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Revision as of 23:35, 27 June 2022

The density of a pattern is the proportion of live cells within its bounding box.

Still Lifes

The general problem of finding the maximum density of an nxn square pseudo still life was solved for all n in 2012.[1] The maximum densities of square still lifes up to 20×20 were known ten years earlier.[2]

The following table contains maximum density still life information for various side lengths.

Side length Max. density Max. population†
(example)		 Example pattern(s)
1 0 0
2 1 4
block
x = 2, y = 2, rule = B3/S23 2o$2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBSIZE 3 ]]
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RLE: here Plaintext: here
3 0.6 6
ship
x = 3, y = 3, rule = B3/S23 b2o$obo$2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBSIZE 3 ]]
(click above to open LifeViewer)
RLE: here Plaintext: here
4 0.5 8
pond
long ship
x = 4, y = 4, rule = B3/S23 b2o$o2bo$o2bo$b2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBSIZE 3 ]]
(click above to open LifeViewer)
RLE: here Plaintext: here

x = 4, y = 4, rule = B3/S23 2b2o$bobo$obo$2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBSIZE 3 ]]
(click above to open LifeViewer)
RLE: here Plaintext: here
5 0.64 16
four blocks
x = 5, y = 5, rule = B3/S23 2ob2o$2ob2o2$2ob2o$2ob2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBSIZE 3 ]]
(click above to open LifeViewer)
RLE: here Plaintext: here
6 0.5 18
two long shillelaghs
trans-snake-tie with ship
three blocks with ship
x = 6, y = 6, rule = B3/S23 2o2b2o$o2bobo$b2o$3b2o$obo2bo$2o2b2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBSIZE 3 ]]
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RLE: here Plaintext: here

x = 6, y = 6, rule = B3/S23 2o2b2o$o2bobo$bob2o$2o$2b2obo$2bob2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBSIZE 3 ]]
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RLE: here Plaintext: here

