User:Tropylium/Tie

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A tie is a stationary pattern (a still life or an oscillator) that can be partitioned into two parts that are connected to each other, but are still stable stationary patterns on their own. The points of connection may be, at least, stable cells with 2 neighbors in the isolated half, 3 when connected to their counterpart; two spark cells joining to form a larger spark; or a pair of dying edge cells with 4 neighbors in the isolated half, 6 when connected to their counterpart.

It is not necessarily unambiguous how a pattern is a tie. The simplest example is the long long barge which can be considered a tub-barge tie in two different ways, despite not being a 3-element tie. The simplest example divisible in ways yielding different constituents is the long^3 boat, divisible into a boat and barge, or a long boat and a tub. In general, any long^n vessel (barge, boat or ship) with n ≥ 3 can be partitioned into smaller vessels. (Note that since the barge is a long tub, the long long barge is still a long^3 vessel.) Another kind of example is long barge with four tails, divisible lengthways into two cis-fuses with two tails, or crossways into two D2d tubs with two tails.

Small ties

These are all still lifes; the smallest oscillator ties are probably boat tie bipole (13 cells, 1-bit tie) and mold with bi-loaf (19 cells, 2-bit tie).

Orthogonal ties:

  • 12 cells (6+6): snake link snake (name?)
  • 13 cells (6+7): snake link eater (?), snake link long snake
  • 14 cells
    • 6+8: snake link canoe, snake link hook with tail, snake link long eater, snake link long long snake
    • 7+7: eater link eater, eater link long snake, long snake link long snake

("Snake link snake" looks like this:)

O.OO...
OO.O...
...O.OO
...OO.O

One-point diagonal ties:

  • 10 cells (5+5): Boat-tie
  • 11 cells (5+6): Boat-ship-tie, boat tie snake, boat tie carrier
  • 12 cells
    • 5+7: Boat tie eater (two isomers), boat tie long snake, boat tie long boat
    • 6+6: Ship-tie, ship tie snake, ship tie carrier, snake-tie (D2d and S2 isomers), snake bridge snake, snake tie carrier, snake bridge carrier, carrier-tie (D2d and S2 isomers), carrier bridge carrier

Odder connections: long canoe, vlong eater (similarly for 2-bit connections etc.)

.O..OO  .O....
O.O..O  O.O...
.OO.O.  .OO...
...O..  ...OO.
O.O...  O.O..O
OO....  OO..OO

Two-point diagonal ties other than long^n barges/boats/ships:

  • 12 cells (6+6): Bi-beehive
  • 13 cells (6+7): Beehive at loaf
  • 14 cells
  • 15 cells
    • 6+9: Beehive at hat, beehive at hook with long tail
    • 7+8: Loaf at pond, loaf at mango, loaf at hook with tail
  • 16 cells
    • 6+10: Bi-beehive with tail (2 isomers), beehive at loop, beehive at hook with vlong tail (cis, up trans, and down trans isomers), beehive at claw with tail, beehive at hook with hook-tail, beehive at fuse with two tails
    • 7+9: Loaf at hat, loaf at hook with long tail
    • 8+8: Bi-pond, bi-mango 1, bi-hook-with-tail, mango at hook with tail

Three-point diagonal ties:

  • 14 cells (7+7): Capsule (name?)
  • 16 cells (8+8): Bi-mango 2
  • 19 cells (8+11): Mango at eleven loop

Tie hypothesis

As N → ∞, the number of strict still lifes of N cells that are ties grows faster than the number of strict still lifes that are not.

(Probably shouldn't call this a "conjecture" just yet.)