Rare emergence in the Ice Nine rule: a 24 hour experiment
George Maydwell
The Ice Nine rule is a rule based upon the four state Star Wars rule
(2/345). When viewed for short or medium periods of time it appears
similar to Star Wars, but without the long term stable stationary
structures which form in Star Wars.
What's unique about the Ice Nine rule is the amount of time it takes a
particular emergent behavior to manifest when starting from a random
configuration. Given dozens of hours of computer simulation time on a
fast computer large roundish stationary structures ("iceballs")
sometimes emerge. The iceballs seem to grow very slowly, at a rate of
less than 10 cells/hour in each direction.
Apparently the iceballs need to achieve a critical mass of sorts
before their continued existence becomes likely. Achieving this
critical mass must be a rare event, as evidenced by the huge amounts
of simulation time generally required before even a single iceball
emerges.
The rule uses Star Wars' two refractory states. It adds a second cycle
of three refractory states which contribute to neighbor counting. Here
are the transitions which differ from Star Wars, all transitions from
a live cell:
3 adjacent, 0 diagonal --> 3 counting "refractory" states --> dead
3 adjacent, 4 diagonal --> 3 counting "refractory" states --> dead
4 adjacent, 3 diagonal --> 3 counting "refractory" states --> dead
4 adjacent, 4 diagonal --> 3 counting "refractory" states --> dead
I've set up a page with a stand alone Ice Nine experiment which may be
run, should you have a fast computer and 24 to 48 hours of
uninterrupted computer time available:
www.collidoscope.com/modernca/icenine.html
I haven't run the experiment a great number of times, but I have seen
it succeed in forming iceballs twice while failing to form iceballs
once.
George Maydwell
--
Modern Cellular Automata: www.collidoscope.com/modernca
Collidoscope Hexagonal Screensaver: www.collidoscope.com
Dean Hickerson
> The Ice Nine rule is a rule based upon the four state Star Wars rule
> (2/345). When viewed for short or medium periods of time it appears
> similar to Star Wars, but without the long term stable stationary
> structures which form in Star Wars.
>
> What's unique about the Ice Nine rule is the amount of time it takes a
> particular emergent behavior to manifest when starting from a random
> configuration. Given dozens of hours of computer simulation time on a
> fast computer large roundish stationary structures ("iceballs")
> sometimes emerge. The iceballs seem to grow very slowly, at a rate of
> less than 10 cells/hour in each direction.
>
> Apparently the iceballs need to achieve a critical mass of sorts
> before their continued existence becomes likely. Achieving this
> critical mass must be a rare event, as evidenced by the huge amounts
> of simulation time generally required before even a single iceball
> emerges.
There's a totalistic rule on the Moore neighborhood, B25678/S5678, that
has similar behaviour, although the iceballs form and grow faster. (I
assume so anyway; how many generations are in an hour in your Ice Nine
simulation?)
Try this 3-cell starting pattern, for example:
x = 3, y = 2, rule = B25678/S5678
2o$2bo!
This begins growing at the speed of light, chaotically filling a
diamond-shaped region with density about 25%. The first iceball forms
around generation 600, the second around gen 780. By gen 2000 there are
about a dozen of them. Iceballs grow at about c/9, remaining roughly
diamond-shaped, and eventually merge with each other to fill space with
density very close to 1. (I doubt that the limiting density is exactly 1,
since there are small stable holes that could form when two iceballs meet.
But I haven't actually seen such a hole form.)
It's hard to run patterns in the plane for very long, since their populations
grow rapidly. It's easier to run them on a small torus, in which case the
first iceball usually grows to fill the whole space. For example, the
pattern above, run on a 100x100 torus, forms an iceball about gen 31300,
which fills the torus at gen 31951.
Dean Hickerson
de...@math.ucdavis.edu
George Maydwell
<de...@math.ucdavis.edu> wrote:
>George Maydwell <geo...@collidoscope.com> wrote:
>
>> The Ice Nine rule is a rule based upon the four state Star Wars rule
>> (2/345). When viewed for short or medium periods of time it appears
>> similar to Star Wars, but without the long term stable stationary
>> structures which form in Star Wars.
>>
>> What's unique about the Ice Nine rule is the amount of time it takes a
>> particular emergent behavior to manifest when starting from a random
>> configuration. Given dozens of hours of computer simulation time on a
>> fast computer large roundish stationary structures ("iceballs")
>> sometimes emerge. The iceballs seem to grow very slowly, at a rate of
>> less than 10 cells/hour in each direction.
>>
>> Apparently the iceballs need to achieve a critical mass of sorts
>> before their continued existence becomes likely. Achieving this
>> critical mass must be a rare event, as evidenced by the huge amounts
>> of simulation time generally required before even a single iceball
>> emerges.
>
>There's a totalistic rule on the Moore neighborhood, B25678/S5678, that
>has similar behaviour, although the iceballs form and grow faster. (I
>assume so anyway; how many generations are in an hour in your Ice Nine
>simulation?)
It takes about 25 seconds to run 1000 generations on the computer on
which I performed the tests, which works out to about 150,000
generations per hour. The iceballs in B25678/S5678 form more often and
grow faster.
>
>Try this 3-cell starting pattern, for example:
>
>x = 3, y = 2, rule = B25678/S5678
>2o$2bo!
>
>This begins growing at the speed of light, chaotically filling a
>diamond-shaped region with density about 25%. The first iceball forms
>around generation 600, the second around gen 780. By gen 2000 there are
>about a dozen of them. Iceballs grow at about c/9, remaining roughly
>diamond-shaped, and eventually merge with each other to fill space with
>density very close to 1. (I doubt that the limiting density is exactly 1,
>since there are small stable holes that could form when two iceballs meet.
>But I haven't actually seen such a hole form.)
Thanks. Here's an iceball formation experiment for B25678/S5678:
www.collidoscope.com/modernca/iceballs.html