New neighborhood: PentaWorld

[p46beth]

I recently thought of a new neighborhood for cellular automata in which each cell has five neighbors. For an idea of what it looks like, see:

http://i808.photobucket.com/albums/zz1/ ... orld-1.png

Imagine the entire plane is tessellated with that image: each 3 by 3 square is divided into four 2 by 1 white rectangles and 1 black square.
Now, each white rectangle is considered ONE cell. The black squares are considered void, they are not part of the PentaWorld universe.
Every cell has five neighbors orthogonally, as shown in the image (note that in addition, each cell also has 1 neighbor diagonally, this is not counted for birth/survival purposes). A k-state rule using pentaworld can easily be emulated by a k^4 state rule in a Von Neumann neighborhood.

The next step is to look at the simplest rules that pentaworld allows for: PentaLife-like rules. The first question is to look for spaceships. In order for a (penta)life-like rule to allow for spaceships, it must contain either B0 or B2. It also cannot contain B1 without B0, otherwise all patterns would expand in all directions (note that it IS possible for a rule that has B0 AND B1 to contain spaceships, see B017/S1). This reduces the search down to 6*(2^9)=3072 rules that COULD contain spaceships. So far, I have not found any,though I have not done extensive searches.

[Axaj]

This looks to be pretty neat. You said you have been searching for spaceship containing rules; can you post the rule table/tree(s) you are using?

[apg]

This reduces the search down to 6*(2^7)=768 rules that COULD contain spaceships.

I think you mean 6*(2^9) = 3072 rules. Don't forget that there are 2^12 semi-totalistic Pentaworld rules.


B0-not-S8 (or B0-not-S5 in your neighbourhood) rules are very unnatural, since they have no ground state. B0+S8 (resp. B0+S5) rules can be inverted to remove the B0.

So, when you eliminate B0 and B1, and enforce B2, there are 2^9 = 512 rules to traverse.

You also cannot have S0123, as the trailing edge cannot die, reducing the number of non-pathological possible spaceship-supporting rules to 480.


By the way, as well as a von Neumann rule with k^4 states, it can be emulated somewhat more economically in a Moore rule with 4k states.

[p46beth]

I think you mean 6*(2^9) = 3072 rules. Don't forget that there are 2^12 semi-totalistic Pentaworld rules.

Yes, that is what I meant;)

[ebcube]

You can probably simulate this with TiledCA. As soon as I get Windows working, I'll try.

[p46beth]

By the way, as well as a von Neumann rule with k^4 states, it can be emulated somewhat more economically in a Moore rule with 4k states.

I have not been able to figure out how you did this. Could you give a sample rule table for something like B2/S2?

[apg]

I have not been able to figure out how you did this.

You can deform your diagram until the cells are all squares, where each one is connected to a subset of 5 of the 8 neighbours. Hence, there are 4k states -- 4 for the position multiplied by k for the emulated state.

[p46beth]

Following your advice, I made an 8-state table called pentalife8, which models the rule pentalife (B2/S2).
Actually, the neighborhood modeled is a mirror image of the picture I linked to in the first post of this thread, sorry for the inconsistency.

Code: Select all

#Pentalife
#Format: C,N,NE,E,SE,S,SW,W,NW,C'
#0,1,2, and 3 are dead
#4,5,6, and 7 are alive
#All cells are connected orthogonally
#0 and 4 connect to the southwest
#1 and 5 connect to the southeast
#2 and 6 connect to the northeast
#3 and 7 connect to the northwest
#
n_states:8
neighborhood:Moore
symmetries:none
var a={0,1,2,3,4,5,6,7}
var b={0,1,2,3,4,5,6,7}
var c={0,1,2,3,4,5,6,7}
var d={0,1,2,3,4,5,6,7}
var e={0,1,2,3,4,5,6,7}
#
var dead={0,1,2,3}
var debd={0,1,2,3}
var decd={0,1,2,3}
var dedd={0,1,2,3}
#
var alive={4,5,6,7}
var blive={4,5,6,7}
var clive={4,5,6,7}
var dlive={4,5,6,7}
#
var notfour={0,1,2,3,5,6,7}
var notfive={0,1,2,3,4,6,7}
var notsix={0,1,2,3,4,5,7}
var notseven={0,1,2,3,4,5,6}
#
#0 in the center
0,alive,a,blive,b,dead,notsix,debd,c,4
0,alive,a,dead,b,blive,notsix,debd,c,4
0,alive,a,dead,b,debd,6,decd,c,4
0,alive,a,dead,b,debd,notsix,blive,c,4
#
0,dead,a,alive,b,blive,notsix,debd,c,4
0,dead,a,alive,b,debd,6,decd,c,4
0,dead,a,alive,b,debd,notsix,blive,c,4
#
0,dead,a,debd,b,alive,6,decd,c,4
0,dead,a,debd,b,alive,notsix,blive,c,4
0,dead,a,debd,b,decd,6,alive,c,4
#
#1 in the center
1,alive,a,blive,notseven,dead,b,debd,c,5
1,alive,a,dead,7,debd,b,decd,c,5
1,alive,a,dead,notseven,blive,b,debd,c,5
1,alive,a,dead,notseven,debd,b,blive,c,5
#
1,dead,a,alive,7,debd,b,decd,c,5
1,dead,a,alive,notseven,blive,b,debd,c,5
1,dead,a,alive,notseven,debd,b,blive,c,5
#
1,dead,a,debd,7,alive,b,decd,c,5
1,dead,a,debd,7,decd,b,alive,c,5
1,dead,a,debd,notseven,alive,b,blive,c,5
#
#2 in the center
2,alive,4,dead,a,debd,b,decd,c,6
2,alive,notfour,blive,a,dead,b,debd,c,6
2,alive,notfour,dead,a,blive,b,debd,c,6
2,alive,notfour,dead,a,debd,b,blive,c,6
#
2,dead,4,alive,a,debd,b,decd,c,6
2,dead,4,debd,a,alive,b,decd,c,6
2,dead,4,debd,a,decd,b,alive,c,6
#
2,dead,notfour,alive,a,blive,b,debd,c,6
2,dead,notfour,alive,a,debd,b,blive,c,6
2,dead,notfour,debd,a,alive,b,blive,c,6

