One-cell-thick pattern
A unidimensional 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.
Infinite growth
In May 1998, Stephen Silver produced a unidimensional 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 probably the most well-known unidimensional pattern, and Callahan showed that it is the smallest such unidimensional pattern (in terms of its bounding box) to exhibit infinite growth. It contains 28 alive cells and has a 39×1 bounding box.
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.
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 simplest example is the y=3 case, which is simply the blinker (the only known oscillator that is unidimensional in one of its phases). The next interesting such pattern comes when y=10, which rapidly evolves into a pentadecathlon.
The shortest unidimensional pattern with all of its cells alive that produces an escaping glider has a width of 56.[2] It produces four gliders at about generation 100.
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.
Other examples
Some other interesting examples of unidimensional 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.[3]
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.
The smallest unidimensional pattern to emit a glider contains 12 live cells and fits in a 15×1 bounding box.[4] It runs for 3183 generations before stabilizing and has a final population of 1059 cells.
References
- ↑ Eric Weisstein. "Infinite Growth". Eric Weisstein's Treasure Trove of Life C.A.. Retrieved on May 27, 2009.
- ↑ "One cell thick patterns". ConwayLife.com forums (March 29, 2009). Retrieved on May 27, 2009.
- ↑ Jason Summers' jslife pattern collection.
- ↑ "Unidimensional Patterns (2)". Infinite Growth Weblog (April 2, 2009). Retrieved on May 27, 2009.