<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; to n=4 this is the optimal solution, as can be seen in the above table.
*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.
*The maximum density of an nxn still life is greater than or equal to 1/2 for all finite n>1.{{citation needed}}
== Densities of other objects ==
== Densities of other objects ==
Line 93:
Line 94:
{{LinkLexicon|lex_d.htm#density}}
{{LinkLexicon|lex_d.htm#density}}
*[https://conwaylife.com/book/ Conway's Game of Life: Mathematics and Construction, Section 2.5-Still Life Density]
*[https://conwaylife.com/book/ Conway's Game of Life: Mathematics and Construction, Section 2.5-Still Life Density]
*[https://golhobby.com/patman Maximum Density Still Lifes-PatMan pattern collection (over 200 RLEs)]
*[https://golhobby.com/patman Maximum Density Still Lifes-PatMan pattern collection] all n=2...200
*[https://oeis.org/A055397 OEIS: A055397-Maximum population of an n X n stable pattern in Conway's Game of Life]
*[https://oeis.org/A055397 OEIS: A055397-Maximum population of an n X n stable pattern in Conway's Game of Life]
Revision as of 18:58, 5 July 2022
The density of a pattern is the proportion of live cells within its bounding box.
The general problem of finding the maximum density of an nxn square pseudostill life was solved for all n in 2012.[1] The maximum densities of square still lifes up to 20×20 were known ten years earlier.[2]
The following table contains maximum density still life information for side lengths through 25.
†After the first few terms, the numbers in this sequence closely approximate the triangular number sequence.[3]
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.
The maximum density of an nxn still life is greater than or equal to 1/2 for all finite n>1.[citation needed]
Densities of other objects
The maximum density of a still life agar was proven to be 1/2 by Noam Elkies[4] (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.[5]