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 tile0 0 0 0 dimer1 0 0 0 | 2 0 0 0 cross1 1 1 1 | 2 0 0 0 catherine wheel1 2 3 0 | 4 0 0 0 T structure in video3 1 0 1 | 4 0 1 0 | 2 0 0 0 extended catherine wheel1 2 3 0 | 4 0 5 0 | 6 0 0 0 infinite line1 0 2 0 infinite tiling1 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 structure1 1 1 1 | 2 0 0 0 | 2 0 0 0 nondeterministic but likely bound structure1 0 2 0 | 1 2 0 0 nondeterministic unbound structure1 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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