Phoenices

[Ntsimp]

Just to keep things simple, let's say a "phoenix" is an oscillator in which no cells ever survive.
In which rules can they exist, and which periods can they have?
Since removing survival rules can never affect a phoenix, finding phoenices is equivalent to finding oscillators in rules without survival.
The LifeWiki article claims a proof has been found that no phoenix in Conway's Game of Life (so in B3/S) can have period 3.
In fact, all the phoenices I can find in any rules have even periods. I've found periods 2, 4, 6, 8, 10, and 14 so far.
I've found phoenices whose simplest rules are B2/S, B24/S, B25/S, B3/S, B34/S, B346/S, B35/S, and B36/S.

Then there's the rule B345/S. There are many oscillators in this rule, including periods 18, 22, 24, 26, 30, 32, 34, 38, 40, 42, 62, 66, 72,

[velcrorex]

There are p3 oscillators in B2/S, and likely other similar rules. They're much bigger than the smallest p2 or p4 oscillators. I think they're rarer because each phase must not share any cells with the other two phases.

Here's a few from my own collection. There's probably some slightly smaller examples around.

Code: Select all

x = 142, y = 118, rule = /2
31$26bo50bo$24boo6bo9bo18bo9bo6boo$22bobo10bo4bo22bo4bo10bobo$22boboo
4boo5bobbo9bobbo9bobbo5boo4boobo$21bo6bo8bo5bo16bo5bo8bo6bo$25bobo12b
oo3bobbo6bobbo3boo12bobo$26b3ob7obbo3bo7boo7bo3bobb7ob3o$22bo5bobbo7bo
7bo8bo7bo7bobbo5bo$23bob3o7bo4bo9bobbo9bo4bo7b3obo$23bobbo4bo3bo15boo
15bo3bo4bobbo$26bo3boobo4bo3bo5bo6bo5bo3bo4boboo3bo$26booboo7bo5b3o10b
3o5bo7booboo$26bo3boobo4bo3bo5bo6bo5bo3bo4boboo3bo$23bobbo4bo3bo15boo
15bo3bo4bobbo$23bob3o7bo4bo9bobbo9bo4bo7b3obo$22bo5bobbo7bo7bo8bo7bo7b
obbo5bo$26b3ob7obbo3bo7boo7bo3bobb7ob3o$25bobo12boo3bobbo6bobbo3boo12b
obo23bo$21bo6bo8bo5bo16bo5bo8bo6bo15bo4boo$22boboo4boo5bobbo9bobbo9bo
bbo5boo4boobo14boo$22bobo10bo4bo22bo4bo10bobo21boobo$24boo6bo9bo18bo9b
o6boo15b3o4b3obo$26bo50bo17bo5bobo3bo$94bobbobobbo$99bo$100boo$106bo$
93bobbo4bobobo$91bobobbobo4bobo$82bo4bo5b4o5boo$43bo27bo6bo4boo7boobb
oo3bo$38bobbo26bo3boobboo7boo4bo6bo$40boobo22boo5b4o5bo4bo$38bo3bo21bo
bo4bobobbobo$34bobobbo24bobobo4bobbo$43bobo4bo12bo$35b3obo28boo$48boo
20bo$34b4obboo4bo20bobbobobbo$33bobbo8bo16bo3bobo5bo$45b3o15bob3o4b3o$
36bo4bo6bo14boboo36bo6bo$33b4o35boo19bo6bo3boobboo3bo$32b3oboobboo4bob
o16boo4bo18bo3boobboo5b4o5boo$32bo3boobbo3bo22bo20boo5b4o4bobobbobo4bo
bo$32bo3boobbo3bo41bobo4bobobbobo4bobbo4bobobo$32b3oboobboo4bobo37bobo
bo4bobbo19bo$33b4o48bo26boo$36bo4bo6bo41boo19bo$45b3o44bo13bobbobobbo$
33bobbo8bo43bobbobobbo9bo5bobo3bo$34b4obboo4bo37bo3bobo5bo10b3o4b3obo$
48boo35bob3o4b3o18boobo$35b3obo45boboo19boo$43bobo4bo43boo14bo4boo$34b
obobbo47boo4bo20bo$38bo3bo46bo$40boobo$38bobbo$43bo!

