Density: Difference between revisions
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m x = 0, y = 0, rule = B3/S23 23b$b2ob2obob2ob2obob2ob2ob$b2obob3obobob3obob2ob$4bo5bobo5bo4b$b3ob5obob5ob3ob$bo2bo4bobobo4bo2bob$2bobob2obobobob2obobo2b$b2obobobobobobobobob2ob$bo2bobobobobobobobobo2b$2bobobobobobobobobobo2b$b2obobobobobobobobob2ob$4bobobobobobobobo4b$b2obobobobobobobobob2ob$2bobobobobobobobobo2bob$2bobobobobobobobobobo2b$b2obobobobobobobobob2ob$2bobob2obobobob2obobo2b$bo2bo4bobobo4bo2bob$b3ob5obob5ob3ob$4bo5bobo5bo4b$b2obob3obobob3obob2ob$b2ob2obob2ob2obob2ob2ob! |
171, 190, 210 — I recognize those numbers |
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| Line 50: | Line 50: | ||
| 21 || ? || >=230 | | 21 || ? || >=230 | ||
|} | |} | ||
The numbers in this sequence are always very close to the triangular number sequence. | |||
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. | 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. | ||
Revision as of 13:24, 8 March 2022
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(s) |
|---|---|---|---|
| 1 | 0 | 0 | |
| 2 | 1 | 4 (block) | |
| 3 | 0.6 | 6 (ship) | |
| 4 | 0.5 | 8 (pond, long ship) | |
| 5 | 0.64 | 16 (four blocks) | |
| 6 | 0.5 | 18 (two long shillelaghs, trans-snake-tie with ship) |
|
| 7 | 0.5714 | 28 | |
| 8 | 0.5625 | 36 (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 | |
| 21 | ? | >=230 |
The numbers in this sequence are always very close to the triangular number sequence.
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 0.02877 (1 in 35).
- The density of a space dusty oscillator or spaceship is usually between 0.25 and 0.3.