob3obo$21booboo8$3booboob
oo10bo5bo$bbob6obo8booboboboo$bobobboobbobo6bobbooboobbo$ob3o4b3obo4b
oobb5obboo$oobooboobooboo6boobbobboo$bo3b4o3bo6b3obbobb3o$3oboobboob3o
7b7o$3oboobboob3o5b3obbobb3o$bo3b4o3bo7boobbobboo$oobooboobooboo4boobb
5obboo$ob3o4b3obo5bobbooboobbo$bobobboobbobo7booboboboo$bbob6obo9bo5bo
$3boobooboo7$3boobooboo11booboboboo$bbob6obo9bob7obo$boboo4boobo7boboo
boboboobo$ob10obo6b4obobob4o$4o6b4o7bo3bobo3bo$bobob4obobo7b6ob6o$oobo
bobboboboo7bo9bo$oobobobboboboo6b6ob6o$bobob4obobo8bo3bobo3bo$4o6b4o6b
4obobob4o$ob10obo6bobooboboboobo$boboo4boobo8bob7obo$bbob6obo10boobobo
boo$3boobooboo6$3boobooboo$bbob6obo$boboobooboobo$ob3o4b3obo$4ob4ob4o$
bobboboobobbo$3oboobboob3o$3oboobboob3o$bobboboobobbo$4ob4ob4o$ob3o4b
3obo$boboobooboobo$bbob6obo$3boobooboo4$3boobooboo$bbob6obo$bobobobbob
obo$ob10obo$oobob4oboboo$b4o4b4o$ooboo4booboo$ooboo4booboo$b4o4b4o$oob
ob4oboboo$ob10obo$bobobobbobobo$bbob6obo$3boobooboo!

Haven't found a spaceship yet, but while searching I did find this creature which extends at c/2.

Code: Select all

x = 28, y = 27, rule = B34/S03456
5bobobobobobobobobo$oobobobobobobobobobobobooboo$5ob17ob4o$6obobobobob
obobobob5o$oob4ob14obb4o$b3obobobboboboboboo5boo$b3obobbobobobobob3ob
ooboo$bboobob14ob4o$bboobobobobobobobobo3b3o$3b3oboo3b4oboboboboo$3b3o
bbo3boobobo3boboo$4boobbo3b4obobob3o$4boobobobobobobobob3o$5b18o$4b3ob
oboboboboboboboo$4b3obobob4o3bobboo$3boobo3boboboo3bobb3o$3boobobobob
4o3boob3o$bb3o3boboboboboboboboboo$bb4ob14oboboo$booboob3obobobobobobb
ob3o$boo5boobobobobobbobob3o$4obb14ob4oboo$5obobobobobobobobob6o$4ob
17ob5o$oobooboboboboboboboboboboboo$6bobobobobobobobobo!

[checkman]

Here's a few oscillators I've found of periods, 3,4,5,6, and 7.

Have you found any phoenixes? (Phoenices?)

Haven't found a spaceship yet, but while searching I did find this creature which extends at c/2.

Neat ... It's something that grows, yet it isn't "tar" ...

What software did you use to search?

[wildmyron]

I've been playing with the oscillators, extending their boundaries and trying to fill them with interesting patterns. But this rule is pretty reliable at turning random patterns into still life with the odd p2 scattered about. Occasionally I found a p3 or p4 but none of them seem particularly interesting.

While thinking about agars, fuses and phoenix patterns I did find this p6

Code: Select all

x = 17, y = 17, rule = B34/S03456
3b2ob2ob2ob2o$2bob9obo$bobobo2bo2bobobo$ob13obo$2obo9bob2o$b4o7b4o$2ob
o9bob2o$2ob2o7b2ob2o$b3o9b3o$2ob2o7b2ob2o$2obo9bob2o$b4o7b4o$2obo9bob
2o$ob13obo$bobobo2bo2bobobo$2bob9obo$3b2ob2ob2ob2o!

