User:H. V. McIntosh/Herparium

Examination of the patterns exhibited in the numerous catalogs of Life artifacts readily leads to the conclusion that patterns frequently occur in families. While designs resulting from mere repetition of smaller patterns are routinely excluded from catalogs, there are types of close contact impeding further dissection yet still preserving a discernible shape. Although often referred to as extensible patterns, the use of regular expressions permits an exact yet compact description of the family. They may be discerned empirically once having examined several configurations, or be found more systematically by examining loops in de Bruijn diagrams.

The Snake is a common, seemingly innocent, lifeform which nevertheless participates in at least three kinds of extensions; moreover they may be mixed, resulting in ever more elaborate snakes -- long snakes, siamese snakes, and so on. Several extensions and their regular expressions are discussed below.

Image gallery

Horizontal snake extension

• 

  a simple lonely snake

• 

  but here's another

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  and even a siamese pair

From this simple sequence a pattern can already be discerned. The nucleus, N, of the pattern is a square with occupied cells on its diagonal and empty cells on the antidiagonal. Those squares are bounded to the left and right by vertical live dominoes D, but only one single domino separates two succesive nuclei. Therefore the regular expression D (N D)*, equivalently (D N)* D, represents a snake with an arbitrarily long row of segments (vertebrae?). However, * = 0 leaves only a single standing domino which is not a still life; regular expression syntax would avoid this possibility by writing + rather than *, iteration then starting from 1, not 0.

Diagonal snake extension

• 

  a simple lonely snake

• 

  but this one's grown a bit

• Longlongsnake.png

  and still some more

This time the nucleus is a single live cell, s, but there are two end pieces, an upper 3/4 block L and a lower 3/4 block R; altogether producing the regular expression L s* R. This time * = 0 is valid, correctly revealing the primordial snake. Interestingly, when Adam named the animals, it is this menangerie which he seems to have had in mind.

Diagonal ship extension

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  a lonely ship

• 

  but this one's grown a bit

• Longlongship.png

  very long ship

Diagonal tub extension

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  a lonely tub

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  but now it's a barge

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  and a barge for ever after

The difference between ships and snakes is the difference between a double or single diagonal; both have the same tail and head, which are just differently notched blocks. A doubly notched block is also feasible, leading to the tub-barge-long barge-longer barge sequence. In turn, selection of the body and the terminals determines the alphabet to be used by the regular expression describing the combination. The actual tiles represented by an alphabet of roman letters can be connected in various ways, all resulting in a linear structure or frieze. So points of contact must be designated along with the shape of the tiles.

Oblique aircraft carrier extension

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  a ship once again

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  split by a diagonal skip

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  then by repeatable spacers

Possible connections already have names: bridge, tie, weld, but even these need to be specified in greater detail. Every exterior boundary edge in a still life is a potential edge, just as every exterior vertex could be the site of a tie. More precisely, if there are only two live neighbors of a cell just inside the boundary, that cell could also live if there were an additional live neighbor right outside the boundary.

Shims give additional flexibility to description by regular expression; although lacking cells they can carry points for ties to other tiles or marked edges for bridge placement. By marking a strategic tie point on the two bounding 3/4 blocks, L and R, a single live cell with diagonally opposite tie points s will generate snakes via the regular expression L s* R, although * = 0 just gives back a ship which might not want to be called a snake. A three cell high vertical line H with the top line left sensitized and the bottom segment right sensitized will generate a stairstep. Together with a welded RL tread T = (0 1 1, 1 1 0), they will generate the relatively unnamed aircraft carrier sequence L (H T)* H R.

Parseltongue

Several letters for a regular expression alphabet have now been introduced:

• L is a notched block w/ notch at lower right; tips tie sensitive, right edge bridge sensitive
• R is a notched block w/ notch at upper left; tips tie sensitive, left edge bridge sensitive
• s is a single live cell; opposite diagonal corners tie sensitive
• S is twin live cells, joined antidiagonally; diagonally opposite corners tie sensitive
• T is a tread; equivalent to a welded LR
• H is a riser; three cells high, top bridges to left, bottom to right
• D is a vertical domino;

Introducing a backspace could induce welding; in any event there is enough grammar present to allow a parselmouth to speak with a forked tongue. So far the discussion has not exhausted the possible letters for an extended alphabet, but enough has been shown to recognize many families amongst the catalogued still life patterns.