## Phoenices

For discussion of other cellular automata.

### Phoenices

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,
Ntsimp

Posts: 46
Joined: June 8th, 2010, 9:11 am

### Re: Phoenices

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.
`x = 142, y = 118, rule = /231\$26bo50bo\$24boo6bo9bo18bo9bo6boo\$22bobo10bo4bo22bo4bo10bobo\$22boboo4boo5bobbo9bobbo9bobbo5boo4boobo\$21bo6bo8bo5bo16bo5bo8bo6bo\$25bobo12boo3bobbo6bobbo3boo12bobo\$26b3ob7obbo3bo7boo7bo3bobb7ob3o\$22bo5bobbo7bo7bo8bo7bo7bobbo5bo\$23bob3o7bo4bo9bobbo9bo4bo7b3obo\$23bobbo4bo3bo15boo15bo3bo4bobbo\$26bo3boobo4bo3bo5bo6bo5bo3bo4boboo3bo\$26booboo7bo5b3o10b3o5bo7booboo\$26bo3boobo4bo3bo5bo6bo5bo3bo4boboo3bo\$23bobbo4bo3bo15boo15bo3bo4bobbo\$23bob3o7bo4bo9bobbo9bo4bo7b3obo\$22bo5bobbo7bo7bo8bo7bo7bobbo5bo\$26b3ob7obbo3bo7boo7bo3bobb7ob3o\$25bobo12boo3bobbo6bobbo3boo12bobo23bo\$21bo6bo8bo5bo16bo5bo8bo6bo15bo4boo\$22boboo4boo5bobbo9bobbo9bobbo5boo4boobo14boo\$22bobo10bo4bo22bo4bo10bobo21boobo\$24boo6bo9bo18bo9bo6boo15b3o4b3obo\$26bo50bo17bo5bobo3bo\$94bobbobobbo\$99bo\$100boo\$106bo\$93bobbo4bobobo\$91bobobbobo4bobo\$82bo4bo5b4o5boo\$43bo27bo6bo4boo7boobboo3bo\$38bobbo26bo3boobboo7boo4bo6bo\$40boobo22boo5b4o5bo4bo\$38bo3bo21bobo4bobobbobo\$34bobobbo24bobobo4bobbo\$43bobo4bo12bo\$35b3obo28boo\$48boo20bo\$34b4obboo4bo20bobbobobbo\$33bobbo8bo16bo3bobo5bo\$45b3o15bob3o4b3o\$36bo4bo6bo14boboo36bo6bo\$33b4o35boo19bo6bo3boobboo3bo\$32b3oboobboo4bobo16boo4bo18bo3boobboo5b4o5boo\$32bo3boobbo3bo22bo20boo5b4o4bobobbobo4bobo\$32bo3boobbo3bo41bobo4bobobbobo4bobbo4bobobo\$32b3oboobboo4bobo37bobobo4bobbo19bo\$33b4o48bo26boo\$36bo4bo6bo41boo19bo\$45b3o44bo13bobbobobbo\$33bobbo8bo43bobbobobbo9bo5bobo3bo\$34b4obboo4bo37bo3bobo5bo10b3o4b3obo\$48boo35bob3o4b3o18boobo\$35b3obo45boboo19boo\$43bobo4bo43boo14bo4boo\$34bobobbo47boo4bo20bo\$38bo3bo46bo\$40boobo\$38bobbo\$43bo!`
-Josh Ball.

velcrorex

Posts: 339
Joined: November 1st, 2009, 1:33 pm

### Re: Phoenices

This p3 works in B2 through B25678

`x = 91, y = 97, rule = /2567830\$28bo\$30bo4bo\$24bo5bobbo\$33bo5bo\$25boo\$27bo9boo\$20bo6bo8bo\$22bo6bo6bo6bo\$22bo11bo6bo\$23boo16bo\$18bo20boo\$25bo19bo\$19boo17bo\$43boo\$\$20boo\$26bo17boo\$19bo19bo\$24boo20bo\$23bo16boo\$23bo6bo11bo\$21bo6bo6bo6bo\$28bo8bo6bo\$26boo9bo\$38boo\$25bo5bo\$31bobbo5bo\$29bo4bo\$36bo!`
-Josh Ball.

velcrorex

Posts: 339
Joined: November 1st, 2009, 1:33 pm

### Re: Phoenices

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.

Lewis

Posts: 320
Joined: March 17th, 2009, 5:26 pm
Location: UK

### Re: Phoenices

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'.
Paul Tooke

Posts: 111
Joined: May 19th, 2010, 7:35 am
Location: Cambridge, UK

### Re: Phoenices

Ntsimp wrote: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:
`x = 3, y = 6, rule = B345/S3o2\$bo\$bo2\$3o!`
checkman

Posts: 12
Joined: February 25th, 2014, 5:33 pm
Location: Tempe, AZ (Not Phoenix, Bob Schieffer!)

### Re: Phoenices

B34/S03456 has no phoenixes with period = 3.

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

Posts: 12
Joined: February 25th, 2014, 5:33 pm
Location: Tempe, AZ (Not Phoenix, Bob Schieffer!)

### Re: Phoenices

Ntsimp wrote: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.
This post was brought to you by the Element of Magic.

Plz correct my grammar mistakes. I'm still studying English.

Working on:

Nothing.

Favorite gun ever:
`#C Favorite Gun. Found by me.x = 4, y = 6, rule = B2e3i4at/S1c23cijn4ao2bo\$4o3\$4o\$o2bo!`
Hunting

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### Re: Phoenices

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

testitemqlstudop

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### Re: Phoenices

testitemqlstudop wrote: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.
This post was brought to you by the Element of Magic.

Plz correct my grammar mistakes. I'm still studying English.

Working on:

Nothing.

Favorite gun ever:
`#C Favorite Gun. Found by me.x = 4, y = 6, rule = B2e3i4at/S1c23cijn4ao2bo\$4o3\$4o\$o2bo!`
Hunting

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Joined: September 11th, 2017, 2:54 am
Location: Ponyville, Equestria

### Re: Phoenices

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.

testitemqlstudop

Posts: 769
Joined: July 21st, 2016, 11:45 am
Location: very very very very boats

### Re: Phoenices

testitemqlstudop wrote: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.
This post was brought to you by the Element of Magic.

Plz correct my grammar mistakes. I'm still studying English.

Working on:

Nothing.

Favorite gun ever:
`#C Favorite Gun. Found by me.x = 4, y = 6, rule = B2e3i4at/S1c23cijn4ao2bo\$4o3\$4o\$o2bo!`
Hunting

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Joined: September 11th, 2017, 2:54 am
Location: Ponyville, Equestria

### Re: Phoenices

Hunting wrote:
testitemqlstudop wrote: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.
Last edited by PkmnQ on April 15th, 2019, 12:54 pm, edited 1 time in total.
`x = 3, y = 3, rule = B26/S2|B358/S32o\$2bo\$obo!`
PkmnQ

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### Re: Phoenices

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

testitemqlstudop

Posts: 769
Joined: July 21st, 2016, 11:45 am
Location: very very very very boats

### Re: Phoenices

This post was brought to you by the Element of Magic.

Plz correct my grammar mistakes. I'm still studying English.

Working on:

Nothing.

Favorite gun ever:
`#C Favorite Gun. Found by me.x = 4, y = 6, rule = B2e3i4at/S1c23cijn4ao2bo\$4o3\$4o\$o2bo!`
Hunting

Posts: 1058
Joined: September 11th, 2017, 2:54 am
Location: Ponyville, Equestria