Density: Difference between revisions
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<sup>†</sup>After the first few terms, the numbers in this sequence closely approximate the triangular number sequence. | <sup>†</sup>After the first few terms, 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. | 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.{{citation needed}} | ||
== Densities of other objects == | == Densities of other objects == | ||
* The maximum density of a still life agar was proven to be 1/2 by [[Noam Elkies]].<ref>{{cite web|url=https://arxiv.org/abs/math/9905194 |title=The still-Life density problem and its generalizations|author=Noam D. Elkies| accessdate=May 26, 2022| date=May 31, 1999}}</ref> ('''still life conjecture''') | * The maximum density of a still life agar was proven to be 1/2 by [[Noam Elkies]].<ref>{{cite web|url=https://arxiv.org/abs/math/9905194 |title=The still-Life density problem and its generalizations|author=Noam D. Elkies| accessdate=May 26, 2022| date=May 31, 1999}}</ref> ('''still life conjecture''') | ||
* 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 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}}.{{citation needed}} | ||
* 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" /> | ||
* By observation, the density of a [[space dust]]y oscillator or spaceship is usually between 0.25 and 0.3. | * By observation, the density of a [[space dust]]y oscillator or spaceship is usually between 0.25 and 0.3. | ||
Revision as of 18:19, 30 May 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 |
†After the first few terms, 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.[citation needed]
Densities of other objects
- The maximum density of a still life agar was proven to be 1/2 by Noam Elkies.[1] (still life conjecture)
- 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.[citation needed]
- The density of ash is approximately 0.0287 (1 in 35).[2]
- By observation, the density of a space dusty oscillator or spaceship is usually between 0.25 and 0.3.
References
- ↑ Noam D. Elkies (May 31, 1999). "The still-Life density problem and its generalizations". Retrieved on May 26, 2022.
- ↑ 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