Difference between revisions of "OCA:Gems"

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==Patterns==
==Patterns==
No still lifes exist in Gems, and no odd period oscillators have been found in Gems.
No odd period [[oscillator]]s have been found in Gems.


==Spaceships==
===Still lives===
Gems has a single known [[spaceship]] moving orthogonally at a speed of [[c]]/5648, [[David Eppstein]]'s glider 13047. This is the slowest known elementary spaceship in any two-state [[outer-totalistic]] rule. It was also the slowest known across all two-state [[isotropic]] rules until it was surpassed by a c/5953 spaceship in the rule B2-an3acqry4cnw5-cj6cei78/S1c2n3-nqy4acknqtw5-ijkr6ei78, found by [[Arie Paap]] in July 2019.<ref name="post80019" />
 
No finite [[still life]]s can exist in Gems.
Assume (for contradiction) that a finite still life exists, and let C denote the leftmost alive cell in the first nonempty row. For the cell C to survive under the rule B3457/S4568 in the next tick, it must have at least 4 alive neighbours; the only possibility is that the alive neighbours are the cell to the right of C and three neighbouring cells in the next row:
 
    0 0 0 0 0 0 0 0
    0 0 0 1 1 ? ? ? < the first nonempty row; C is the leftmost alive cell
    ? x 1 1 1 ? ? ?  state of cells denoted 'x' or '?' are unspecified
    ? ? ? ? ? ? ? ?
 
Then the dead cell to the left of C will be born in the next generation, either via B3 (if the cell 'x' is dead) or via B4 (if the cell 'x' is alive). This contradicts the assumption that C is the leftmost alive cell in the first nonempty row of a finite still life.
 
===Spaceships===
Gems has a single known [[spaceship]] moving orthogonally at a [[speed]] of c/5648, [[David Eppstein]]'s glider 13047.<ref>[https://web.archive.org/web/20150422210543/http://fano.ics.uci.edu/ca/rules/b3457s4568/g1.html Glider 13047]</ref> This is the slowest known elementary spaceship in any two-state [[outer-totalistic]] rule. It was also the slowest known across all two-state [[isotropic]] rules until it was surpassed by a c/5953 spaceship in the rule <tt>B2-an3acqry4cnw5-cj6cei78/S1c2n3-nqy4acknqtw5-ijkr6ei78</tt>, found by [[Arie Paap]] in July 2019.<ref name="post80019" />


{{EmbedViewer
{{EmbedViewer
|pname = gemsc5648
|pname = gemsc5648
|position = center
|position = center
|caption = Glider 13047 ({{LinkCatagolue|xq5648_0oou4114uoozvbmdvbbvdmbvzw33e77e33|rule=b3457s4568|style=brief}})<br>
|caption = Glider 13047
|apgcode = xq5648_0oou4114uoozvbmdvbbvdmbvzw33e77e33/b3457s4568
|viewerconfig = #C [[ THUMBSIZE 2 ]]
}}
 
==Similar rules==
A related rule is '''Gems Minor''' which removes eight from the survive list, but adds it to the birth list.  It also has many high period oscillators and a slow spaceship, this one moving orthogonally at c/2068.
{{Rule
|name            = Gems Minor
|char            = Explosive
|b              = 34578
|s              = 456
|reversal        = B015678/S2678
|ruleinteger    = 57784
}}
 
{{EmbedViewer
|pname = gemsmc2068
|position = center
|caption = c/2068 ship
|apgcode = xq2068_k4rjvvk4z21bedb/b34578s456
|viewerconfig = #C [[ THUMBSIZE 2 ]]
|viewerconfig = #C [[ THUMBSIZE 2 ]]
}}
}}
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==External links==
==External links==
* [http://www.foundalis.com/mat/life/unigems.htm Gems: Long Live the Oscillators, and the Slowest Glider]
* [http://www.foundalis.com/mat/life/unigems.htm Gems: Long Live the Oscillators, and the Slowest Glider]
{{LinkCatagolueRule|b3457s4568}}
* {{LinkCatagolueRule|b3457s4568}}
{{LinkEppsteinRule|b3457s4568}}
* {{LinkEppsteinRule|b3457s4568}}
* {{LinkForumThread|t=3063|title=Gems (B3457/S4568)}}
* {{LinkCatagolueRule|b34578s456|rulename=Gems Minor}}
 