[velcrorex]

This p3 works in B2 through B25678

Code: Select all

x = 91, y = 97, rule = /25678
30$28bo$30bo4bo$24bo5bobbo$33bo5bo$25boo$27bo9boo$20bo6bo8bo$22bo6bo6b
o6bo$22bo11bo6bo$23boo16bo$18bo20boo$25bo19bo$19boo17bo$43boo$$20boo$
26bo17boo$19bo19bo$24boo20bo$23bo16boo$23bo6bo11bo$21bo6bo6bo6bo$28bo
8bo6bo$26boo9bo$38boo$25bo5bo$31bobbo5bo$29bo4bo$36bo!

[Lewis]

I know this is slightly off-topic, but does the 12-cell phoenix in Life ever occur naturally? I've never checked all the way through Andrzej Okrasinski's census, but I expect that it shouldn't be too rare due to it's size and symmetry.

[Paul Tooke]

Lewis wrote:

I know this is slightly off-topic, but does the 12-cell phoenix in Life ever occur naturally?

FWIW, I've been testing modifications to my random soup/agar program and it has turned up twice in test runs in Life. It may have shown up in others but I wasn't specifically looking for it. Furthermore, my progam only outputs one copy of anything that it sees, so I have no idea how frequently this pattern appears. Whether the patterns appearance in this context qualifies as 'occuring naturally' depends on your definition of 'natural'. Soups in symmetrical bounded universes yeild symmetrical patterns that may have little chance of appearing in an asymmetrical soup in an unbounded universe. The latter being closer to my interpretation of 'natural'.

[checkman]

Just to keep things simple, let's say a "phoenix" is an oscillator in which no cells ever survive.
In which rules can they exist, and which periods can they have?

Are there any Life-like CAs where you'd like this question answered in particular, for period 2?

Then there's the rule B345/S. There are many oscillators in this rule, including periods 18, 22, 24, 26, 30, 32, 34, 38, 40, 42, 62, 66, 72,

And the simplest phoenix with period 2 in B345/S appears to be one of width 3:

Code: Select all

x = 3, y = 6, rule = B345/S
3o2$bo$bo2$3o!

[checkman]

B34/S03456 has no phoenixes with period = 3.

See viewtopic.php?f=11&t=1323#p11194

[Hunting]

The LifeWiki article claims a proof has been found that no phoenix in Conway's Game of Life (so in B3/S) can have period 3.

Sorry for extremenecroposting, but it is wrong. Removing S23 can't affect a phoneix, but it can affect those not-a-phoneix oscillators, and even unstable objects.

[testitemqlstudop]

If a p3 phoenix exists in B3/S, then duh it exists in B3/S23!

[Hunting]

If a p3 phoenix exists in B3/S, then duh it exists in B3/S23!

No, that's wrong. The p3 phoenix can has some cells which has 2 or 3 neighbours, when run in B3/S23 they will fail.

[testitemqlstudop]

Whoops. I mean to say its converse:

If a p3 phoenix does NOT exist in B3/S, then a p3 phoenix cannot exist in B3/S23.

[Hunting]

Whoops. I mean to say its converse:

If a p3 phoenix does NOT exist in B3/S, then a p3 phoenix cannot exist in B3/S23.

That's right, but it does not contradicts my sentence.

[PkmnQ]

Whoops. I mean to say its converse:

If a p3 phoenix does NOT exist in B3/S, then a p3 phoenix cannot exist in B3/S23.

That's right, but it does not contradicts my sentence.

In fact, using first-order logic, I can prove that one leads to the other.
A implies B, original statement.
Goal: Not B implies Not A.

A implies B.
I will assume Not B.
I will assume A.
I will deduce B from A implies B and A, using Modus Ponens (given).
I will deduce Not A from B and Not B, using Reductio Ad Absurdum (which i can break down, but it would be too long here)
I will deduce Not B implies Not A, using Deduction Theorem (given).

