Density

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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
1 0 0
2 1 4 (block)
3 0.6̄ 6 (ship)
4 0.5 8 (pond, long ship)
5 0.64 16 (four blocks)
6 0.5 18 (eg. two long shillelaghs)
7 0.5714 28
8 0.5625 36 (eg. nine blocks)
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