Block

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Block
x = 2, y = 2, rule = B3/S23 2o$2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C Still life
Pattern type Strict still life
Number of cells 4
Bounding box 2 × 2
Frequency class 0.0
Discovered by John Conway
Year of discovery 1969

Block is an extremely well-known and common still life that was found by John Conway in 1970.[1] In terms of its 4 cells it is tied with tub as the smallest still life, and in terms of its 2 × 2 bounding box it is the outright smallest. It is also the only known still life that is a polyomino, and the only finite strict still life where all living cells have three neighbors. Its small size, ability to act as a catalyst, and simple glider syntheses make it extremely useful in the construction of larger patterns.

The block was one of the patterns described in the original 1970 article by Martin Gardner that introduced the Game of Life to the world.

Use in higher still lifes

Blocks serve useful in the construction of larger still lifes. Due to its high density, orthogonal and diagonal connections (like those seen in ship-tie) would overpopulate any present cells, preventing any patterns with this arrangement from being useful in a still life (this reaction can instead be used for oscillators such as beacon and variants of star). As such, its uses are limited to acting as an induction coil.

Given the high symmetry of the block (the highest a pattern can have on {4,4}), cis-, trans- and other isomeric variants only arise if the object the block is stabilising is itself of a sufficiently low symmetry.

Examples of known still lifes which use the block as an induction coil are as follows:

Blocks can also be used to stabilise certain oscillators in a non-catalytic fashion, with examples as follows:

Uses in catalysis

The block sees a diverse array of uses in larger patterns. For example, it can work as a reflector for two gliders via the interchange synthesis, as can be seen in 106P135.

There are multiple ways in which the block may act as a catalyst. These fall into roughly two camps, according to how the block will be regenerated (since the block's cells already have three neighbors, it cannot act as a rock, except through induction, such as the p47 pre-pulsar shuttle.). Oscillators such as blocker, p56 B-heptomino shuttle, queen bee shuttle, twin bees shuttle and unix showcase mechanisms which leave a pre-block or a hook, while the mechanisms seen in Coe's p8, eater 5 and octagon 4 leave a grin. A different mechanism yet, where only a single cell of the original block survives the whole ordeal, appears in 37P7.1. It has been seen eating mangos, boats, loaves, and beehives.

Some more complex eaters can be based on the block as well. Eater 2 is a construction allowing a block to eat gliders: while the collision would normally destroy both, the addition of further still life(s) or inductees allows the remaining three-cell spark to regenerate into a block.

Since the block is the most common object to emerge from soups, it is also the most common object to show transparent debris effect, as described above. Several larger oscillators including p54 shuttle, 78P70, two blockers hassling R-pentomino and Achim's p144 are examples. A particular reaction with a B-heptomino is commonly used in the construction of Herschel tracks.

Blockic constellation

Main article: Blockic

A blockic constellation is a constellation consisting entirely of blocks. It's possible to arrange blocks in a way that can be triggered by a single glider to produce any glider constructible pattern.[2]

Block arrays

Pseudo still lifes exist consisting of an arbitrary array of blocks[3] placed next to each other, which may be referred to as block agars. On December 3, 2015, Martin Grant found a method of constructing a 3 × n array via glider synthesis,[4] and on May 6, 2020, Goldtiger997 found a method of constructing a 4 × n array.[5]

Commonness

The block is the most common object in the Catagolue census, comprising 30.9% of all object occurrences.

In Achim Flammenkamp's census it occurred almost twice as often as beehive, the next most common still life. However, the blinker is slightly more common than the block in Achim's census, unlike on Catagolue. This can be explained by Herschels, which have a lot of empty space to evolve in 16 × 16 soups but not on a torus, leaving two blocks behind.[6]

Occurrence

The block is the most common object, and thus the most common still life, that occurs as a result of random starting patterns. As such, it is the reference element for calculating frequency class.

The chance of a block in a specific location in an infinite random soup is approximately 1 in 473. This number is the frequency of on cells (0.0287) divided by the average object size (4.388) times the chance that a particular object will be a block (0.323). As blocks have 8-way symmetry, there is no symmetry factor to take into account. This number is helpful in determining how often transparent blocks occur; transparent blocks are twice as common as transparent blinkers, almost four times as common as transparent beehives, and more than twenty times as common as anything else.

Glider synthesis

The block is the most common object produced by 2-glider collisions; three head-on instances and three perpendicular ones create a clean block, and there are also other examples that produce solely blocks (in the form of bi-block, half-blockade, four skewed blocks or blockade).

See also

References

  1. Dean Hickerson's oscillator stamp collection. Retrieved on March 14, 2020.
  2. Nathaniel Johnston and Dave Greene (March 11, 2022). "Conway's Game of Life: Mathematics and Construction, Chapter 5". Retrieved on May 25, 2022.
  3. Block Array Constructions at Game of Life News. Posted by Heinrich Koenig on April 20, 2005.
  4. Martin Grant (December 3, 2015). Re: Synthesising Oscillators (discussion thread) at the ConwayLife.com forums
  5. Goldtiger997 (May 6, 2020). Re: Synthesising Oscillators (discussion thread) at the ConwayLife.com forums
  6. Achim Flammenkamp (September 7, 2004). "Most seen natural occurring ash objects in Game of Life". Retrieved on January 15, 2009.

External links