Density: Difference between revisions

From LifeWiki
Jump to navigation Jump to search
← Older editNewer edit →
Book (talk | contribs)
Trusted users
8,810 edits
→Densities of other objects: label still life conjecture as such
a little reformatting
Line 4: Line 4:
The maximal densities of square (pseudo) [[still life]]s up to 20×20 are:
The maximal densities of square (pseudo) [[still life]]s up to 20×20 are:


{| class="wikitable"
{|class="wikitable" style="text-align:center"
|-
|-
! Side length !! Max. density !! Max. population<br/>(example) !! Pattern(s)
! Side length !! Max. density !! Max. population<br/>(example) !! Pattern(s)
Line 10: Line 10:
| 1 || 0 || 0
| 1 || 0 || 0
|-
|-
| 2 || 1 || 4 ([[block]]) || {{EmbedViewer|pname=block|viewerconfig= #C [[ THUMBSIZE 3 ]]}}
| 2 || 1 || 4<br/>[[block]] || {{EmbedViewer|pname=block|viewerconfig= #C [[ THUMBSIZE 3 ]]}}
|-
|-
| 3 || 0.6 || 6 ([[ship]]) || {{EmbedViewer|pname=ship|viewerconfig= #C [[ THUMBSIZE 3 ]]}}
| 3 || 0.6 || 6<br/>[[ship]] || {{EmbedViewer|pname=ship|viewerconfig= #C [[ THUMBSIZE 3 ]]}}
|-
|-
| 4 || 0.5 || 8 ([[pond]], [[long ship]]) || {{EmbedViewer|pname=pond|viewerconfig= #C [[ THUMBSIZE 3 ]]}}<br>{{EmbedViewer|pname=longship|viewerconfig= #C [[ THUMBSIZE 3 ]]}}
| 4 || 0.5 || 8<br/>[[pond]]<br/>[[long ship]]) || {{EmbedViewer|pname=pond|viewerconfig= #C [[ THUMBSIZE 3 ]]}}<br>{{EmbedViewer|pname=longship|viewerconfig= #C [[ THUMBSIZE 3 ]]}}
|-
|-
| 5 || 0.64 || 16 (four blocks) || {{EmbedViewer|pname=4blocks|viewerconfig= #C [[ THUMBSIZE 3 ]]}}
| 5 || 0.64 || 16<br/>four blocks || {{EmbedViewer|pname=4blocks|viewerconfig= #C [[ THUMBSIZE 3 ]]}}
|-
|-
| 6 || 0.5 || 18 (two [[long shillelagh]]s,<br/>trans-snake-tie with ship) || {{EmbedViewer|pname=twolongshillelaghs|viewerconfig= #C [[ THUMBSIZE 3 ]]}}<br/>{{EmbedViewer|pname=transsnaketiewithship|viewerconfig= #C [[ THUMBSIZE 3 ]]}}
| 6 || 0.5 || 18<br/>two [[long shillelagh]]s<br/>trans-snake-tie with ship) || {{EmbedViewer|pname=twolongshillelaghs|viewerconfig= #C [[ THUMBSIZE 3 ]]}}<br/>{{EmbedViewer|pname=transsnaketiewithship|viewerconfig= #C [[ THUMBSIZE 3 ]]}}
|-
|-
| 7 || 0.5714 || 28 || {{EmbedViewer|pname=xs28db8n9arz3123032|viewerconfig= #C [[ THUMBSIZE 3 ]]}}
| 7 || 0.5714 || 28 || {{EmbedViewer|pname=xs28db8n9arz3123032|viewerconfig= #C [[ THUMBSIZE 3 ]]}}
|-
|-
| 8 || 0.5625 || 36 (nine blocks) || {{EmbedViewer|pname=9blocks|viewerconfig= #C [[ THUMBSIZE 3 ]]}}
| 8 || 0.5625 || 36<br/>nine blocks || {{EmbedViewer|pname=9blocks|viewerconfig= #C [[ THUMBSIZE 3 ]]}}
|-
|-
| 9 || 0.5309 || 43 || {{EmbedViewer|pname=43cell9x9pseudostilllifes|viewerconfig= #C [[ THUMBSIZE 3 ]]}}
| 9 || 0.5309 || 43 || {{EmbedViewer|pname=43cell9x9pseudostilllifes|viewerconfig= #C [[ THUMBSIZE 3 ]]}}