It is extensible in space (horizontally) and I was hoping it would also be extensible in period by extending the pattern vertically. However, for all the heights I have manually tried (up to internal height of 45) the familiar still life patterns return.

[checkman]

Here's a phoenix of width 6 and period 2. (The narrowest phoenix in regular Life has width 8.)

Code: Select all

x = 6, y = 12, rule = B34/S03456
2bo$2bobo$o3bo$obobo$2bo2bo$2o$4b2o$o2bo$bobobo$bo3bo$bobo$3bo!

and one of width 7 and period 2 which looks extendable:

Code: Select all

x = 7, y = 33, rule = B34/S03456
2b2o2$b2ob2o$bo$5bo$2o3b2o$o$5b2o$2o4bo$o$4b3o$2o$o3bo$5b2o$2o$o4b2o$
6bo$2o$bo3b2o$6bo$b2o$bo3b2o$5bo$2o$5b2o$bo4bo$b2o$5b2o$bo2bo$2bobobo$
2bo3bo$2bobo$4bo!

I still have to code a bit to generate the RLE automatically, before I post the one of width 8. (The RLEs here are via Golly, so they should be correct.)

EDIT: Here's the RLE for the phoenix of width 8 that my program produced.

Code: Select all

x = 8, y = 446, rule = B34/S03456
3b2o$$b2ob2o$5bo$bo$2o3b2o$6bo$bo$bo4b2o$3bobo$b2o3bo$o4bo$$2o3bobo$2bo2bobo$$2b2o2bo$bo4b
o$6bo$b2o3bo$bo$4bobo$2o2bobo$o$5bo$2o3bo$o4bo$5bo$2o$o3bobo$4bobo$2o$2bo3bo$6bo$b2o3bo$bo
4bo$$b2o2bobo$bo3bobo$$b2o2bo$o4bo$5bo$2o3bo$2bo$5bobo$b2o2bobo$bo$6bo$b2o3bo$bo4bo$6bo$b
2o$bo3bobo$5bobo$b2o$bo4bo$6bo$2o4bo$2bo3bo$$3o2bobo$5bobo$o$b2o3bo$6bo$bo4bo$b2o3bo$$bo2b
obo$2o2bobo$$bo4bo$b2o3bo$6bo$bo4bo$b2o$5bobo$bo3bobo$b2o$6bo$bo4bo$b2o3bo$6bo$bo$b2o2bob
o$5bobo$bo$b2o3bo$6bo$2o4bo$bo4bo$$2o2bobo$o3bobo$$2o3bo$2bo2bo$2bo2bo$2bo3b2o$$bobo2bo$bo
2bobo$$2o4bo$2bo3bo$6bo$3o3bo$$o4bobo$b2o2bobo$$b2o3bo$bo4bo$6bo$2o4bo$o$5bobo$2o3bobo$2b
o$2bo2bo$2bo2bo$5bo$obo3bo$obo3bo$5bo$2bo4bo$2bo3bo$2bo4bo$2bo2bo$6bo$bobo2bo$bobo3bo$7bo
