OCA:Iceballs

This article is about a Life-like rule. For a hypothetical unstoppable quadratic growth pattern, see IceNine.

Iceballs
x=64, y = 64, rule = B25678/S5678 ! #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	5678/25678 B25678/S5678
Rule integer	246244
Character	Explosive
Black/white reversal	B45678/S4578

Iceballs, also known as Ice nine, is a Life-like cellular automaton with rulestring B25678/S5678. It is named for its tendency to produce patches of solid agar with density 100%, as exemplified below.

x = 20, y = 20, rule = B25678/S5678:T20,20 $8b2o$10bo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBNAIL ZOOM 20 WIDTH 600 HEIGHT 600 ]]

A 3-cell pattern on the 20 × 20 torus runs chaotically, with average density about 25%, until about gen 435175. Then an iceball forms, which slowly grows, filling the torus completely in gen 435374. The same pattern, run on a 100 × 100 torus, forms an iceball about gen 31300, which fills the torus at gen 31951. (click above to open LifeViewer)

No finite still lifes can exist in Iceballs. 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 B25678/S5678 in the next tick, it must have exactly 5, 6, 7 or 8 neighbours, including at least one of the positions marked 'x' below:

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

This contradicts the assumption that C is the leftmost alive cell in the first nonempty row of a finite still life.

Spaceships

There are known spaceships of speeds c, 4c/4, 5c/5, c/2o, c/3o, c/5o, c/3d, c/4d.

x = 49, y = 9, rule = B25678/S5678 b2o5b2o5b2o5b2o9b2o9b2o$o2bo10bo6bo10bo2bo7bo2bo$7bo2bo6bo6bo6bob2obo 5bob2obo$25bo17b4o$23bo7bob2obo4bob4obo$24bo19b2o$32bo2bo7b4o2$42bo4bo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBSIZE 3 ZOOM 18 WIDTH 1200 HEIGHT 600 Y -10 GPS 5 AUTOSTART T 0 PAUSE 3 T 20 PAUSE 2 LOOP 21 ]]

Photons, left to right: p1 (moon), p1, p1, p4, p5, p1 (click above to open LifeViewer)

x = 71, y = 33, rule = B25678/S5678 7b7o19bo5bo$10bo47bo5bo$bo3b2o2b3o2b2o3bo11bo3b3o3bo$2bobobobo3bobobob o9b3o3bo3bo3b3o11bo3b3o3bo$8b5o14bobo5b3o5bobo7b3o3bo3bo3b3o$4bo4b3o4b o13bo4b3o4bo9bobo5b3o5bobo$4bo4b3o4bo12bobo3b3o3bobo11bo4b3o4bo$4bo2b 7o2bo16b7o14bobo3b3o3bobo$3bo5b3o5bo13bo3b3o3bo16b7o$b2o5b5o5b2o10bo3b 5o3bo13bo3b3o3bo$o2bo2b9o2bo2bo9bo2bob3obo2bo12bo3b5o3bo$4bo3b5o3bo13b 3ob5ob3o12bo2bob3obo2bo$4b13o17b5o16b3ob5ob3o$3b3o2b5o2b3o14b9o18b5o$ 4b13o13bo2b7o2bo15b7o$3b3o2b5o2b3o15b7o14bo4b5o4bo$5bob7obo12b2obo2b5o 2bob2o10b2ob7ob2o$6bob5obo12bobo3bob3obo3bobo7bo4b7o4bo$2b2o2bob5obo2b 2o14bob3obo16bob7obo$7bob3obo15b3obo2bo2bob3o14b7o$4bobob5obobo11bo2bo bo5bobo2bo14b5o$30bo11bo10bo3bob5obo3bo$5bo2bo3bo2bo17bo2bo2bo12bo3bob ob3obobo3bo$32bo7bo15bo4bo4bo$35bobo22bobo$29bo5b3o5bo15b5o$28bo3b2o5b 2o3bo16bo$32bobo3bobo17b7o$32b2o2bo2b2o19b3o$57bob5obo$30b3o2bobo2b3o 18bo$56bo9bo$29bo4bo3bo4bo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBSIZE 3 ZOOM 15 WIDTH 1200 HEIGHT 650 GPS 4 AUTOSTART TRACKLOOP 3 0/3 -1/3 ]]

c/3 orthogonal spaceships, left to right: 176 cells, outer-totalistically endemic; 182 cells (Rocknlol, 2020); 180 cells (Rocknlol, 2020) (click above to open LifeViewer)

x = 59, y = 38, rule = B25678/S5678 bo31bo$ob31obo15bo$b33obob2o9bobobo$ob33ob2o2bo3bo8bo$ob5obob9obo4b10o b2o2bobo4b2o9bo$bo2bob3o3bob5o5bo3bobob3obo5bobo10b3o$2b3obo4bo2bobo 19bo5b3o4b5obo$bobo7bobo2bo6b2o6bo3bobo3b6obob5obo$bobo7bobo2bo6b2o6bo 3bobo3b6obob5obo$2b3obo4bo2bobo19bo5b3o4b5obo$bo2bob3o3bob5o5bo3bobob 3obo5bobo10b3o$ob5obob9obo4b10ob2o2bobo4b2o9bo$ob33ob2o2bo3bo8bo$b33ob ob2o9bobobo$ob31obo15bo$bo31bo7$bo32bo$ob32ob4o$b34o4bo13bo$ob8obobobo bob19o5bo7b2o$ob6o2bo4b2o2b2obo4b9o2b2o4b2o8b2o$bo5bo2bo4b2o12bobobobo 7bobobo5bo4bo$2b2ob3o2bo11bobo4b2o11b7o2b2obobo$bobo4b2o4bo5bo3bobo3bo bobobobobob4o3b7obo$bobo4b2o4bo5bo3bobo3bobobobobobob4o3b7obo$2b2ob3o 2bo11bobo4b2o11b7o2b2obobo$bo5bo2bo4b2o12bobobobo7bobobo5bo4bo$ob6o2bo 4b2o2b2obo4b9o2b2o4b2o8b2o$ob8obobobobob19o5bo7b2o$b34o4bo13bo$ob32ob 4o$bo32bo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBSIZE 3 ZOOM 15 WIDTH 1200 HEIGHT 800 GPS 4 AUTOSTART TRACKLOOP 5 1/5 0/5 ]]

