Density

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 (example)	Pattern
1	0	0
2	1	4 (block)	x = 2, y = 2, rule = B3/S23 2o$2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] (click above to open LifeViewer) RLE: here Plaintext: here
3	0.6̄	6 (ship)	x = 3, y = 3, rule = B3/S23 b2o$obo$2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] (click above to open LifeViewer) RLE: here Plaintext: here
4	0.5	8 (pond, long ship)	x = 4, y = 4, rule = B3/S23 b2o$o2bo$o2bo$b2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] (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 ]] (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$2ob2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] (click above to open LifeViewer) RLE: here Plaintext: here
6	0.5	18 (two long shillelaghs, trans-snake-tie with ship)	x = 6, y = 6, rule = B3/S23 2o2b2o$o2bobo$b2o$3b2o$obo2bo$2o2b2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] (click above to open LifeViewer) RLE: here Plaintext: here x = 6, y = 6, rule = B3/S23 2o2b2o$o2bobo$bob2o$2o$2b2obo$2bob2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] (click above to open LifeViewer) 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 ]] (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$2ob2ob2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] (click above to open LifeViewer) RLE: here Plaintext: here
9	0.5309	43	x = 89, y = 29, rule = B3/S23 2o2b2ob2o11b2o2b2ob2o11bob2ob2obo11bob2ob2obo11bob2obob2o$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 ]] (click above to open LifeViewer) RLE: here Plaintext: here
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.

Densities of other objects

• The maximum density of a still life agar is proven to be 1/2.
• The maximum average density of an oscillating agar is conjectured but not proven to be 1/2.
• The density of ash is 0.0287 (1 in 35).
• The density of a space dusty oscillator or spaceship is usually between 0.25 and 0.3.

References

• Density at the Life Lexicon
• http://www.ai.sri.com/~nysmith/life/