$2bo3bo$2bo4bo$2bo4bo$2bo3bo$7bo$bobo3bo$bobo2bo$6bo$2bo2bo$2bo4bo$2bo3bo$2bo4bo$7bo$bobo
2bo$bobo2bo$7bo$bo3bo$bo4bo$bo4bo$bo3bo$6bo$obo3bo$obo2bo$6bo$bo4bo$bo3bo$bo4bo$bo4bo$5bo
$obo2bo$obo3bo$6bo$bo3bo$bo5bo$bo4bo$bo5bo$6bo$bobo2bo$2bo2bo$o4bo$bo$bo2bobo$2bo2bo$2bo
4bo$6bo$bobo3bo$bobo3bo$6bo$2bo4bo$2bo2bo$bo2bobo$bo$o4bo$2bo2bo$bobo2bo$6bo$bo5bo$bo3bob
o$bo3bo$bo4bo$6bo$obo2bo$obo4bo$6bo$2bo4bo$bo3bo$2bobobo$2o$5bo$o4bo$2o3bo$5bo$2o$2bo2bob
o$2bo2bobo$2bo$3bo2bo$bobo2bo$bo4bo$2bo3bo$bo$bo2bobo$o3bobo$o$bo3bo$o4bo$o4bo$bo3bo$bo$2b
o2bobo$bo3bobo$bo$2bo2bo$o4bo$bo3bo$o4bo$2bo$bo3bobo$bo3bobo$2bo$bo4bo$bo4bo$2bo3bo$bo4bo
$2bo$o4bobo$bo3bobo$bo$2bo3bo$bo4bo$bo4bo$o5bo$o$bo2bobo$bo2bobo$o$o4bo$bo3bo$o4bo$2bo2bo
$bo$bo3bobo$2bo2bobo$bo$bo4bo$2bo3bo$bo4bo$bo4bo$o$o3bobo$bo2bobo$bo$2bo3bo$o5bo$bo4bo$bo
4bo$o$bo3bobo$o4bobo$o$bo4bo$o5bo$o5bo$bo4bo$bo$o3bobo$o3bobo$bo$o5bo$bo4bo$bo4bo$3bo2bo$2
bo$2bo2bobo$o4bobo$bo$o4bo$2bo3bo$bobobo$6b2o$2bo$2bo3b2o$2bo4bo$2bo$6b2o$bobo3bo$bobo$6b2
o$2bo4bo$2bo$2bo3b2o$2bo2bo$$obo2b2o$obo3bo$$2bo2b2o$2bo4bo$2bo$2bo3b2o$7bo$bobo$bobo2b2o$
7bo$bo$bo4b2o$bo3bo$bo$5b2o$obo3bo$obo$5b2o$bo4bo$bo$bo3b2o$bo4bo$$bobo2b2o$bobo3bo$$2bo3b
2o$2bo2bo$2bo2bo$2o3bo$$bo2bobo$bobo2bo$$bo4b2o$bo3bo$bo$bo3b2o$$bobo2bo$bobo2b2o$$2bo4bo$
2bo2b2o$bo2bo$bo4bo$o5bo$2bo$2bo3b2o$3bo3bo$2bo$2bo3b2o$bo3bo$bo$2bo2b2o$o3bo$o3bobo$2bo2bo
$bo5bo$bo4bo$2bo3bo$2bo2bo$3bo3bo$2bo3bo$2bo3bo$bo3bo$bobo3bo$bobo3bo$bo2b2o$2bo3bo$o5bo$bo
5bo$bo3bo$2b2o2bo$o3bobo$o3bobo$2b2o2bo$bo3bo$bo5bo$o5bo$2bo3bo$bobobo$bo3bo$3bobo$2bobo$2b
o3bo$2bobobo$bo2bo$5b2o$b2o$bo3b2o$6bo$b2o$2bo2b2o$7bo$2b2o$2bo3b2o$5bo$b2o$bo3b3o$$b2o4bo$
bo3b2o$$b2o2b2o$bo4bo$$b2o3b2o$bo3bo$$b2o2b3o$bo$7bo$b2o2b2o$bo$5b2o$b2o3bo$bo$5b2o$2o4bo$
2bo$5b2o$b2o2bo$bo$5b3o$b2o$bo5bo$5b2o$b3o$6bo$3bo2b2o$2b2o$6b2o$2bo2bo$3bobobo$3bo3bo$3bob
o$5bo!

