Density: Difference between revisions
Jump to navigation
Jump to search
added date to ref |
added 20 ex in viewer |
||
| Line 26: | Line 26: | ||
| 9 || 0.5309 || 43 || {{EmbedViewer|pname=43cell9x9pseudostilllifes|viewerconfig= #C [[ THUMBSIZE 3 ]]}} | | 9 || 0.5309 || 43 || {{EmbedViewer|pname=43cell9x9pseudostilllifes|viewerconfig= #C [[ THUMBSIZE 3 ]]}} | ||
|- | |- | ||
| 10 || 0.5400 || 54 | | 10 || 0.5400 || 54 | ||
|- | |- | ||
| 11 || 0.5289 || 64 | | 11 || 0.5289 || 64 | ||
| Line 46: | Line 46: | ||
| 19 || 0.526 || 190 | | 19 || 0.526 || 190 | ||
|- | |- | ||
| 20 || 0.525 || 210 | | 20 || 0.525 || 210 || {{EmbedViewer|viewerconfig= #C [[ THUMBSIZE 3 ]] | rle=x = 20, y = 20, rule = B3/S23 | ||
2o2b2ob2obob2ob2ob2o$o2bobobob3obob2obob$bobobobo5bo5bo$2obobob6ob6o$b | |||
obobo6bobo5b$o2bob6obobob4o$4o6bobobobo2bo$4b5obobobobobob$4o4bobobobo | |||
bob2o$o2bob2obobobobobobob$bobobobobobob2obo2bo$2obobobobobo4b4o$bobob | |||
obobob5o4b$o2bobobobo6b4o$4obobob6obo2bo$5bobo6bobobob$6ob6obobob2o$o | |||
5bo5bobobobob$bob2obob3obobobo2bo$2ob2ob2obob2ob2o2b2o!}} | |||
|- | |- | ||
| 21 || ? || >=231 | | 21 || ? || >=231 | ||
Revision as of 21:51, 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 | |
| 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. For example, see block (n=1), four blocks (n=2) and nine blocks (n=3) in the above table. 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[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