#3 in the center
#
3,alive,a,blive,b,dead,c,debd,notfive,7
3,alive,a,dead,b,blive,c,debd,notfive,7
3,alive,a,dead,b,debd,c,blive,notfive,7
3,alive,a,dead,b,debd,c,decd,5,7
#
3,dead,a,alive,b,blive,c,debd,notfive,7
3,dead,a,alive,b,debd,c,blive,notfive,7
3,dead,a,alive,b,debd,c,decd,5,7
#
3,dead,a,debd,b,alive,c,blive,notfive,7
3,dead,a,debd,b,alive,c,decd,5,7
3,dead,a,debd,b,decd,c,alive,5,7

#State-specific deaths
#
#
#4 in the center
#isolation
4,alive,a,dead,b,debd,notsix,decd,c,0
4,dead,a,alive,b,debd,notsix,decd,c,0
4,dead,a,debd,b,alive,notsix,decd,c,0
4,dead,a,debd,b,decd,notsix,alive,c,0
#overcrowding
4,alive,a,blive,b,clive,c,d,e,0
4,alive,a,blive,b,c,clive,d,e,0
4,alive,a,blive,b,c,d,clive,e,0
4,alive,a,b,c,blive,clive,d,e,0
4,alive,a,b,c,blive,d,clive,e,0
4,alive,a,b,c,d,blive,clive,e,0
#
4,a,b,alive,c,blive,clive,d,e,0
4,a,b,alive,c,blive,d,clive,e,0
4,a,b,alive,c,d,blive,clive,e,0
#
4,a,b,c,d,alive,blive,clive,e,0



#5 in the center
#isolation
5,alive,a,dead,notseven,debd,b,decd,c,1
5,dead,a,alive,notseven,debd,b,decd,c,1
5,dead,a,debd,7,decd,b,dedd,c,1
5,dead,a,debd,notseven,alive,b,decd,c,1
5,dead,a,debd,notseven,decd,b,alive,c,1
#overcrowding
5,alive,a,blive,7,b,c,d,e,1
5,alive,a,blive,b,clive,c,d,e,1
5,alive,a,blive,b,c,d,clive,e,1
5,alive,a,b,7,blive,c,d,e,1
5,alive,a,b,7,c,d,blive,e,1
5,alive,a,b,c,blive,d,clive,e,1
#
5,a,b,alive,7,blive,c,d,e,1
5,a,b,alive,7,c,d,blive,e,1
5,a,b,alive,c,blive,d,clive,e,1
#
5,a,b,c,7,alive,d,blive,e,1


#6 in the center
#isolation
6,alive,notfour,dead,a,debd,b,decd,c,2
6,dead,4,debd,a,decd,b,dedd,c,2
6,dead,notfour,alive,a,debd,b,decd,c,2
6,dead,notfour,debd,a,alive,b,decd,c,2
6,dead,notfour,debd,a,decd,b,alive,c,2
#overcrowding
6,alive,4,blive,a,b,c,d,e,2
6,alive,4,a,b,blive,c,d,e,2
6,alive,4,a,b,c,d,blive,e,2
6,alive,a,blive,b,clive,c,d,e,2
6,alive,a,blive,b,c,d,clive,e,2
6,alive,a,b,c,blive,d,clive,e,2
#
6,a,4,alive,b,blive,c,d,e,2
6,a,4,alive,b,c,d,blive,e,2
6,a,4,b,c,alive,d,blive,e,2
#
6,a,b,alive,c,blive,d,clive,e,2