[wildmyron]

Those phoenix patterns are fascinating. the width 8 one seems to have a few errors though. Here's a version that doesn't explode:

Code: Select all

x = 8, y = 446, rule = B34/S03456
3b2o2$b2ob2o$5bo$bo$2o3b2o$6bo$bo$bo4b2o$3bobo$b2o3bo$o4bo2$2o3bobo$2b
o2bobo2$2b2o2bo$bo4bo$6bo$b2o3bo$bo$4bobo$2o2bobo$o$5bo$2o3bo$o4bo$5bo
$2o$o3bobo$4bobo$2o$2bo3bo$6bo$b2o3bo$bo4bo2$b2o2bobo$bo3bobo2$b2o2bo$
o4bo$5bo$2o3bo$2bo$5bobo$b2o2bobo$bo$6bo$b2o3bo$bo4bo$6bo$b2o$bo3bobo$
5bobo$b2o$bo4bo$6bo$2o4bo$2bo3bo2$3o2bobo$5bobo$o$b2o3bo$6bo$bo4bo$b2o
3bo2$bo2bobo$2o2bobo2$bo4bo$b2o3bo$6bo$bo4bo$b2o$5bobo$bo3bobo$b2o$6bo
$bo4bo$b2o3bo$6bo$bo$b2o2bobo$5bobo$bo$b2o3bo$6bo$2o4bo$bo4bo2$2o2bobo
$o3bobo2$2o3bo$2bo2bo$2bo2bo$2bo3b2o2$bobo2bo$bo2bobo2$2o4bo$2bo3bo$6b
o$3o3bo2$o4bobo$b2o2bobo2$b2o3bo$bo4bo$6bo$2o4bo$o$5bobo$2o3bobo$2bo$
2bo2bo$2bo2bo$5bo$obo3bo$obo3bo$5bo$2bo4bo$2bo3bo$2bo4bo$2bo2bo$6bo$bo
bo2bo$bobo3bo$7bo$2bo3bo$2bo4bo$2bo4bo$2bo3bo$7bo$bobo3bo$bobo2bo$6bo$
2bo2bo$2bo4bo$2bo3bo$2bo4bo$7bo$bobo2bo$bobo2bo$7bo$bo3bo$bo4bo$bo4bo$
bo3bo$6bo$obo3bo$obo2bo$6bo$bo4bo$bo3bo$bo4bo$bo4bo$5bo$obo2bo$obo3bo$
6bo$bo3bo$bo5bo$bo4bo$bo5bo$6bo$bobo2bo$2bo2bo$o4bo$bo$bo2bobo$2bo2bo$
2bo4bo$6bo$bobo3bo$bobo3bo$6bo$2bo4bo$2bo2bo$bo2bobo$bo$o4bo$2bo2bo$bo
bo2bo$6bo$bo5bo$bo3bobo$bo3bo$bo4bo$6bo$obo2bo$obo4bo$6bo$2bo4bo$bo3bo
$2bobobo$2o$5bo$o4bo$2o3bo$5bo$2o$2bo2bobo$2bo2bobo$2bo$3bo2bo$bobo2bo
$bo4bo$2bo3bo$bo$bo2bobo$o3bobo$o$bo3bo$o4bo$o4bo$bo3bo$bo$2bo2bobo$bo
3bobo$bo$2bo2bo$o4bo$bo3bo$o4bo$2bo$bo3bobo$bo3bobo$2bo$bo4bo$bo4bo$2b
o3bo$bo4bo$2bo$o4bobo$bo3bobo$bo$2bo3bo$bo4bo$bo4bo$o5bo$o$bo2bobo$bo
2bobo$o$o4bo$bo3bo$o4bo$2bo2bo$bo$bo3bobo$2bo2bobo$bo$bo4bo$2bo3bo$bo
4bo$bo4bo$o$o3bobo$bo2bobo$bo$2bo3bo$o5bo$bo4bo$bo4bo$o$bo3bobo$o4bobo
$o$bo4bo$o5bo$o5bo$bo4bo$bo$o3bobo$o3bobo$bo$o5bo$bo4bo$bo4bo$2bo3bo$b
o$2bo2bobo$o4bobo$bo$o4bo$2bo3bo$bobobo$6b2o$2bo$2bo3b2o$2bo4bo$2bo$6b
2o$bobo3bo$bobo$6b2o$2bo4bo$2bo$2bo3b2o$2bo2bo2$obo2b2o$obo3bo2$2bo2b
2o$2bo4bo$2bo$2bo3b2o$7bo$bobo$bobo2b2o$7bo$bo$bo4b2o$bo3bo$bo$5b2o$ob
o3bo$obo$5b2o$bo4bo$bo$bo3b2o$bo4bo2$bobo2b2o$bobo3bo2$2bo3b2o$2bo2bo$
2bo2bo$2o3bo2$bo2bobo$bobo2bo2$bo4b2o$bo3bo$bo$bo3b2o2$bobo2bo$bobo2b
2o2$2bo4bo$2bo2b2o$bo2bo$bo4bo$o5bo$2bo$2bo3b2o$3bo3bo$2bo$2bo3b2o$bo
3bo$bo$2bo2b2o$o3bo$o3bobo$2bo2bo$bo5bo$bo4bo$2bo3bo$2bo2bo$3bo3bo$2bo
3bo$2bo3bo$bo3bo$bobo3bo$bobo3bo$bo2b2o$2bo3bo$o5bo$bo5bo$bo3bo$2b2o2b
o$o3bobo$o3bobo$2b2o2bo$bo3bo$bo5bo$o5bo$2bo3bo$bobobo$bo3bo$3bobo$2bo
bo$2bo3bo$2bobobo$bo2bo$5b2o$b2o$bo3b2o$6bo$b2o$2bo2b2o$7bo$2b2o$2bo3b
2o$5bo$b2o$bo3b3o2$b2o4bo$bo3b2o2$b2o2b2o$bo4bo2$b2o3b2o$bo3bo2$b2o2b
3o$bo$7bo$b2o2b2o$bo$5b2o$b2o3bo$bo$5b2o$2o4bo$2bo$5b2o$b2o2bo$bo$5b3o
$b2o$bo5bo$5b2o$b3o$6bo$3bo2b2o$2b2o$6b2o$2bo2bo$3bobobo$3bo3bo$3bobo$
5bo!

