One-cell-thick pattern
A one-cell-thick pattern is a pattern that is only one cell thick; that is, it is contained entirely within one dimension of the Life plane. Put another way, it is a pattern with bounding box of the form y × 1 for some natural number y. Because of their size restriction, exhaustive computer searches have been carried out to explore unidimensional patterns up to size about 40 × 1. Despite their inherent limitations, unidimensional patterns can exhibit quite complex behavior, even at reasonably small sizes.
Naively, one would assume that searching all y × 1 patterns would require O(2y) time. However, all such patterns containing one-cell and two-cell islands can be discarded, which reduces the search time to O(φy).[note 1] Paul Callahan employed this optimisation in his search for unidimensional infinite-growth patterns.
Infinite growth
In May 1998, Stephen Silver produced a one-cell-thick pattern that exhibits infinite growth, following a conjecture of Nick Gotts that such patterns exist. This pattern was extremely large (12470 × 1 in the first version, reduced to 5447 × 1 the following day).[1]
In October 1998, Paul Callahan performed an exhaustive computer search to find the following pattern that exhibits infinite growth. It is one of the most well-known one cell thick patterns, and Callahan showed that it is the smallest such one cell thick pattern (in terms of its bounding box) to exhibit infinite growth. It contains 28 alive cells and has a 39 × 1 bounding box.
| Paul Callahan's one cell thick infinite growth pattern (click above to open LifeViewer) RLE: here Plaintext: here |
Indeed, this pattern produces two block-laying switch engines at about generation 700. The following image shows what it looks like at generation 2000:
Quadratic growth
In April 2011, Stephen Silver constructed a unidimensional pattern based on a breeder by Nick Gotts, over a million cells long, which displays quadratic growth.[2]
In November 2014, Chris Cain constructed a 7242 × 1 quadratic growth pattern.[3] In October 2015, he reduced this pattern to 2596 × 1.[4]
With all cells alive
Making the restriction that all cells in the unidimensional pattern must be alive (that is, the pattern is a y × 1 rectangle of alive cells) still leaves some interesting patterns. The majority of small patterns of this type quickly decay into constellations made up of blinkers, blocks and other common still lifes and oscillators, but some interesting cases are considered here. The simplest interesting such pattern is the case y = 3, which is simply the blinker (the only known finite oscillator that is one cell thick in one of its phases). The next interesting such pattern comes when y = 10, which rapidly evolves into a pentadecathlon.
| A pentadecathlon predecessor (click above to open LifeViewer) RLE: here Plaintext: here |
The 41 × 1 box creates four pulsars after about 200 generations -- this is the smallest one-cell thick pattern with all of its cells alive that creates an oscillator of period other than 2 or 15 (pulsars are also created by a line of length 135[5]). The shortest such pattern that creates a toad has length 96, and the shortest such pattern that creates a spark coil has length 72. The shortest such pattern that produces an escaping glider has a width of 56.[6] It produces four gliders at about generation 100.
| Unidimensional pattern that produces four gliders (click above to open LifeViewer) RLE: here Plaintext: here |
The period of the pattern that results from the evolution of a y × 1 rectangle for y = 1, 2, 3, ... is given by 1, 1, 2, 1, 2, 1, 1, 1, 2, 15, 2, 1, 2, 1, 1, ... (Sloane's 
A061342). As the length of the rectangle increases, its evolution becomes increasingly predictable. A long row of live cells will eventually form a shape resembling two copies of the Sierpiński triangle[7]. In the process of making those triangles, it will send away several gliders and form bi-blocks in the middle of the triangles.
Rectangles of size 1 × 1 (dot), 2 × 1 (domino), 6 × 1 (line-of-six spark), 14 × 1, 15 × 1, 18 × 1, 19 × 1, 23 × 1 and 24 × 1 die out completely when evolved. It is conjectured (and strongly believed) that these are the only such rectangles.
Other examples
Some other interesting examples of one cell thick patterns include the following predecessor of a tumbler, which was found during Paul Callahan's computer search in October 1998. It has 27 live cells and fits in a 36 × 1 bounding box.[8]
| Paul Callahan's unidimensional tumbler predecessor (click above to open LifeViewer) RLE: here Plaintext: here |
The following die hard pattern, also due to Callahan, has 66 live cells and fits in a 149 × 1 bounding box, yet dies out completely after 233 generations. Additionally, it is interesting because each half produces two gliders and two lightweight spaceships.
| Paul Callahan's unidimensional pattern that dies completely (click above to open LifeViewer) RLE: here Plaintext: here |
The smallest one cell thick pattern to emit a glider contains 12 live cells and fits in a 15 × 1 bounding box.[9] It runs for 3183 generations before stabilizing and has a final population of 1059 cells.
