Methuselah

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A methuselah is a pattern that takes a large number of generations in order to stabilize (known as its lifespan) and becomes much larger than its initial configuration at some point during its evolution. There is no consensus on the exact definition,[1] but things that stabilize in less than 100 generations are not generally called methuselahs.

Generation 1103 of R-pentomino (excluding six gliders).

Martin Gardner defined methuselahs as patterns of fewer than ten cells that take longer than 50 generations to stabilize.[2] Some other interpretations allow for more cells while requiring a longer lifespan, or characterize the size of an initial configuration by the size of its bounding box instead of the number of cells. Others use more complex metrics to measure the "quality" of methuselahs (see Measuring methuselahs below).

The time when a pattern is considered to have stabilized is commonly agreed upon to be the first generation such that the pattern can be resolved into still lifes, oscillators and escaping spaceships, provided such a generation exists. For infinitely growing patterns, no agreed-upon definition is known, although the Life Lexicon describes a particular ark as stabilizing at generation 736692.[3] Most interpretations exclude such patterns.

There is no limit to the lifespan of a pattern with 8 or more cells, as a methuselah consisting of a glider heading towards an arbitrarily distant blinker can be trivially constructed. Therefore, patterns with excessively large bounding boxes are generally implicitly excluded.

Methuselahs which eventually disappear are known as diehards.

Examples

The smallest and most well-known methuselah is the R-pentomino, a pattern of five cells first considered by John Conway[4] that takes 1103 generations before stabilizing as a pattern of eight blocks, six gliders, four beehives, four blinkers, one boat, one loaf, and one ship. This methuselah is particularly notable since almost all other patterns of similar size stabilize within 10 generations.

Martin Gardner gave the first well-known definition of a methuselah along with some examples. Among the examples are pi-heptomino, thunderbird, B-heptomino and acorn.[5] The acorn, a pattern of seven cells developed by Charles Corderman, takes 5206 generations to stabilize.

Because they are very active, frequently-appearing methuselahs can be used as conduit objects. Known methuselahs of this type include B-heptomino, century, Herschel, pi-heptomino, queen bee, R-pentomino, and wing (also known as Block and glider).

Acorn
B-heptomino
Pi-heptomino

Soup searches

#O Adam P. Goucher, February 10, 2019 #C A methuselah with a lifespan of 47575 generations, found using apgsearch. x = 16, y = 16, rule = B3/S23 5b2ob2o2bob2o$2bo2bo4b3obo$o2b2obobob2o3bo$bob2o3b2o2b3o$3o8b2o2bo$bob 3o4b3ob2o$bob4obob3o$ob2obo2bo3b3o$2b4o2b4o2bo$6b2o2bobob2o$o3bo2bo2b 5o$2b4o3b7o$o2b2o2bob2ob3o$o2bo3b2o3bobo$2ob2obob8o$6o3bobo! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ WIDTH 480 HEIGHT 480 ]]
47575M (click above to open LifeViewer)
RLE: here Plaintext: here

Soup searching is a popular method of finding methuselahs fitting within a given bounding box. The Online Life-Like CA Soup Search, for example, collected the longest-lasting soups found using Nathaniel Johnston's search script. The longest-lasting soup found in this census was Fred, which takes 35426 generations to stabilize and was found by Schneelocke on May 15, 2010.[6]

Versions v4.54 and above of apgsearch report soups lasting at least 25,000 generations, allowing the results to be tabulated on Catagolue.[7] As of early 2019, the longest-lasting methuselah found using apgsearch takes 47575 generations to stabilize and was found by Adam P. Goucher on February 10, 2019.[8] Versions v4.69 and above also report diehards lasting at least 500 generations, referring to them as "messless methuselahs".[9][note 1]

Due to the difficulty of testing a soup's ash for stability, both of these censuses estimate the lifespan of methuselahs found.[note 2][note 3]

Measuring methuselahs

Various metrics have been proposed to measure methuselahs so as to reward patterns such as the R-pentomino and acorn while penalizing trivial examples such as the glider-and-blinker construction mentioned above. Oscar Cunningham suggested using the minimum covering polyplet size (MCPS) for this as a compromise between population and bounding box,[10] resulting in the L/MCPS metric, the quotient of the methuselah's lifespan and its MCPS.

Other quotient-based metrics include F/I, F/L, and L/I, with F, I, and L standing for final population, initial population, and lifespan respectively.

See also

Notes

  1. Methuselahs and diehards are only reported by apgsearch in symmetries of Conway's Game of Life.
  2. Long-lived soups found as part of TOLLCASS had their exact lifespan verified manually.
  3. apgsearch automatically tests the lifespan of a soup more precisely if its estimated lifespan is sufficiently high, but is not guaranteed to detect all methuselahs with a lifespan of less than 26,000 generations.

References

  1. Methuselah Definition (discussion thread) at the ConwayLife.com forums
  2. Gardner, M. (1983). "The Game of Life, Part III". Wheels, Life and Other Mathematical Amusements: 246, W.H. Freeman. 
  3. "Ark". The Life Lexicon. Stephen Silver. Retrieved on March 14, 2016.
  4. Gardner, M. (1983). "The Game of Life, Part III". Wheels, Life and Other Mathematical Amusements: 219, 223, W.H. Freeman. 
  5. Gardner, M. (1983). "The Game of Life, Part III". Wheels, Life and Other Mathematical Amusements: 246, W.H. Freeman. 
  6. "Long-Lived Patterns in Conway's Life". Online Life-Like CA Soup Search. Retrieved on March 2, 2019. (archived from the original)
  7. Adam P. Goucher. Re: apgsearch v4.0 (discussion thread) at the ConwayLife.com forums
  8. Ian07. Re: Soup search results (discussion thread) at the ConwayLife.com forums
  9. Ian07. Re: apgsearch v4.0 (discussion thread) at the ConwayLife.com forums
  10. Oscar Cunningham. Re: Largest and oldest methuselah ever found! (discussion thread) at the ConwayLife.com forums

External links