[[Category:Life-like cellular automata]]
[[Category:Rules without still lives]]
__NOTOC__
__NOTOC__

Latest revision as of 07:26, 18 December 2023

Gems
x=0, y = 0, rule = B3457/S4568 ! #C [[ THEME Inverse ]] #C [[ RANDOMIZE2 RANDSEED 1729 THUMBLAUNCH THUMBNAIL THUMBSIZE 2 GRID ZOOM 6 WIDTH 600 HEIGHT 600 LABEL 90 -20 2 "#G" AUTOSTART PAUSE 2 GPS 8 LOOP 256 ]]
LifeViewer-generated pseudorandom soup
View static image
Rulestring 4568/3457
B3457/S4568
Rule integer 188600
Character Explosive
Black/white reversal B15678/S02678

Gems is a Life-like cellular automaton in which dead cells are born if they have precisely three, four, five or seven live neighbors, and in which live cells will survive if they have precisely four, five, six or eight live neighbors. Small random initial configurations in Gems tend to turn into high-period oscillators (the "gems" the rule is named after); larger random initial configurations explode slowly.

Patterns

No odd period oscillators have been found in Gems.

Still lives

No finite still lifes can exist in Gems. Assume (for contradiction) that a finite still life exists, and let C denote the leftmost alive cell in the first nonempty row. For the cell C to survive under the rule B3457/S4568 in the next tick, it must have at least 4 alive neighbours; the only possibility is that the alive neighbours are the cell to the right of C and three neighbouring cells in the next row: 

   0 0 0 0 0 0 0 0
   0 0 0 1 1 ? ? ? < the first nonempty row; C is the leftmost alive cell
   ? x 1 1 1 ? ? ?   state of cells denoted 'x' or '?' are unspecified
   ? ? ? ? ? ? ? ?

Then the dead cell to the left of C will be born in the next generation, either via B3 (if the cell 'x' is dead) or via B4 (if the cell 'x' is alive). This contradicts the assumption that C is the leftmost alive cell in the first nonempty row of a finite still life.

Spaceships

Gems has a single known spaceship moving orthogonally at a speed of c/5648, David Eppstein's glider 13047.[1] This is the slowest known elementary spaceship in any two-state outer-totalistic rule. It was also the slowest known across all two-state isotropic rules until it was surpassed by a c/5953 spaceship in the rule B2-an3acqry4cnw5-cj6cei78/S1c2n3-nqy4acknqtw5-ijkr6ei78, found by Arie Paap in July 2019.[2]

x = 12, y = 14, rule = B3457/S4568 5b2o$3bo4bo$3b2o2b2o$b3o4b3o$b3o4b3o$2ob6ob2o$3ob4ob3o$ob3o2b3obo$2ob6ob2o$ob obo2bobobo$2b2ob2ob2o$2b8o$4b4o$4bo2bo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBSIZE 2 ]]
Glider 13047
(click above to open LifeViewer)
RLE: here Plaintext: here
Catagolue: here

Similar rules

A related rule is Gems Minor which removes eight from the survive list, but adds it to the birth list. It also has many high period oscillators and a slow spaceship, this one moving orthogonally at c/2068.

Gems Minor
x=0, y = 0, rule = B34578/S456 ! #C [[ THEME Inverse ]] #C [[ RANDOMIZE2 RANDSEED 1729 THUMBLAUNCH THUMBNAIL THUMBSIZE 2 GRID ZOOM 6 WIDTH 600 HEIGHT 600 LABEL 90 -20 2 "#G" AUTOSTART PAUSE 2 GPS 8 LOOP 256 ]]
LifeViewer-generated pseudorandom soup
Rulestring 456/34578
B34578/S456
Rule integer 57784
Character Explosive
Black/white reversal B015678/S2678
#C Population: 31 x = 0, y = 0, rule = B34578/S456 5bo$2b3ob2o$2b2o3bo$2o6bo$bo5bo$2o4b3o$2b2o3bo$2b2o3b2o$4b3o$4bobo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBSIZE 2 ]]
c/2068 ship
(click above to open LifeViewer)
RLE: here Plaintext: here
Catagolue: here

References

  1. ↑ Glider 13047
  2. ↑ Arie Paap (July 24, 2019). Re: Smallest Spaceships Supporting Specific Speeds (5s) Project (discussion thread) at the ConwayLife.com forums

External links

Retrieved from "https://conwaylife.com/w/index.php?title=OCA:Gems&oldid=143303"