Density: Difference between revisions

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| 4 || 0.5 || 8 ([[pond]], [[long ship]]) || {{EmbedViewer|pname=pond}}<br>{{EmbedViewer|pname=longship}}
| 4 || 0.5 || 8 ([[pond]], [[long ship]]) || {{EmbedViewer|pname=pond}}<br>{{EmbedViewer|pname=longship}}
|-
|-
| 5 || 0.64 || 16 (four blocks) || {{EmbedViewer|rle=2ob2o$2ob2o2$2ob2o$2ob2o!}}
| 5 || 0.64 || 16 (four blocks) || {{EmbedViewer|pname=4blocks}}
|-
|-
| 6 || 0.5 || 18 (eg. two [[long shillelagh]]s) || {{EmbedViewer|pname=twolongshillelaghs}}<br>{{EmbedViewer|rle=2o2b2o$o2bobo$bob2o$2o$2b2obo$2bob2o!}}
| 6 || 0.5 || 18 (two [[long shillelagh]]s) || {{EmbedViewer|pname=twolongshillelaghs}}
|-
|-
| 7 || 0.5714 || 28 || {{EmbedViewer|rle=ob2ob2o$2obobo$3bo2bo$3ob3o$o2bo$bobob2o$2ob2obo!}}
| 7 || 0.5714 || 28 || {{EmbedViewer|pname=Xs28db8n9arz3123032}}
|-
|-
| 8 || 0.5625 || 36 (eg. nine blocks) || {{EmbedViewer|rle=2ob2ob2o$2ob2ob2o2$2ob2ob2o$2ob2ob2o2$2ob2ob2o$2ob2ob2o!}}
| 8 || 0.5625 || 36 (nine blocks) || {{EmbedViewer|pname=9blocks}}
|-
|-
| 9 || 0.5309 || 43 || {{EmbedViewer|rle=2o2b2ob2o11b2o2b2ob2o11bob2ob2obo11bob2ob2obo11bob2obob2o$o2bobob2o11b
| 9 || 0.5309 || 43 || {{EmbedViewer|pname=densitypoint5309examples}}
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!}}
|-
|-
| 10 || 0.5400 || 54
| 10 || 0.5400 || 54

Revision as of 22:19, 24 October 2021

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 Pattern
1 0 0
2 1 4 (block)
x = 2, y = 2, rule = B3/S23 2o$2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]]
(click above to open LifeViewer)
RLE: here Plaintext: here
3 0.6̄ 6 (ship)
x = 3, y = 3, rule = B3/S23 b2o$obo$2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]]
(click above to open LifeViewer)
RLE: here Plaintext: here
4 0.5 8 (pond, long ship)
x = 4, y = 4, rule = B3/S23 b2o$o2bo$o2bo$b2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]]
(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 ]]
(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$2ob2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]]
(click above to open LifeViewer)
RLE: here Plaintext: here
6 0.5 18 (two long shillelaghs)
x = 6, y = 6, rule = B3/S23 2o2b2o$o2bobo$b2o$3b2o$obo2bo$2o2b2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]]
(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 ]]
(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 ]]
(click above to open LifeViewer)
RLE: here Plaintext: here
9 0.5309 43
[[:RLE:Densitypoint5309examples]] #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]]
(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

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 is proven to be 1/2.
  • The maximum average density of an oscillating agar is conjectured but not proven to be 1/2.
  • The density of ash is 0.0287 (1 in 35).
  • The density of a space dusty oscillator or spaceship is usually between 0.25 and 0.3.

References

Retrieved from "https://conwaylife.com/w/index.php?title=Density&oldid=83581"