# Density

From LifeWiki

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

- Dense Stable Patterns at the Life Lexicon
- http://www.ai.sri.com/~nysmith/life/