#7 in the center
#isolation
7,alive,a,dead,b,debd,c,decd,notfive,3
7,dead,a,alive,b,debd,c,decd,notfive,3
7,dead,a,debd,b,alive,c,decd,notfive,3
7,dead,a,debd,b,decd,c,alive,notfive,3
#overcrowding
7,alive,a,blive,b,clive,c,d,e,3
7,alive,a,blive,b,c,d,clive,e,3
7,alive,a,blive,b,c,d,e,5,3
7,alive,a,b,c,blive,d,clive,e,3
7,alive,a,b,c,blive,d,e,5,3
7,alive,a,b,c,d,e,blive,5,3
#
7,a,b,alive,c,blive,d,clive,e,3
7,a,b,alive,c,blive,d,e,clive,3
7,a,b,alive,c,d,e,blive,5,3
#
7,a,b,c,d,alive,e,blive,5,3
#
#universal deaths
4,dead,a,debd,b,decd,c,dedd,d,0
5,dead,a,debd,b,decd,c,dedd,d,1
6,dead,a,debd,b,decd,c,dedd,d,2
7,dead,a,debd,b,decd,c,dedd,d,3

I suggest setting states 0-3 as dark colors, and states 4-7 as light colors.
One draw-back to this approach is that you can't just draw anywhere. If you want to try patterns, first fill a sufficiently large area with the pattern:

Code: Select all

x = 2, y = 2, rule = pentalife8
.C$AB!

This is easy to do with the tile-with-clip script.

Then, adjust some cells to their live forms (i.e. 0 becomes 4, 1 becomes 5, ect.). This means that random fill won't give you a valid simulation of pentalife. Additionally, if a pattern expands to the edge of the dead cell grid, then it will come into contact with contiguous state 0 cells--an invalid formation. This could be improved slightly by adding a ninth state that remains dead (and have that be state 0) while shifting states 0-7 up one value (i.e. state 0 becomes state 1, state 1 becomes state 2, ect.). I saved this as pentalife9. This allows you to accurately model any FINITE pattern, but does not allow for infinite growth. If this is not what you had in mind, I would appreciate if you would specify what you meant. As it is, my idea for a 16-state emulation in a VN neighborhood is the best way to accurately model infinite growth in pentalife, assuming that infinite growth exists at all.

[ebcube]

Well, I've tried the TiledCA approach:

http://cl.ly/00d951a321e00d8f1a82

The only nontrivial pattern I've found in B2/S2 is this p12 oscillator:

http://cl.ly/4d90027cd44de333614b

I'm doing screenshots because TiledCA is confusing.

[p46beth]

I'd expect spaceships, if any exist, to be quite rare. For example, in B2/S2H (the closest thing to pentalife), most patterns explode, but there is a rather unnatural glider:

Code: Select all

x = 5, y = 5, rule = B2/S2H
o2b2o$3b2o$2bo2$4bo!

If pentalife spaceships do exist, then it would be the simplest neighborhood with spaceships (life-like rules in a VN neighborhood either always explode or never escape their bounding box). However, given that a relativity small portion of 2-state hexagonal rules have spaceships, the simplest pentalife spaceship may be quite large.

By the way, I also found the p12 oscillator...but that seems to be the highest natural period.

[velcrorex]

I tried TiledCA as well. The chaos in B2S14 looked promising, but nothing found.

[apg]

then it would be the simplest neighborhood with spaceships

Correction: smallest neighbourhood with spaceships. The hexagonal neighbourhood is simpler, being the Voronoi tesselation of the A2 lattice.

And spaceships exist in 3-state von Neumann neighbourhood rules, such as Serizawa.

[ebcube]

B23S02C3 (02/23/3) has more complex behaviour. I haven't found a spaceship, but this p20 oscillator shows how a pattern can grow and shrink:

http://cl.ly/e156d01807cd880401e5

[p46beth]

Calycman: Thank you for the correction.

B23S02C3

What does the C3 mean? Sorry if its a dumb question, I'm not very familiar with TiledCA.

[ebcube]

C is the number of possible states. It works like Generation rules in Golly.

[p46beth]

I see. Of course, it seems natural that rules with more states would often have more complex behavior. My hope was to find a 2-state pentalife-like CA that contained spaceships.

Actually, does anyone know if it would be possible to write a script which allows easy construction of pentalife rule tables (similar to the tri-life script, which gives B/S rules in a triangular neighborhood). That would probably make it a lot easier to search for rules with more complex behavior.