| A unidimensional pattern that emits six gliders (click above to open LifeViewer) RLE: here Plaintext: here |
On October 11, 2018, carybe found a 256 × 1 soup using apgsearch which produces a middleweight Schick engine along with some junk.[10]
| A reduced version (63 × 1) of the one-cell-thick Schick engine soup (click above to open LifeViewer) RLE: here Plaintext: here |
Methuselahs
In 2017, Simon Ekström discovered 14911M, the longest-lived unidimensional methuselah found to date.[11]
| Unidimensional methuselah with lifespan 14,911 generations (click above to open LifeViewer) RLE: here Plaintext: here |
Spaceships and oscillators
It is unknown whether or not there exists a spaceship that is one cell thick in one of its phases, (research has been conducted in finding such a spaceship[12]) though it can be shown via symmetry arguments that any such spaceship, if they exist, would have to move in the direction that it "points". The blinker is the only known finite oscillator that is one cell thick in one or more of its phases (tiling the plane with oooooo.. gives an infinite period-9 oscillator based on the worker bee), and in April 1992, Allan Wechsler used a search program to show that there are no oscillators of period 3, 5 or 7 that are one cell thick. The situation is unknown for other periods.[13] However, the pentadecathlon has a simple one cell thick predecessor, and the oscillator's evolution sequence involves it evolving into a pattern that evolves very similar to a one cell thick pattern.
A plausible strategy for building such unidimensional oscillators/spaceships is to create a synthesisable unidimensional pattern that is a predecessor to a universal constructor. Creating a unidimensional constructor predecessor is the easy part; synthesising it is much more difficult, as all sparks must disappear before it enters its unidimensional phase.
One cell thick patterns evolving into one cell thick patterns
A weaker problem is to find a unidimensional pattern that is the predecessor of another non-trivial unidimensional pattern. Non-trivial means that it must contain at least one line that is not the blinker, a spark, or a hypothetical one-cell-thick spaceship. For example, this rules out the pattern oo.o.oo.ooo, which becomes a single blinker without any of the separate line segments ever having the opportunity to interact. By contrast, the pattern ooooo.ooo also produces a single blinker, but the interaction between the two segments makes it a non-trivial unidimensional pattern.
In August 2016, toroidalet presented a unidimensional predecessor of a unidimensional pattern;[14] it evolves into a different one cell thick pattern in generation 9, and into two beehives in generation 22:
| (click above to open LifeViewer) RLE: here Plaintext: here |
A different example was given by M. I. Wright in August 2017,[15] based on a suitable reaction identified by Wojowu in January 2012;[16] their pattern evolves into different one-cell thick pattern (albeit on a different axis) in generation 128, and subsequently into two traffic lights in generation 148:
| (click above to open LifeViewer) RLE: here Plaintext: here |
See also
Notes
- ↑ Here, φ = 0.5 · (1 + 5½) ≈ 1.6180339887498948482... is the golden ratio.
References
- ↑ Dave Greene (December 21, 2022). Re: Can we substantiate this claim? (discussion thread) at the ConwayLife.com forums
- ↑ Quadratic population growth from one row of cells at Game of Life News. Posted by Dave Greene on May 07, 2011.
- ↑ Chris Cain (November 9, 2014). Re: Making switch-engines (discussion thread) at the ConwayLife.com forums
- ↑ Chris Cain (October 3, 2017). Re: Making switch-engines (discussion thread) at the ConwayLife.com forums
- ↑ Richard Hendricks (June 20, 2008). "Sierpiński triangle in Life". Retrieved on June 16, 2009.
- ↑ Oscar Cunningham (March 29, 2009). One cell thick patterns (discussion thread) at the ConwayLife.com forums
- ↑ Sierpiński triangle at Wikipedia
- ↑ Jason Summers' jslife pattern collection.
- ↑ "Unidimensional Patterns (2)". Infinite Growth Weblog (April 2, 2009). Retrieved on May 27, 2009.
- ↑ wwei23 (October 11, 2018). Re: Soup search results (discussion thread) at the ConwayLife.com forums
- ↑ Re: Long-lived methuselahs (discussion thread) at the ConwayLife.com forums
- ↑ Adam P. Goucher (February 14, 2016). How about a unidimensional spaceship? (discussion thread) at the ConwayLife.com forums
- ↑ E-mail sent from Allan Wechsler to LifeList
- ↑ toroidalet (August 7, 2016). Re: Thread For Your Useless Discoveries (discussion thread) at the ConwayLife.com forums
- ↑ M.I. Wright (August 20, 2017). Re: 1-Dimensional Patterns (discussion thread) at the ConwayLife.com forums
- ↑ Wojowu (January 13, 2012). Re: Thread For Your Accidental Discoveries (discussion thread) at the ConwayLife.com forums
External links
- 1-Dimensional Patterns (discussion thread) at the ConwayLife.com forums