[checkman]

Those phoenix patterns are fascinating. the width 8 one seems to have a few errors though.

Aaaaargh! That means the rle routine isn't right in my code. ... Or that the phoenix finding part is bad. (I had checked the top of the pattern for a few generations, but evidently not for enough (446 * 2 or so).

It's interesting that there's a fuse of width 5 in the middle, after about 150 generations. (After a few moments) No, there isn't; I had changed the rule in Golly, and that fuse works in B345/S. Look for it under "accidental discoveries".

ADDENDUM: The bug's in the rle code; I had my Java program print out the phoenix in o-and-dot notation, and copied it into Golly, where it didn't blow up. Golly gives me the following RLE:

Code: Select all

x = 8, y = 446, rule = B34/S03456
3b2o2$b2ob2o$5bo$bo$2o3b2o$6bo$bo$bo4b2o$3bobo$b2o3bo$o4bo2$2o3bobo$2b
o2bobo2$2b2o2bo$bo4bo$6bo$b2o3bo$bo$4bobo$2o2bobo$o$5bo$2o3bo$o4bo$5bo
$2o$o3bobo$4bobo$2o$2bo3bo$6bo$b2o3bo$bo4bo2$b2o2bobo$bo3bobo2$b2o2bo$
o4bo$5bo$2o3bo$2bo$5bobo$b2o2bobo$bo$6bo$b2o3bo$bo4bo$6bo$b2o$bo3bobo$
5bobo$b2o$bo4bo$6bo$2o4bo$2bo3bo2$3o2bobo$5bobo$o$b2o3bo$6bo$bo4bo$b2o
3bo2$bo2bobo$2o2bobo2$bo4bo$b2o3bo$6bo$bo4bo$b2o$5bobo$bo3bobo$b2o$6bo
$bo4bo$b2o3bo$6bo$bo$b2o2bobo$5bobo$bo$b2o3bo$6bo$2o4bo$bo4bo2$2o2bobo
$o3bobo2$2o3bo$2bo2bo$2bo2bo$2bo3b2o2$bobo2bo$bo2bobo2$2o4bo$2bo3bo$6b
o$3o3bo2$o4bobo$b2o2bobo2$b2o3bo$bo4bo$6bo$2o4bo$o$5bobo$2o3bobo$2bo$
2bo2bo$2bo2bo$5bo$obo3bo$obo3bo$5bo$2bo4bo$2bo3bo$2bo4bo$2bo2bo$6bo$bo
bo2bo$bobo3bo$7bo$2bo3bo$2bo4bo$2bo4bo$2bo3bo$7bo$bobo3bo$bobo2bo$6bo$
2bo2bo$2bo4bo$2bo3bo$2bo4bo$7bo$bobo2bo$bobo2bo$7bo$bo3bo$bo4bo$bo4bo$
