Polyomino Assembly By Iain Johnston, University of Birmingham; please see (and cite!) Phys. Rev. E 83 066105 (2011) [link; free link] and Phys. Rev. E 82 026117 (2010) [link; free link].

Enter a "ruleset" in the box below, and click Assemble. Rulesets are series of numbers (separated by spaces) describing the "colours" on the four edges of a set of square tiles. For example, the ruleset 1 1 1 1 | 2 0 0 0 describes a set of two tiles, one with colour 1 (say yellow) on each of its edges, and one with colour 2 (say green) on one edge and all other edges blank. If green bonds to yellow, shaking a pool of these tiles together will produce cross-shaped structures. We use bonds 1<->2, 3<->4, etc, with 0 always neutral (nonbonding).

Assembly Grid
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Assembly Log

Ruleset:


Fast Slow
Don't log Log only UND events Keep log (may be slow for large polyominoes)
Small grid Big grid
Some examples: (click links to use ruleset)
single tile
0 0 0 0
dimer
1 0 0 0 | 2 0 0 0
cross
1 1 1 1 | 2 0 0 0
catherine wheel
1 2 3 0 | 4 0 0 0
T structure in video
3 1 0 1 | 4 0 1 0 | 2 0 0 0
extended catherine wheel
1 2 3 0 | 4 0 5 0 | 6 0 0 0
infinite line
1 0 2 0
infinite tiling
1 1 2 2
IC!
0 1 0 0 | 2 13 3 0 | 4 0 5 0 | 6 0 7 0 | 8 0 9 0 | 10 23 37 0 | 14 0 15 0 | 16 37 17 37 | 18 0 19 0 | 20 0 21 0 | 22 37 0 37 | 24 0 25 0 | 26 27 0 37 | 28 0 0 29 | 30 0 31 0 | 32 0 33 0 | 34 0 0 35 | 36 0 37 0 | 38 0 0 0
infinite tiling (two block)
1 1 1 1 | 2 2 2 2
nondeterministic fixed structure
1 1 1 1 | 2 0 0 0 | 2 0 0 0
nondeterministic but likely bound structure
1 0 2 0 | 1 2 0 0
nondeterministic unbound structure
1 3 0 2 | 2 4 0 0
UoB! (needs big grid)
0 1 0 0 | 0 2 0 3 | 0 5 41 4 | 0 7 0 6 | 0 9 0 8 | 0 11 0 10 | 0 13 0 12 | 0 15 67 14 | 0 17 0 16 | 0 19 0 18 | 0 21 0 20 | 0 23 43 22 | 0 25 0 24 | 0 0 0 26 | 0 39 0 42 | 33 0 31 40 | 0 32 0 29 | 0 30 27 0 | 0 75 0 28 | 0 35 0 34 | 0 0 37 36 | 0 75 0 38 | 0 76 0 23 | 0 69 0 68 | 71 0 0 70 | 0 0 73 72 | 0 71 0 74 | 0 45 0 44 | 47 0 51 46 | 0 49 0 48 | 0 0 25 50 | 0 52 53 0 | 0 55 0 54 | 23 57 0 56 | 0 59 0 58 | 61 0 0 60 | 0 63 0 62 | 0 65 0 64 | 25 0 0 66
Some videos:


Some questions:
  • Can you design a ruleset to produce the first letter of your name?
  • What features of a ruleset produce unbound polyominoes?
  • What features of a ruleset produce non-deterministic polyominoes?
  • What do you notice about the colours around the edge of a bound structure?
  • What's the biggest bound, deterministic polyomino you can produce with two tiles?
  • What kind of structures can be encoded with few tiles, and what structures need many? Think about the symmetry of the cross compared to the "IC" structure.
  • We're assuming that the first tile in the ruleset is always placed first. What changes if a randomly-chosen tile is placed first?
  • All the grey structures in the top figure on the right can be produced using just two tiles. Can you find the rulesets for each?
  • The blue pictures in the bottom figure on the right show some types of unbound and/or non-deterministic structures that can be formed using only two tiles. Which can you reproduce?
Two-tile structures:


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