And here is Reductio Ad Absurdum seperately:

Assuming A, B and not B.
I will assume not A.
I will deduce B from B and not B, using Conjuntion Elimination.
I will deduce Not A or B from B, using Disjunction Construction.
I will deduce not B from B and not B, using Conjunction Elimination.
I will deduce Not A from Not A or B and not B, using Disjunction Elimination.
I will deduce A or not A, using Law of excluded Middle.
I will deduce Not A, assuming A or Not A, using Case Analysis.
I will deduce A or Not A implies Not A, using Deduction Theorem.
I will deduce Not A from A or Not A implies Not A and A or Not A.

[testitemqlstudop]

In fact, that's called a contrapositive, and it will always imply the original statement.

[Hunting]

逆否命题

[kaarel]

If there was a p3 phoenix in B3/S, it would also work in B3/S/G3 because the cells would be dead for 2 generations in a row. This also applies to all 'super-phoenices' (every cell alive only once per period) in B3/S/G(period). There would also be P3 phoenices that would work in the generations rule but not B3/S because of birth into the wrong spot.

No P3 or higher finite super-phoenices can exist in the rules B3/S/Gxxx.

Proof:

Suppose there was one above this gray line here. Consider the blue cell here (blue because it is alive at time t):

Code: Select all

x = 17, y = 2, rule = LifeHistory
8FB$7.10F!

This would require 3 green cells (alive at time t-1) in one of these 4 configurations:

Code: Select all

x = 105, y = 13, rule = LifeHistory
8.2A24.A.A27.2A29.3A$8FBA17.8FBA20.8FBA21.8FB$7.10F17.10F20.10F21.10F
5$94.C.C$7.2C26.3C25.3C28.C.C$8.C28.C27.C28.3C$8.C26.3C25.3C30.C$8.C
26.C29.C30.C$7.3C25.3C25.3C!

Configuration 1 can be ruled out because the lower right green cell doesn't have 3 empty spaces for red (t-2) cells around it for it to be born.

Configuration 2 has enough:

Code: Select all

x = 17, y = 4, rule = LifeHistory
7.ADAD$8FBAD$7.10F!

It doesn't work because the lower right red cell cant be born for the same reason.

Configuration 3 has the same problem:

Code: Select all

x = 17, y = 3, rule = LifeHistory
7.2A2D$8FBAD$7.10F!

Now we are left with config. 4:

Code: Select all

x = 17, y = 3, rule = LifeHistory
7.3A$8FB$7.10F!

Considering the left green cell, it has 4 more configurations for red cells to give birth to it:

Code: Select all

x = 96, y = 14, rule = LifeHistory
6.2D26.D.D24.2D22.3D$6.D3A24.D3A22.D3A22.3A$8FB19.8FB17.8FB16.8FB$7.
10F18.10F16.10F15.10F6$6.2C26.3C23.3C25.C.C$7.C28.C25.C25.C.C$7.C26.
3C23.3C25.3C$7.C26.C27.C27.C$6.3C25.3C23.3C27.C!

The first 3 can be ruled out the same way as the first 3 configurations for the green cells. Now let's consider the middle green cell. It has 2 ways for it to be born:

Code: Select all

x = 53, y = 11, rule = LifeHistory
6.3D33.4D$7.3A33.3A$8FBD26.8FB$7.10F26.10F3$6.2C34.3C$7.C36.C$7.C34.
3C$7.C34.C$6.3C33.3C!

Let's consider configuration 2.
Case 1: the period is greater than 3:
This would make the lower right red cell be unable to exist, due to a lack of neighbors
Case 2: p=3
The cell would imply this configuration of blue cells (t-3n):

Code: Select all

x = 17, y = 4, rule = LifeHistory
6.3D$7.3AB$8FBDB$7.10F!

The lower right blue cell would require this:

Code: Select all

x = 18, y = 4, rule = LifeHistory
7.3D$8.3ABA$.8FBDBA$8.10F!

Where the lower right green cell would be unable to be born.

So, this configuration would give birth to the middle green cell:

Code: Select all

x = 17, y = 4, rule = LifeHistory
6.4D$7.3A$8FB$7.10F!

The middle-right red cell would now require 3 (t-3) cells above it to be born:

Code: Select all

x = 17, y = 5, rule = LifeHistory
7.3E$6.4D$7.3A$8FB$7.10F!

The middle yellow cell would require the same thing with (t-4), which would require more (t-5), and so on indefinetely. Thus, no finite P3+ super-phoenix can exist in B3/S/Gxfxttxfcscd.