bo3bo$6bo$obo3bo$obo2bo$6bo$bo4bo$bo3bo$bo4bo$bo4bo$5bo$obo2bo$obo3bo$
6bo$bo3bo$bo5bo$bo4bo$bo5bo$6bo$bobo2bo$2bo2bo$o4bo$bo$bo2bobo$2bo2bo$
2bo4bo$6bo$bobo3bo$bobo3bo$6bo$2bo4bo$2bo2bo$bo2bobo$bo$o4bo$2bo2bo$bo
bo2bo$6bo$bo5bo$bo3bobo$bo3bo$bo4bo$6bo$obo2bo$obo4bo$6bo$2bo4bo$bo3bo
$2bobobo$2o$5bo$o4bo$2o3bo$5bo$2o$2bo2bobo$2bo2bobo$2bo$3bo2bo$bobo2bo
$bo4bo$2bo3bo$bo$bo2bobo$o3bobo$o$bo3bo$o4bo$o4bo$bo3bo$bo$2bo2bobo$bo
3bobo$bo$2bo2bo$o4bo$bo3bo$o4bo$2bo$bo3bobo$bo3bobo$2bo$bo4bo$bo4bo$2b
o3bo$bo4bo$2bo$o4bobo$bo3bobo$bo$2bo3bo$bo4bo$bo4bo$o5bo$o$bo2bobo$bo
2bobo$o$o4bo$bo3bo$o4bo$2bo2bo$bo$bo3bobo$2bo2bobo$bo$bo4bo$2bo3bo$bo
4bo$bo4bo$o$o3bobo$bo2bobo$bo$2bo3bo$o5bo$bo4bo$bo4bo$o$bo3bobo$o4bobo
$o$bo4bo$o5bo$o5bo$bo4bo$bo$o3bobo$o3bobo$bo$o5bo$bo4bo$bo4bo$3bo2bo$
2bo$2bo2bobo$o4bobo$bo$o4bo$2bo3bo$bobobo$6b2o$2bo$2bo3b2o$2bo4bo$2bo$
6b2o$bobo3bo$bobo$6b2o$2bo4bo$2bo$2bo3b2o$2bo2bo2$obo2b2o$obo3bo2$2bo
2b2o$2bo4bo$2bo$2bo3b2o$7bo$bobo$bobo2b2o$7bo$bo$bo4b2o$bo3bo$bo$5b2o$
obo3bo$obo$5b2o$bo4bo$bo$bo3b2o$bo4bo2$bobo2b2o$bobo3bo2$2bo3b2o$2bo2b
o$2bo2bo$2o3bo2$bo2bobo$bobo2bo2$bo4b2o$bo3bo$bo$bo3b2o2$bobo2bo$bobo
2b2o2$2bo4bo$2bo2b2o$bo2bo$bo4bo$o5bo$2bo$2bo3b2o$3bo3bo$2bo$2bo3b2o$b
o3bo$bo$2bo2b2o$o3bo$o3bobo$2bo2bo$bo5bo$bo4bo$2bo3bo$2bo2bo$3bo3bo$2b
o3bo$2bo3bo$bo3bo$bobo3bo$bobo3bo$bo2b2o$2bo3bo$o5bo$bo5bo$bo3bo$2b2o
2bo$o3bobo$o3bobo$2b2o2bo$bo3bo$bo5bo$o5bo$2bo3bo$bobobo$bo3bo$3bobo$
2bobo$2bo3bo$2bobobo$bo2bo$5b2o$b2o$bo3b2o$6bo$b2o$2bo2b2o$7bo$2b2o$2b
o3b2o$5bo$b2o$bo3b3o2$b2o4bo$bo3b2o2$b2o2b2o$bo4bo2$b2o3b2o$bo3bo2$b2o
2b3o$bo$7bo$b2o2b2o$bo$5b2o$b2o3bo$bo$5b2o$2o4bo$2bo$5b2o$b2o2bo$bo$5b
3o$b2o$bo5bo$5b2o$b3o$6bo$3bo2b2o$2b2o$6b2o$2bo2bo$3bobobo$3bo3bo$3bob
o$5bo!

