One cell thick pattern

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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(phiy). Callahan employed this optimisation in his search for unidimensional infinite-growth patterns.

Infinite growth

Generation 2000 of Callahan's pattern
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
Download RLE: click here

Indeed, this pattern produces two block-laying switch engines at about generation 700. The image to the right shows what it looks like at generation 2000.

In April 2011, Stephen Silver constructed a unidimensional pattern based on a breeder by Nick Gotts, which displays quadratic growth.[2]

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 y=3 case, which is simply the blinker (the only known 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
RLE: 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[3]). The shortest such pattern that creates a toad has length 96, and the shortest such pattern that creates a spark coil has length 102. The shortest such pattern that produces an escaping glider has a width of 56.[4] It produces four gliders at about generation 100.

Unidimensional pattern that produces four gliders
Download RLE: click 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 Sierpinski Triangle. In the process of making those triangles, it will send away several gliders and form bi-blocks in the middle of the triangles.

Generation 100,000 of a row of 60,000 live cells

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.[5]

Paul Callahan's unidimensional tumbler predecessor
Download RLE: click here

The following 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
Download RLE: click here

The smallest one cell thick pattern to emit a glider contains 12 live cells and fits in a 15×1 bounding box.[6] It runs for 3183 generations before stabilizing and has a final population of 1059 cells.

A unidimensional pattern that emits six gliders
RLE: here

Spaceships and oscillators

It is unknown whether or not there exists a spaceship that is one cell thick in one of its phases, 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 oscillator that is one cell thick in one or more of its phases, 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.[7]

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.

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 of length 4; otherwise it would consist of blinkers and disappearing sparks. For example, this rules out the pattern ooooo.ooo, which becomes a single blinker.

References

  1. Eric Weisstein. "Infinite Growth". Eric Weisstein's Treasure Trove of Life C.A.. Retrieved on May 27, 2009.
  2. Stephen Silver. "Quadratic population growth from one row of cells". Heinrich Koenig, LifeNews weblog. Retrieved on June 9, 2013.
  3. Richard Hendricks (June 20, 2008). "Sierpiński triangle in Life". Retrieved on June 16, 2009.
  4. "One cell thick patterns". ConwayLife.com forums (March 29, 2009). Retrieved on May 27, 2009.
  5. Jason Summers' jslife pattern collection.
  6. "Unidimensional Patterns (2)". Infinite Growth Weblog (April 2, 2009). Retrieved on May 27, 2009.
  7. E-mail sent from Allan Wechsler to LifeList

See Also