QED.

Also, here is a collection of B3/S phoenices found with JLS (including some p4 ones):

Code: Select all

x = 242, y = 127, rule = B3/S
190bo$188bobo$191bo$186b2o$190b2o$40bo146bo$38bobo148bo$41bo146bo$36b
2o150bo$40b2o148bo$36bo44bo16bo87b2o$26bo10bobo39bobo14bobo91b2o$24bob
o10bo44bo16bo86bo$27bo49b2o15b2o91bobo$22b2o23bobo31b2o15b2o87bo$26b2o
17b2o2bo28bo16bo$23bo26bo29bo16bo$25bobo17bo32bo16bo$25bo20bo2b2o30b2o
15b2o$46bobo28b2o15b2o$82bo15bo$79bobo12bobo$81bo14bo$3bo9bo148bo$3bob
o7bobo11bo108bo12bo10bobo24bo$bo9bo14b2obo92bo11bobo10bobo13bo21bobo$
4b2o8b2o8bobo83bo9bobo14bo12bo7b2o28bo$2o8b2o16b2o78bobo12bo8b2o11b2o
15b2o19b2o$bo2bo6bo3bo9bo85bo6b2o16b2o11b2o8bo27b2o$4b2o8b2o9bo3bo24bo
14bo36b2o14b2o9bo12bo14bo22bo$2o8b2o13bo2bo25bobo12bobo38b2o6bo16bobo
10bo11bo25bo$5bo9bo10bobo23bo14bo13bobo22bo12bobobo10bobo10bo14bo22bo$
2bobo7bobo40b2o13b2o9bo2b2o21bobobo6bo2bo15bo11bo8b2o25bo$4bo9bo36b2o
13b2o12bo25bo2bo12b2o8b2o11b2o15b2o23bo$52bo3bo10bo2bo13bo25b2o6b2o16b
2o11b2o7bo24b2o18bo$55b2o13b2o8b2o2b2o20b2o14bo10bo12bo12bobo25b2o13b
2obo$51b2o13b2o13bo29bo6bobo14bobo10bobo8bo24bo15bobo$55bo14bo14bo22bo
bo9bo14bo12bo37bobo15b2o$51bobo12bobo11b2o2bo25bo75bo14bo$53bo14bo13bo
bo116bo3bo$201bo$201b4o$204bo$200bo3bo$160bo12bo30bo$110bo47bobo10bobo
26b2o$108bobo21bo28bo12bo28bobo$111bo18bobo23b2o11b2o29bob2o$106b2o25b
o26b2o11b2o27bo$110b2o16b2o26bo3bo8bo3bo$107bo24b2o22b2o11b2o$109bobo
17bo30b2o11b2o$108bobo20bobo22bo3bo8bo3bo$111bo19bo24b2o11b2o$106b2o
21bo2b2o26b2o11b2o$110b2o45bo11bo$106bo21b2o3bo25bobo8bobo$58b2obobo
43bobo22b2o25bo10bo$56bo4bo2b2o41bo20bobo$56bo2bo70bobo$56bo2bobo3bo$
57b2o3bobo$55bo2bo2bo2bo$55bo4bo$53bobo5bo$55bo4bo128bobo$55bo2bo2bo2b
o122b2o2bo$57b2o3bobo127bo$56bo2bobo3bo122bo$56bo2bo131b2o$56bo4bo2b2o
122bobo$58b2obobo124bo$187bobo$190bo$173bo14bo2b2o$171bobo14bo$174bo
17bo$130bo38b2o16b2o2bo$128b2o6bo36b2o14bobo$126bobo2b2obo2b2o30bo$29b
2obobo91bo5bo3bo33bobo$27bo4bo2b2o87bo9bo3bobo27bo3bo$27bo2bo93bo7bo7b
o29bobo$27bo2bobo3bo88bo6bobobo3bo28bo$28b2o3bobo88bo5bo4bo5bo31b2o$
26bo2bo2bo2bo90bo4bo5bo3bo27b2o$26bo4bo94bo4b2o4bo2bo33bo$24bobo5bo95b
o3bobo4b2o30bobo$24bobo5bo95bo5bo2b2o34bo43bobo16bo2bo$26bo4bo98bo4bo
79b2o2bo2bo11bob2obo$26bo2bo2bo2bo94b2ob2obo83bobo9bo7bo$28b2o3bobo96b
o82bo8bo5bo2bo$27bo2bobo3bo181b4o10bo2bo3b2o$27bo2bo184b3o4b2obob2ob2o
3bobo2bo$27bo4bo2b2o176bobo11bo3bo3bo2bobo$29b2obobo183bobo9bo2bobobo