[checkman]

B34/S03456 has something in common with GoL: There are no phoenixes of period 3 here, either. (Since there are two birth rules, the proof is a little more complicated than Stephen Silver's proof in GoL, but it follows the same idea.)

Theorem: There are no finite phoenixes in B34/S03456 of period >= 3 such that every cell is alive exactly once per period. (Corollary: There are no finite phoenixes in B34/S03456 with period 3.)

Proof: Suppose otherwise. First, we introduce some labelling conventions.

Let the alive cells at any given generation be labelled by a number. Specifically, the cells with label n are alive in the generation immediately after the generation where the cells with label n+1 are alive. Labels should be read as mod p, where p is the period of the phoenix.

Now take a phoenix in the plane, and call (0,0) the cell which is in the bottom-most row, and furthest to the left in this row. Label the cells so that (0,0) is labelled as 0.

The fact we are working with B34/S03456 means:

(*) A cell labelled n has exactly three or four neighbors labelled n+1, and exactly one or two neighbors labelled n. (Note that "die-off" is possible if n were to have seven or eight neighbors labelled n, but this would give the cell at least 7+3 = 10 neighbors.)

Now, consider (0,0). (*) implies that (0,0) has exactly three neighbors labelled 1, and exactly one neighbor labelled 1, since (0,0) has at most four neighbors which are in the phoenix at some time.

Claim: (1,0) is labelled 0. Proof of the claim: Otherwise, (1,0) is labelled 1, and (-1,1), (0,1), and (1,1) are labelled 0. Now, by (*), (1,0) must have at least three neighbors labelled 2, but on the other hand, (1,0) has at most five neighbors, and three of those are already labelled 0 or 1. This proves the claim. (Since the phoenix has period >= 3, the labels 0 and 2 are distinct.)

Thus, (1,0) is labelled 0, and (-1,1), (0,1), and (1,1) are labelled 1. That means (0,1) has three neighbors labelled 2, which cannot be any of (-1,1), (-1,0), (0,0), (1,0), or (1,1). Therefore, (-1,2), (0,2), and (1,2) are all labelled 2.

Similarly, (-1,3), (0,3), and (1,3) are all labelled 3.

However, this is a contradiction; according to B34/S03456, (0,2) will be alive in generation 0, because it has exactly three neighbors labelled 1, and is itself not 1.

Edited since original post (correction and clean-up)

[checkman]

Those phoenix patterns are fascinating. the width 8 one seems to have a few errors though.

Aaaaargh! That means the rle routine isn't right in my code. ... Or that the phoenix finding part is bad. (I had checked the top of the pattern for a few generations, but evidently not for enough (446 * 2 or so).

I figured it out. My RLE was 72 columns wide, and in the fine print in the Wiki, it says that it should be at most 70 columns.