Density: Difference between revisions
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== Densities of other objects == | == Densities of other objects == | ||
* The maximum density of a still life agar | * The maximum density of a still life agar was proven to be 1/2 by [[Noam Elkies]] in {{year|1994}}. | ||
* The maximum average density of an oscillating agar is conjectured but not proven to be 1/2. | * The maximum average density of an oscillating agar is conjectured but not proven to be 1/2. The current upper bound is 8/13, proven by [[Hartmut Holzwart]] in {{year|1992}}. | ||
* The density of [[ash]] is approximately 0.0287 (1 in 35).<ref name="post55185" /> | * The density of [[ash]] is approximately 0.0287 (1 in 35).<ref name="post55185" /> | ||
* The density of a [[space dust]]y oscillator or spaceship is usually between 0.25 and 0.3.{{citation needed}} | * The density of a [[space dust]]y oscillator or spaceship is usually between 0.25 and 0.3.{{citation needed}} | ||
Revision as of 17:25, 10 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 | ? | >=231 |
The numbers in this sequence closely approximate 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 was proven to be 1/2 by Noam Elkies in 1994.
- The maximum average density of an oscillating agar is conjectured but not proven to be 1/2. The current upper bound is 8/13, proven by Hartmut Holzwart in 1992.
- The density of ash is approximately 0.0287 (1 in 35).[1]
- The density of a space dusty oscillator or spaceship is usually between 0.25 and 0.3.[citation needed]
References
- ↑ Oscar Cunningham (January 17, 2018). Re: Life Thermodynamics (discussion thread) at the ConwayLife.com forums
External links
- Density at the Life Lexicon
- Maximum Density Still Life at Neil Yorke-Smith's page