User:Tropylium/Fuse

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Within the context of still lifes, a string is a pattern with a connectivity graph of the shape 8>o-o-(…)-o<8 (ie. with a string of S2 cells stabilized by two "caps"). All string stlifes are primitive.

Indefinitely extendable fuse bodies are common feature of strings. Different structures can be seen in patterns such as hat and loop.

Any fuse body must terminate in a two-cell cap called hook, or join by a turn with another object for stability. Two commonly occurring four-cell terminations are tail, which can be analyzed as a turn + a degenerate 1-cell fuse + hook; and claw, which is a turn + degenerate length-1 boat. These are best treated as independant endings:

  • The "fuse" inside a tail cannot be extended, if joined into a fuse of length > 2.
  • A length-1 boat is not an object by itself and so any string terminated with a claw will be primitive, unlike strings terminated with a longer separate object.

There is also a way combine two parallel fuse bodies three lanes apart, using a turn (notation: u). Both fuses cannot be extended indefinitely; the only way to extend the more ahead one beyond 2 cells is to loop the still life back on itself: a loaf is an u2u3 loop, and a mango is an u3u3 loop. The 2nd fuse can also be replaced with a vessel. Two turns may be combined with a dummy fuse in-between to create a five-lane offset, in two ways. -u1u- is a trimino, while -u2u- is may be called kink.

While a string can be described using turns, hooks and fuse bodies alone, the severe restrictions on the placement of turns means their occurrences can largely be condensed into ___. The following grammar may be used to describe most primitive strings:

  • h: hook
  • L: tail (= h1u)
  • C: claw (= B1u)
  • T: trimino (= u1u)
  • K: kink (= u2u)

A few still lifes can be described under this notation in two ways though: the simplest case is fuse with two tails, h1u2u1h, which can be condensed to either h1K1h or L2L. Similarly, h1u2u1u2u1h will be either h1K1K1h or L2T2L.

Examples:

  • 6 cells: snake (h2h)
  • 7 cells: eater (h1C), long snake (h3h)
  • 8 cells: hook with tail (h2L), shillelagh (h2C), canoe (c-h4h), long long snake (h4h)
  • 9 cells: integral (C1C), long shillelagh (h3C)
  • 10 cells: long integral (C2C), trans-fuse with two tails (L2L), long² shillelagh (h4C), cis shillelagh (c-h4C),