x = 6, y = 6, rule = B3/S23 2ob2o$2ob2o2$2ob2o$2obobo$4b2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBSIZE 3 ]]
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RLE: here Plaintext: here
7 0.5714 28
x = 7, y = 7, rule = B3/S23 ob2ob2o$2obobo$3bo2bo$3ob3o$o2bo$bobob2o$2ob2obo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBSIZE 3 ]]
(click above to open LifeViewer)
RLE: here Plaintext: here
8 0.5625 36
nine blocks
x = 8, y = 8, rule = B3/S23 2ob2ob2o$2ob2ob2o2$2ob2ob2o$2ob2ob2o2$2ob2ob2o$2ob2ob2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBSIZE 3 ]]
(click above to open LifeViewer)
RLE: here Plaintext: here
9 0.5309 43
x = 89, y = 29, rule = B3/S23 2o2b2ob2o11b2o2b2ob2o11bob2ob2obo11bob2ob2obo11bob2obob2o$o2bobob2o11b o2bobob2o11b2obobob2o11b2obobob2o11b2obob2obo$bobobo15bobobo17bobo17bo bo17bo$2obob4o11b2obob4o11b2obob4o11b2obob4o11b3ob4o$o2bo4bo12bobo4bo 12bobo4bo11bo2bo4bo11bo2bo4bo$bob4o14bob4o14bob4o14bob4o14bobob4o$2o4b 2o12b2o4b2o12b2o4b2o12b2o4b2o12b2obobo$o2b2o3bo11bo2b2o3bo11bo2b2o3bo 11bo2b2o3bo11bo2bobob2o$b2ob2ob2o12b2ob2ob2o12b2ob2ob2o12b2ob2ob2o13b 2ob2obo12$ob2ob2obo11bob2ob2obo11bob2ob2obo11bob2ob2obo11bob2ob2obo$2o bobob2o11b2obobob2o11b2obobob2o11b2obobob2o11b2obobob2o$3bobo17bobo17b obo17bobo17bobo$2obob4o11b4ob4o11b2obob4o11b4o2b3o11b4ob4o$2obo4bo11bo 7bo11bob2o4bo11bo3b2o2bo11bo7bo$3b5o13bob5o16b4o13b3o2b2o13b7o$3o17b2o bo17b3o20bo$o2bob2obo11bo2bob2obo11bo2bob2obo11bob2ob2obo11bob2ob2obo$ 2b2obob2o13b2obob2o11b2o2b2ob2o11b2obobob2o11b2obobob2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBSIZE 3 ]]
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RLE: here Plaintext: here
10 0.5400 54
x = 10, y = 10, rule = B3/S23 $ob2ob2ob2o$2ob2obob2o$6bo$2ob3ob3o$bobo2bo2bo$o2bo2bobo$3ob3ob2o$3bo$ 2obob2ob2o$2ob2ob2obo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBSIZE 3 ]]
(click above to open LifeViewer)
RLE: here Plaintext: here
11 0.5289 64
sixteen blocks
x = 11, y = 11, rule = B3/S23 2ob2ob2ob2o$2ob2ob2ob2o2$2ob2ob2ob2o$2ob2ob2ob2o2$2ob2ob2ob2o$2ob2ob2o b2o2$2ob2ob2ob2o$2ob2ob2ob2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBSIZE 3 ]]
(click above to open LifeViewer)
RLE: here Plaintext: here
12 0.5278 76
13 0.5325 90
14 0.5306 104
x = 14, y = 14, rule = B3/S23 2ob2ob2ob2ob2o$2ob2ob2ob2ob2o2$4ob6ob2o$o2bobo4bob2o$bobobob2obo$2obob ob2obob2o$bobobo4bob2o$o2bob6o$4o8b2o$4b7ob2o$2o2bo5bo$obobob3obob2o$b 2ob2obob2ob2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBSIZE 3 ]]
(click above to open LifeViewer)
RLE: here Plaintext: here
15 0.5289 119
x = 15, y = 15, rule = B3/S23 2ob2ob2ob2o2b2o$2ob2ob2obobo2bo$9bobobo$10obob2o$o5bo4bobo$bob2o3b4o2b o$2obob3o4b3o$bobobo2b4o$o2bobobo4b3o$4obob5o2bo$5bo5bobo$4ob5obob2o$o 2bo5bobo$bobob3obobob2o$2ob2obob2ob2obo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBSIZE 3 ]]
(click above to open LifeViewer)
RLE: here Plaintext: here
16 0.531 136
17 0.526 152
18 0.528 171
19 0.526 190
x = 19, y = 19, rule = B3/S23 2ob2ob2ob2obob2ob2o$2ob2ob2obob3obob2o$9bo5bo$2ob6ob5ob3o$2obo4bobo4bo 2bo$3bob2obo2b3obobo$2obobo2b3o2bobob2o$2obo2b2o3b2o2bobo$3b3o2b3o2b3o 2bo$3o3b2o3b2o3b3o$o2b3o2b3o2b3o$bobo2b2o3b2o2bob2o$2obobo2b3o2bobob2o $bobob2obo2b3obo$o2bo4bobo4bob2o$3ob5ob6ob2o$3bo5bo$2obob3obob2ob2ob2o $2ob2obob2ob2ob2ob2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBSIZE 3 ]]
(click above to open LifeViewer)
RLE: here Plaintext: here
20 0.525 210
x = 20, y = 20, rule = B3/S23 2o2b2ob2obob2ob2ob2o$o2bobobob3obob2obob$bobobobo5bo5bo$2obobob6ob6o$b obobo6bobo5b$o2bob6obobob4o$4o6bobobobo2bo$4b5obobobobobob$4o4bobobobo bob2o$o2bob2obobobobobobob$bobobobobobob2obo2bo$2obobobobobo4b4o$bobob obobob5o4b$o2bobobobo6b4o$4obobob6obo2bo$5bobo6bobobob$6ob6obobob2o$o 5bo5bobobobob$bob2obob3obobobo2bo$2ob2ob2obob2ob2o2b2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBSIZE 3 ]]
(click above to open LifeViewer)
RLE: here Plaintext: here
21 0.526 232
x = 21, y = 21, rule = B3/S23 2ob2obob2ob2obob2ob2o$2obob3obobob3obob2o$3bo5bobo5bo$3ob5obob5ob3o$o 2bo4bobo6bo2bo$bobob2obob6obobo$2obob2obo6bobob2o$bobo4b6obobobo$o2b5o 5bobobo2bo$3o5b4obobob4o$3b5o3bobob2o$3o5b2obobo4b3o$o2b4obobobob4o2bo $bobo2bobobobobo2bobo$2obobo2bobobo2bobob2o$bobob4obob4obobo$o2bo6bo6b o2bo$3ob6ob6ob3o$3bo5bobo5bo$2obob3obobob3obob2o$2ob2obob2ob2obob2ob2o ! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBSIZE 3 ]]
(click above to open LifeViewer)
RLE: here Plaintext: here
22 0.523 253
x = 22, y = 22, rule = B3/S23 2ob2ob2obob2ob2ob2ob2o$bob2obob3obob2ob2ob2o$o5bo5bo$6ob5ob9o$5bobo3bo bo7bo$4obo2b2obobob4obo$o2bob2obobo2bobo2bob2o$bobobo2bobob2obo2bobo$ 2obobob2obobo2b4o2bo$bobobo2bobobobo4b3o$o2b2obobobobob5o$3o3b2obobobo 4bob2o$3b3o3bobobob2obob2o$2obo2b3obobobobobo$bobobo2bobobobobobob2o$o 2bob2obobobobobobob2o$4obo2b2obobob2obo$5bobo3bobo4bob2o$6ob5ob6ob2o$o 5bo5bo$bob2obob3obob2ob2ob2o$2ob2ob2obob2ob2ob2ob2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBSIZE 3 ]]
(click above to open LifeViewer)
RLE: here Plaintext: here
23 0.522 276
x = 23, y = 23, rule = B3/S23 2ob2ob2obob2obob2ob2obo$bob2obob3obob2obobob2o$o5bo5bo4bobo$6ob5ob5obo b2o$5bobo3bobo5bob2o$4obo2b2obobob5o$o2bob2obobobobobo3b3o$bobobo2bobo bobo3b2o2bo$2obo2b3obobob4obobo$3b3o3bobobo3bobob2o$3o3b3o2bo2b3o2bobo $o2b3o3b2ob2o3b3o2bo$bobo2b3o2bo2b3o3b3o$2obobo3bobobo3b3o$bobob4obobo b3o2bob2o$o2b2o3bobobobo2bobobo$3o3bobobobobob2obo2bo$3b5obobob2o2bob 4o$2obo5bobo3bobo$2obob5ob5ob6o$3bobo4bo5bo5bo$2obobob2obob3obob2obo$o b2ob2obob2obob2ob2ob2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBSIZE 3 ]]
(click above to open LifeViewer)
RLE: here Plaintext: here

