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.<ref>{{LinkWikipedia|Triangular_number|name=Tringular number|nobullet=nobullet}}</ref> | <sup>†</sup>After the first few terms, the numbers in this sequence closely approximate the triangular number sequence.<ref>{{LinkWikipedia|Triangular_number|name=Tringular number|nobullet=nobullet}}</ref> | ||
For any side length of the form 3n-1, a segment of [[block agar]] can be used to reach a population of 4n² | 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 == | == Densities of other objects == | ||
Revision as of 23:41, 21 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 |
(click above to open LifeViewer) |
| 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 | (click above to open LifeViewer) |
| 21 | ? | >=231 |
†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.[citation needed]
- The density of ash is approximately 0.0287 (1 in 35).[4]
- By observation, the density of a space dusty oscillator or spaceship is usually between 0.25 and 0.3.[5]
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.
- ↑ 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