## Life theorems

For general discussion about Conway's Game of Life.

### Life theorems

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

Tom Mazanec

Posts: 23
Joined: December 18th, 2017, 5:22 pm

### Re: Life theorems

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

Majestas32

Posts: 509
Joined: November 20th, 2017, 12:22 pm
Location: 'Merica

### Re: Life theorems

2. That life can do universal computations

3. That orphans exist

Macbi

Posts: 472
Joined: March 29th, 2009, 4:58 am

### Re: Life theorems

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).

simsim314

Posts: 1539
Joined: February 10th, 2014, 1:27 pm

### Re: Life theorems

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?
x₁=ηx
V ⃰_η=c²√(Λη)
K=(Λu²)/2
Pₐ=1−1/(∫^∞_t₀(p(t)ˡ⁽ᵗ⁾)dt)

$$x_1=\eta x$$
$$V^*_\eta=c^2\sqrt{\Lambda\eta}$$
$$K=\frac{\Lambda u^2}2$$
$$P_a=1-\frac1{\int^\infty_{t_0}p(t)^{l(t)}dt}$$

http://conwaylife.com/wiki/A_for_all

Aidan F. Pierce

A for awesome

Posts: 1620
Joined: September 13th, 2014, 5:36 pm
Location: 0x-1

### Re: Life theorems

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.

Macbi

Posts: 472
Joined: March 29th, 2009, 4:58 am

### Re: Life theorems

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...

simsim314

Posts: 1539
Joined: February 10th, 2014, 1:27 pm

### Re: Life theorems

#4. There are no period-3 phoenixes.
#5. The Block is the only (finite) still life where all cells have three neighbors.
Still drifting.
Bullet51

Posts: 476
Joined: July 21st, 2014, 4:35 am

### Re: Life theorems

Bullet51 wrote:#4. There are no period-3 phoenixes.

I thought it was proven that there were no p3 phoenices.
Things to work on:
- Find a (7,1)c/8 ship in a Non-totalistic rule (someone please search the rules)
- Find a C/10 in JustFriends
- Find a C/10 in Day and Night
AforAmpere

Posts: 707
Joined: July 1st, 2016, 3:58 pm

### Re: Life theorems

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.