†After the first few terms, the numbers in this sequence closely approximate the triangular number sequence.[3]

For any side length of the form 3n-1, a segment of block agar can be used to reach a population of 4n²; to n=4 this is the optimal solution, as can be seen in the above table.

Densities of other objects

  • The maximum density of a still life agar was proven to be 1/2 by Noam Elkies[4] (this is known as the still life conjecture).
  • The maximum average density of an oscillating agar is conjectured but not proven to be 1/2. The current upper bound is 8/13, proven by Hartmut Holzwart in 1992.[5]
  • The density of ash is approximately 0.0287 (1 in 35).[6]
  • By observation, the density of a space dusty oscillator or spaceship is usually between 0.25 and 0.3.[7]

References

  1. ↑ "A Complete Solution to the Maximum Density Still Life Problem". Geoffrey Chu, Peter J. Stuckey, NICTA Victoria Laboratory, Department of Computing and Information Systems, University of Melbourne, Australia (November 8, 2013).
  2. ↑ "Maximum Density Still Life". Neil Yorke-Smith (August 4, 2003).
  3. ↑ Tringular number at Wikipedia
  4. ↑ Noam D. Elkies (May 31, 1999). "The still-Life density problem and its generalizations". Retrieved on May 26, 2022.
  5. ↑ Hartmut Holzwart (June 22, 2022). Re: Can we substantiate this claim? (discussion thread) at the ConwayLife.com forums
  6. ↑ Oscar Cunningham (January 17, 2018). Re: Life Thermodynamics (discussion thread) at the ConwayLife.com forums
  7. ↑ David Raucci (May 29, 2022). Re: Can we substantiate this claim? (discussion thread) at the ConwayLife.com forums
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