Density: Difference between revisions
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let's append patterns |
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{| class="wikitable" | {| class="wikitable" | ||
|- | |- | ||
! Side length !! Max. density !! Max. population | ! Side length !! Max. density !! Max. population !! Pattern | ||
|- | |- | ||
| 1 || 0 || 0 | | 1 || 0 || 0 | ||
|- | |- | ||
| 2 || 1 || 4 ([[block]]) | | 2 || 1 || 4 ([[block]]) || {{EmbedViewer|pname=block}} | ||
|- | |- | ||
| 3 || 0.6̄|| 6 ([[ship]]) | | 3 || 0.6̄|| 6 ([[ship]]) || {{EmbedViewer|pname=ship}} | ||
|- | |- | ||
| 4 || 0.5 || 8 ([[pond]], [[long ship]]) | | 4 || 0.5 || 8 ([[pond]], [[long ship]]) || {{EmbedViewer|pname=pond}}<br>{{EmbedViewer|pname=longship}} | ||
|- | |- | ||
| 5 || 0.64 || 16 (four blocks) | | 5 || 0.64 || 16 (four blocks) || {{EmbedViewer|rle=2ob2o$2ob2o2$2ob2o$2ob2o!}} | ||
|- | |- | ||
| 6 || 0.5 || 18 (eg. two [[long shillelagh]]s) | | 6 || 0.5 || 18 (eg. two [[long shillelagh]]s) || {{EmbedViewer|pname=twolongshillelaghs}}<br>{{EmbedViewer|rle=2o2b2o$o2bobo$bob2o$2o$2b2obo$2bob2o!}} | ||
|- | |- | ||
| 7 || 0.5714 || 28 | | 7 || 0.5714 || 28 | ||
|- | |- | ||
| 8 || 0.5625 || 36 (eg. nine blocks) | | 8 || 0.5625 || 36 (eg. nine blocks) || {{EmbedViewer|rle=2ob2ob2o$2ob2ob2o2$2ob2ob2o$2ob2ob2o2$2ob2ob2o$2ob2ob2o!}} | ||
|- | |- | ||
| 9 || 0.5309 || 43 | | 9 || 0.5309 || 43 | ||
Revision as of 17:01, 8 January 2021
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) | |
| 3 | 0.6̄ | 6 (ship) | |
| 4 | 0.5 | 8 (pond, long ship) | |
| 5 | 0.64 | 16 (four blocks) | (click above to open LifeViewer) |
| 6 | 0.5 | 18 (eg. two long shillelaghs) | (click above to open LifeViewer) |
| 7 | 0.5714 | 28 | |
| 8 | 0.5625 | 36 (eg. nine blocks) | (click above to open LifeViewer) |
| 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.