c/5 orthogonal spaceships, top to bottom: 414 cells, outer-totalistically endemic (Rocknlol, 2020); 380 cells (Rocknlol, 2020) (click above to open LifeViewer)

x = 13, y = 13, rule = B25678/S5678 o11bo$o11bo$o5bo5bo$o5bo5bo$4bobobo$2bo7bo$o4bobo4bo$2b3o3b3o$2bobo3bo bo3$3bobobobo$5bobo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBSIZE 3 ZOOM 15 WIDTH 480 HEIGHT 500 GPS 4 AUTOSTART TRACKLOOP 2 0/2 -1/2 ]]

c/2 orthogonal (click above to open LifeViewer)

x = 59, y = 35, rule = B25678/S5678 8bo24bo$8bobo22bobo$6bo24bo$10b3o22b2o$10bo2bo20bo$8bo27bo$2bo3bo7bo 18bo$7bo2b2o14bo4b2o3b2o$2o3bo5bo19b4o2bo$12bo11b2o4bobo4bo$bob2o2bo3b o16bo3bo3b2o$3bo3b2obobo12bobo18bo$3bo5bobo3bo11bobobo2bo4bo8bo$4bo26b 4o2bobo3bo2bo$6bo31bo2bo4bo$12bo23b2o7b2o$42bo5bo$38bo5bobo$40bo3bobo$ 37bo13bo$41b2o3b2o3bobo$39bo7bobo$35bob3ob2obo2bo5b3o$44b3o6bo2bo$36bo 3bo10bo$45bo3bo7bo$50bo2b2o$43b2o3bo5bo$55bo$44bob2o2bo3bo$46bo3b2obob o$46bo5bobo3bo$47bo$49bo$55bo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBSIZE 3 ZOOM 12 WIDTH 800 HEIGHT 500 GPS 4 AUTOSTART TRACKLOOP 4 -1/4 -1/4 ]]

Two c/4 diagonal spaceships (Josh Ball, 2011) (click above to open LifeViewer)

x = 33, y = 33, rule = B25678/S5678 22bo5bo$20bo6bo$19bo3b2o5bo$17bobobo4bob2o$16b5o6b3o2bo$17b4o4bo2bo2bo $15b6o4b2o2bo$15b4obobo3b2o$13b7o10bo$12b5o3bo2bo6bo$11b6ob2o5bo6bo$9b ob5o2bo10bo$10b6o3bo3bob4o2bo$8bob4obo3b2obobob5o$9b7o5b2ob5o$7bo2b3o 4bo6b6o$8bo3bobo7b5obo$9b3ob3o2b9o$15b2obob5o$2bo5bobo4bo2b7o$o2b3o10b 8o$2b2o2bob4o3bob6o$b6o4bobobob4o$6b2obobo3bo2bo2bo$3b2o2bo3bobo2bo2bo $2bo2bo2b2o7bo$9b3o$2bobo2bo2bobo$b2o2bo2bobobo$3bo4bob3o$bo2b2obo2b2o bo$2bobo5bo$12bo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBSIZE 3 ZOOM 15 WIDTH 800 HEIGHT 700 GPS 4 AUTOSTART TRACKLOOP 3 -1/3 1/3 ]]

c/3 diagonal, outer-totalistically endemic (click above to open LifeViewer)

Rake

x = 37, y = 24, rule = B25678/S5678 18bo$16bobo8bo$6bo10bo3b4o$7b2o7bo4bobo$11bo14bo$3bo2bo10bo4bo5bo$3bo 4b2o3b3o9bobo$9bob2o3bo2bobo$2o5bo5b3o3bo$2bo6b9o$bo3bob11o6bo4bo4bo$ 5bob12o2bo4bo4bo4bo$5bob12o2bo4bo4bo4bo$bo3bob11o6bo4bo4bo$2bo6b9o$2o 5bo5b3o3bo$9bob2o3bo2bobo$3bo4b2o3b3o9bobo$3bo2bo10bo4bo5bo$11bo14bo$ 7b2o7bo4bobo$6bo10bo3b4o$16bobo8bo$18bo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBSIZE 3 ZOOM 12 WIDTH 650 HEIGHT 500 GPS 4 AUTOSTART TRACK -1/4 0/4 ]]

A c/4 orthogonal backwards rake (Josh Ball, 2011) (click above to open LifeViewer)

External links

• Iceballs at Adam P. Goucher's Catagolue
• Iceballs (discussion thread) at the ConwayLife.com forums
• Iceballs at David Eppstein's Glider Database
• David Eppstein. "Interesting Rules". Fano Experimental Web Server. Archived from the original on November 27, 2022.