Revision as of 23:35, 27 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
x = 2, y = 2, rule = B3/S23 2o$2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBSIZE 3 ]]
(click above to open LifeViewer)
RLE: here Plaintext: here
3 0.6 6
ship
x = 3, y = 3, rule = B3/S23 b2o$obo$2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBSIZE 3 ]]
(click above to open LifeViewer)
RLE: here Plaintext: here
4 0.5 8
pond
long ship)
x = 4, y = 4, rule = B3/S23 b2o$o2bo$o2bo$b2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBSIZE 3 ]]
(click above to open LifeViewer)
RLE: here Plaintext: here

x = 4, y = 4, rule = B3/S23 2b2o$bobo$obo$2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBSIZE 3 ]]
(click above to open LifeViewer)
RLE: here Plaintext: here
5 0.64 16
four blocks
x = 5, y = 5, rule = B3/S23 2ob2o$2ob2o2$2ob2o$2ob2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBSIZE 3 ]]
(click above to open LifeViewer)
RLE: here Plaintext: here
6 0.5 18
two long shillelaghs
trans-snake-tie with ship)
x = 6, y = 6, rule = B3/S23 2o2b2o$o2bobo$b2o$3b2o$obo2bo$2o2b2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBSIZE 3 ]]
(click above to open LifeViewer)
RLE: here Plaintext: here

x = 6, y = 6, rule = B3/S23 2o2b2o$o2bobo$bob2o$2o$2b2obo$2bob2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBSIZE 3 ]]
(click above to open LifeViewer)
RLE: here Plaintext: here
7 0.5714 28
x = 7, y = 7, rule = B3/S23 ob2ob2o$2obobo$3bo2bo$3ob3o$o2bo$bobob2o$2ob2obo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBSIZE 3 ]]
(click above to open LifeViewer)
RLE: here Plaintext: here
8 0.5625 36
nine blocks
x = 8, y = 8, rule = B3/S23 2ob2ob2o$2ob2ob2o2$2ob2ob2o$2ob2ob2o2$2ob2ob2o$2ob2ob2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBSIZE 3 ]]
(click above to open LifeViewer)
RLE: here Plaintext: here
9 0.5309 43
x = 89, y = 29, rule = B3/S23 2o2b2ob2o11b2o2b2ob2o11bob2ob2obo11bob2ob2obo11bob2obob2o$o2bobob2o11b o2bobob2o11b2obobob2o11b2obobob2o11b2obob2obo$bobobo15bobobo17bobo17bo bo17bo$2obob4o11b2obob4o11b2obob4o11b2obob4o11b3ob4o$o2bo4bo12bobo4bo 12bobo4bo11bo2bo4bo11bo2bo4bo$bob4o14bob4o14bob4o14bob4o14bobob4o$2o4b 2o12b2o4b2o12b2o4b2o12b2o4b2o12b2obobo$o2b2o3bo11bo2b2o3bo11bo2b2o3bo 11bo2b2o3bo11bo2bobob2o$b2ob2ob2o12b2ob2ob2o12b2ob2ob2o12b2ob2ob2o13b 2ob2obo12$ob2ob2obo11bob2ob2obo11bob2ob2obo11bob2ob2obo11bob2ob2obo$2o bobob2o11b2obobob2o11b2obobob2o11b2obobob2o11b2obobob2o$3bobo17bobo17b obo17bobo17bobo$2obob4o11b4ob4o11b2obob4o11b4o2b3o11b4ob4o$2obo4bo11bo 7bo11bob2o4bo11bo3b2o2bo11bo7bo$3b5o13bob5o16b4o13b3o2b2o13b7o$3o17b2o bo17b3o20bo$o2bob2obo11bo2bob2obo11bo2bob2obo11bob2ob2obo11bob2ob2obo$ 2b2obob2o13b2obob2o11b2o2b2ob2o11b2obobob2o11b2obobob2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ THUMBSIZE 3 ]]
(click above to open LifeViewer)
RLE: here Plaintext: here
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.[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.
  • The density of ash is approximately 0.0287 (1 in 35).[2]
  • The density of a space dusty oscillator or spaceship is usually between 0.25 and 0.3.[citation needed]

References

  1. ↑ Noam D. Elkies (May 31, 1999). "The still-Life density problem and its generalizations". Retrieved on May 26, 2022.
  2. ↑ Oscar Cunningham (January 17, 2018). Re: Life Thermodynamics (discussion thread) at the ConwayLife.com forums
Retrieved from "https://conwaylife.com/w/index.php?title=Density&oldid=108027"