Life theorems

For general discussion about Conway's Game of Life.
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Tom Mazanec
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Life theorems

Post by Tom Mazanec » December 18th, 2017, 5:29 pm

What would you say are the top ten theorems of Life?

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Majestas32
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Re: Life theorems

Post by Majestas32 » December 18th, 2017, 6:02 pm

Probably #1 is that no spaceship can go above speed (M,N)/2(M+N) for M and N pos. integers.
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Macbi
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Re: Life theorems

Post by Macbi » December 18th, 2017, 7:00 pm

2. That life can do universal computations

3. That orphans exist

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simsim314
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Re: Life theorems

Post by simsim314 » December 20th, 2017, 12:32 pm

I would go with: there exist a spaceship of any valid speed and direction i.e. any speed slower than (M,N)/2(M+N).

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praosylen
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Re: Life theorems

Post by praosylen » December 20th, 2017, 12:39 pm

simsim314 wrote:I would go with: there exist a spaceship of any valid speed and direction i.e. any speed slower than (M,N)/2(M+N).
Has that actually been proven? Plus, aren't speeds equal to (M,N)/2(M+N) an open question in any case?
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Macbi
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Re: Life theorems

Post by Macbi » December 20th, 2017, 12:40 pm

simsim314 wrote:I would go with: there exist a spaceship of any valid speed and direction i.e. any speed slower than (M,N)/2(M+N).
I think your wording is ambiguous there. Certainly there exists a ship for any speed strictly slower than (M,N)/2(M+N) (and that is an amazing theorem!), but apart from c/2 and (1,1)c/4 we don't know whether any ships with exactly that speed exist.

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simsim314
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Re: Life theorems

Post by simsim314 » December 24th, 2017, 4:36 pm

Macbi wrote:but apart from c/2 and (1,1)c/4 we don't know whether any ships with exactly that speed exist.
Good point. Interesting enough - no natural nor constructable ship with this property exists, or has been proven to exist.
A for awesome wrote:Has that actually been proven?
Proven yes, by a theoretical construction (actually I've found several ways to construct a universal ship). But no actual construction was made yet...

Bullet51
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Re: Life theorems

Post by Bullet51 » December 25th, 2017, 6:35 am

#4. There are no period-3 phoenixes.
#5. The Block is the only (finite) still life where all cells have three neighbors.
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AforAmpere
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Re: Life theorems

Post by AforAmpere » December 25th, 2017, 1:23 pm

Bullet51 wrote:#4. There are no period-3 phoenixes.
I thought it was proven that there were no p3 phoenices.
I manage the 5S project, which collects all known spaceship speeds in Isotropic Non-totalistic rules. I also wrote EPE, a tool for searching in the INT rulespace.

Things to work on:
- Find (7,1)c/8 and 9c/10 ships in non-B0 INT.
- EPE improvements.

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Majestas32
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Re: Life theorems

Post by Majestas32 » December 25th, 2017, 10:29 pm

AforAmpere wrote:
Bullet51 wrote:#4. There are no period-3 phoenixes.
I thought it was proven that there were no p3 phoenices.
Exactly, this is why this is in the Theorems thread and not Unsolved Conjectures.
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