Polyplet

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A polyplet (or simply a plet) is a finite collection of orthogonally or diagonally connected cells. The form of connectivity allowed by polyplets is sometimes called king-wise connectivity because of the way a king moves on a chess board. King-wise connectivity is a more natural concept in Life-like cellular automata than the orthogonal connectivity of polyominoes.

Sizes of polyplets

Polyplets with n cells for n = 3, 4, 5, ... are called triplets, tetraplets, pentaplets, hexaplets, heptaplets, and n-plets in general. The number of distinct polyplets with n cells for n = 1, 2, 3, ... is given by the sequence 1, 2, 5, 22, 94, 524, 3031, 18770, 118133, ... (Sloane's OEISicon light 11px.pngA030222).

Triplets

There are exactly five distinct triplets, shown below. The first two are the two triominoes (pre-block and blinker), and the other three vanish after two generations (being the banana spark, V spark and a fuse of length 3 respectively).

The five distinct triplets.
Download RLE: click here

Tetraplets

Of the 22 distinct tetraplets:

The twenty-two distinct tetraplets.
Download RLE: click here

Pentaplets

The 94 distinct pentaplets evolve as follows:

  • 39 into nothing
  • 11 into loaves
  • 10 into traffic lights
    • 6 in the phase with "+" on even generations
    • 4 in the phase with "o" on even generations
  • 9 into beehives
  • 7 into boats (including the boat itself)
  • 4 into blocks
  • 4 into gliders (including the two that are themselves phases of the glider)
  • 3 into ponds
  • 2 into tubs
  • 1 into the pi heptomino (after two generations; the first generation evolution is the "<" hexomino)

The remaining four are or become the R-pentomino:

  • one is the R-pentomino itself
  • one becomes the R-pentomino after one generation
  • two become the aforementioned pentaplet after one generation (and thus become the R-pentomino after two).

Hexaplets

Lifeline Volume 4 analysed all six-cell patterns. Of the 524 that are polyplets, here are the results:

  • 171 into nothing
  • 38 into traffic lights
  • 34 into loaves
  • 33 into beehives (including the beehive itself)
  • 30 into blocks
  • 23 into honey farms
  • 21 follow the pi-heptomino sequence
  • 20 into blinkers
  • 19 into boats
  • 19 follow the lumps of muck sequence
  • 17 into ponds
  • 17 follow the R-pentomino sequence
  • 10 follow the block and glider sequence
  • 9 follow the century sequence
  • 7 into ships (including the ship itself)
  • 6 into barges (including the barge itself)
  • 6 into toads (including the toad itself)
  • 6 into beacons
  • 5 follow the teardrop sequence
  • 5 into gliders
  • 5 follow the B-heptomino sequence
  • 4 follow the I-heptomino sequence
  • 3 into two asymmetric loaves
  • 3 follow the two-glider octomino sequence
  • 2 into the long boat, both shown on the long boat page
  • 2 into the moth sequence[1]
  • 1 (a hexomino) into aircraft carrier
  • 1 (a hexomino) into two far blinkers
  • 1 is the snake
  • 1 is the clock
  • 1 follows the butterfly sequence
  • 1 follows the dove sequence
  • 1 is a Herschel without the B's block
  • 1 is the Iwona active region
  • 1 is Thompson's methuselah (3o$3bo$3b2o!)

Tubs are absent from this list.

References

  1. David Raucci (April 4, 2022). Re: Thread for basic questions (discussion thread) at the ConwayLife.com forums

External links