3bo$215bo3bo2bo5bob2o$215bo3bo2bobobo9bo3b2o$215bo11bo4b4o$217bobo6bo
10bobo$218bobo6bo2bo3bob2o$221bob2o3bobo2bo2bo$164bo23b2obobo26bobo2b
2o4bobo$186b2o3bo2b2o$163b2obo17bobo3bo$163b2obobo13bobo9bo$162bo7bo
11bo6b2o3b2o$113bobo49bo4bobo7bo$115bobo5bobo37b2o9bo5bo8bobo3bo$103bo
2bo2bo2b2o5bobobo2b2o46bobobobo8bo2bobo$37bo11bobobo49bob2ob2obo6bo2bo
43bobo24bo2bo$33b3o2b2o5bo5bo2bo42bobo2bo14bo4bo3bo40bo10bo4b2o4bobo2b
o$31b2o4bo5bo2b2ob2o3bo4b3o37bobo13bo2bobo5b2o41bobo7b3o4bo8bo$29b2o4b
obobobo3bo7bo2b2o38b2o5b12o2bo3bobo47bo9bo2bo2b3o4bo$29bo2bo6bobo2bobo
3b2o7bobobo33b18o8bo2bo46bobo2bo2bo5bo4bobo$28bo3bo3bo5bo3bo3bo2b3obo
5bo51b3o3bobo2b2o47bo2bo2bobo3b2o2b2o$28bo2bo4bobobobo4bobo2bobo8bo32b
o23bo56bobo3bobo3b2o$26bobo4bo2bo3bobo2b2o4bo5bo4bo38bob2ob2ob2ob2obo
2b2obo5bo$26bobo3bo3bo3bo3b2o2bo5bobo7bo31b2o2b2o2bo2bo2bo2bo12b2o$26b
obo4bo2bobo13b2o3bobobo2bo33bobo13bo7bobo$28bo2bo2bo2bo2bo5bobo5b2o5b
2o39bo2bo2bo2bo2bo5b2o2bobo$28bo5b2o2bobobo5bo14bo38bob2ob2ob2obo2bobo
$30b3o5bo9bo6bo7bo54bobo$29bo5b2o2bo2bo7bobobobo6bo$29bo2bo6bob2obo5bo
2bo6b2o$29bo4bobo7b2obob2o2b2o2b2o$31b2obo11bobo2bo3b2o3bo!

[breaker's glider gun]

I'd actually say that a pheonix is something where it starts as an explosion, turns into still lives/oscillators, and a spaceship. The spaceship then slams into one of the still lives/oscillators, which reignites it. "Revives it from the ashes."

EDIT:
Like this:

Code: Select all

x = 17, y = 2, rule = Life
5A3$2.5A!

EDIT: Sorry, only works on a sphere.

[GUYTU6J]

I'd actually say that a pheonix is something where it starts as an explosion, turns into still lives/oscillators, and a spaceship. The spaceship then slams into one of the still lives/oscillators, which reignites it. "Revives it from the ashes."

EDIT:
Like this:

Code: Select all

x = 17, y = 2, rule = Life
5A3$2.5A!

EDIT: Sorry, only works on a sphere.

However, that is not the commonly accepted meaning of the term phoenix. Also by sphere we mean a certain geometry of the finite universe, which does not seem like to be the case with your pattern.

The type of pattern you are describing is more interesting if it works on an infinite plane and shows periodicity.

[breaker's glider gun]

Whoops, yeah I didn't mean sphere. I meant torus.

EDIT:(got the answer for how to do it in the thread for basic questions.)

Code: Select all

x = 7, y = 4, rule = B3/S23:T150,150
5o3$2b5o!