Density: Difference between revisions
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== 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> (this is known as the '''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> (this is known as the '''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}}.<ref name="post147457" /> | ||
* 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.<ref name="post146282" /> | * By observation, the density of a [[space dust]]y oscillator or spaceship is usually between 0.25 and 0.3.<ref name="post146282" /> | ||
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==References== | ==References== | ||
<references> | <references> | ||
<ref name="post147457">{{LinkForumThread | |||
|format = ref | |||
|title = Re: Can we substantiate this claim? | |||
|p = 147457 | |||
|author = Hartmut Holzwart | |||
|date = June 22, 2022 | |||
}}</ref> | |||
<ref name="post55185">{{LinkForumThread | <ref name="post55185">{{LinkForumThread | ||
|format = ref | |format = ref | ||
Revision as of 17:11, 23 June 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:[1]
| 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 sixteen blocks |
|
| 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 |
†After the first few terms, the numbers in this sequence closely approximate the triangular number sequence.[2]
For any side length of the form 3n-1, a segment of block agar can be used to reach a population of 4n²; to n=4 this is the optimal solution, as can be seen in the above table.
Densities of other objects
- The maximum density of a still life agar was proven to be 1/2 by Noam Elkies[3] (this is known as the 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.[4]
- The density of ash is approximately 0.0287 (1 in 35).[5]
- By observation, the density of a space dusty oscillator or spaceship is usually between 0.25 and 0.3.[6]
References
- ↑ "Maximum Density Still Life". Neil Yorke-Smith (Auguat 4, 2003).
- ↑ Tringular number at Wikipedia
- ↑ Noam D. Elkies (May 31, 1999). "The still-Life density problem and its generalizations". Retrieved on May 26, 2022.
- ↑ Hartmut Holzwart (June 22, 2022). Re: Can we substantiate this claim? (discussion thread) at the ConwayLife.com forums
- ↑ Oscar Cunningham (January 17, 2018). Re: Life Thermodynamics (discussion thread) at the ConwayLife.com forums
- ↑ David Raucci (May 29, 2022). Re: Can we substantiate this claim? (discussion thread) at the ConwayLife.com forums
External links
- Density at the Life Lexicon