The density of a pattern is the proportion of live cells within its bounding box.
The maximal densities of square (pseudo) still lifes up to 20×20 are:
| Side length |
Max. density |
Max. population |
Pattern
|
| 1 |
0 |
0
|
| 2 |
1 |
4 (block) |
x = 2, y = 2, rule = B3/S23
2o$2o!
#C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]]
|
| 3 |
0.6̄ |
6 (ship) |
x = 3, y = 3, rule = B3/S23
b2o$obo$2o!
#C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]]
|
| 4 |
0.5 |
8 (pond, long ship) |
x = 4, y = 4, rule = B3/S23
b2o$o2bo$o2bo$b2o!
#C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]]
x = 4, y = 4, rule = B3/S23
2b2o$bobo$obo$2o!
#C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]]
|
| 5 |
0.64 |
16 (four blocks) |
2ob2o$2ob2o2$2ob2o$2ob2o!
#C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]]
|
| 6 |
0.5 |
18 (eg. two long shillelaghs) |
x = 6, y = 6, rule = B3/S23
2o2b2o$o2bobo$b2o$3b2o$obo2bo$2o2b2o!
#C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]]
2o2b2o$o2bobo$bob2o$2o$2b2obo$2bob2o!
#C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]]
|
| 7 |
0.5714 |
28
|
| 8 |
0.5625 |
36 (eg. nine blocks) |
2ob2ob2o$2ob2ob2o2$2ob2ob2o$2ob2ob2o2$2ob2ob2o$2ob2ob2o!
#C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]]
|
| 9 |
0.5309 |
43
|
| 10 |
0.5400 |
54
|
| 11 |
0.5289 |
64
|
| 12 |
0.5278 |
76
|
| 13 |
0.5325 |
90
|
| 14 |
0.5306 |
104
|
| 15 |
0.5289 |
119
|
| 16 |
0.531 |
136
|
| 17 |
0.526 |
152
|
| 18 |
0.528 |
171
|
| 19 |
0.526 |
190
|
| 20 |
0.525 |
210
|
For any side length of the form 3n-1, a segment of block agar can be used to reach a population of 4n². Up to n = 4 this is the optimal solution.
References