Lake

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A lake is a still life that is a simple closed curve made from diagonally connected dominoes. The smallest example is the pond and the next smallest is lake 2. The seven smallest lakes are shown below. The number of alive cells in a lake is always divisible by 8, and for any multiple of 8 other than 16 there is a lake with that many alive cells. The number of distinct lakes with 8n alive cells for n = 1, 2, 3, ... is given by the sequence 1, 0, 1, 1, 4, 7, 31, 98, 446, 1894, 9049, 43151, ... (Sloane's OEISicon light 11px.png A156228).

Examples

The sequence of L's and R's notate which direction the edge of the still life turns, starting from the leftmost top domino and moving counterclockwise.

  • Pond/Lake 1 (8 cells) (LLLL)

  • Lake 2 (24 cells) (LLRLLRLLRLLR)

  • The 32-cell lake (LLRLLRLRLLRLLRLR)

  • Lake 3 (40 cells) (LLRLRLLRLRLLRLRLLRLR)

  • A 40-cell lake (LLRLLRLRLRLLRLLRLRLR)

  • A 40-cell lake (LLRLRLLRRLLRLRLRLLRR)

  • A 40-cell lake (LLRLLRLRLLRLRLLRLLRR)

Semilakes

Smaller still lifes can be based on lakes by contracting their extremal dominoes into single cells, two at a time. Contracting the pond creates the beehive and the tub. The small lake is a twice contracted lake 2. Larger lakes with complex outlines may be contracted several times, e.g. the 6th lake listed above yields 16 different semilakes.

Counting lakes

Finding the next terms of the OEIS sequence has proven a very difficult task, Nathaniel Johnston wrote a blog post with a C program, and there was a discussion regarding it.[1]

Construction

x = 4, y = 4, rule = B3/S23 b2o$o2bo$o2bo$b2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ HEIGHT 600 WIDTH 600 THUMBSIZE 3 ZOOM 60 ]]
Pond/Lake 1 (8 bits), constructible with 2 gliders
(click above to open LifeViewer)
Catagolue: here
x = 10, y = 10, rule = B3/S23 4b2o$3bo2bo$3bo2bo$b2o4b2o$o8bo$o8bo$b2o4b2o$3bo2bo$3bo2bo$4b2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ HEIGHT 600 WIDTH 600 THUMBSIZE 3 ZOOM 48 ]]
Lake 2 (24 bits), constructible with 10 gliders
(click above to open LifeViewer)
Catagolue: here
x = 13, y = 13, rule = B3/S23 7b2o$6bo2bo$6bo2bo$4b2o4b2o$3bo8bo$3bo8bo$b2o7b2o$o8bo$o8bo$b2o4b2o$3b o2bo$3bo2bo$4b2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ HEIGHT 600 WIDTH 600 THUMBSIZE 3 ZOOM 36 ]]
The 32-bit lake, constructible with 24 gliders[2]
(click above to open LifeViewer)
Catagolue: here
x = 16, y = 16, rule = B3/S23 7b2o$6bo2bo$6bo2bo$4b2o4b2o$3bo8bo$3bo8bo$b2o10b2o$o14bo$o14bo$b2o10b 2o$3bo8bo$3bo8bo$4b2o4b2o$6bo2bo$6bo2bo$7b2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ HEIGHT 600 WIDTH 600 THUMBSIZE 3 ZOOM 30 ]]
Lake 3 (40 bits), constructible with 20 gliders[3]
(click above to open LifeViewer)
Catagolue: here
x = 16, y = 16, rule = B3/S23 10b2o$9bo2bo$9bo2bo$7b2o4b2o$6bo8bo$6bo8bo$4b2o7b2o$3bo8bo$3bo8bo$b2o 7b2o$o8bo$o8bo$b2o4b2o$3bo2bo$3bo2bo$4b2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ HEIGHT 600 WIDTH 600 THUMBSIZE 3 ZOOM 30 ]]
A 40-bit lake, no known synthesis
(click above to open LifeViewer)
Catagolue: here
x = 16, y = 10, rule = B3/S23 4b2o4b2o$3bo2bo2bo2bo$3bo2bo2bo2bo$b2o4b2o4b2o$o14bo$o14bo$b2o4b2o4b2o $3bo2bo2bo2bo$3bo2bo2bo2bo$4b2o4b2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ HEIGHT 600 WIDTH 600 THUMBSIZE 3 ZOOM 30 ]]
A 40-bit lake, no known synthesis
(click above to open LifeViewer)
Catagolue: here
x = 16, y = 13, rule = B3/S23 7b2o$6bo2bo$6bo2bo$4b2o4b2o$3bo8bo$3bo8bo$b2o10b2o$o14bo$o14bo$b2o4b2o 4b2o$3bo2bo2bo2bo$3bo2bo2bo2bo$4b2o4b2o! #C [[ THUMBSIZE 2 THEME 6 GRID GRIDMAJOR 0 SUPPRESS THUMBLAUNCH ]] #C [[ HEIGHT 600 WIDTH 600 THUMBSIZE 3 ZOOM 30 ]]
A 40-bit lake, no known synthesis
(click above to open LifeViewer)
Catagolue: here


See also

References

  1. ↑ Elithrion (2009-02-06). Counting lakes (discussion thread) at the ConwayLife.com forums
  2. ↑ shinjuku (#4865264220) (August 14, 2023). Job triggered by Adam P. Goucher at GitLab Catagolue project.
  3. ↑ shinjuku (#4880764226) (August 16, 2023). Job triggered by Adam P. Goucher at GitLab Catagolue project.

External links


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