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Self-Assembly with Geometric Tiles

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  1. ICALP 2012 Self-Assembly with Geometric Tiles • Bin Fu University of Texas – Pan American Matt Patitz University of Arkansas Robert Schweller (Speaker) University of Texas – Pan American Robert Sheline University of Texas – Pan American

  2. Outline • Basic Tile Assembly Model • Geometric Tile Assembly Model • Basic Model • Planar Model • More efficient n x n squares • Future Directions

  3. Tile Assembly Model (Rothemund, Winfree, Adleman) a b c x d e G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 T = Glue Function: Tile Set: Temperature:

  4. Tile Assembly Model (Rothemund, Winfree, Adleman) a b c x d e G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 T = e d

  5. Tile Assembly Model (Rothemund, Winfree, Adleman) a b c x d e G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 T = e d

  6. Tile Assembly Model (Rothemund, Winfree, Adleman) a b c x d e G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 T = e d b c

  7. Tile Assembly Model (Rothemund, Winfree, Adleman) a b c x d e G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 T = e d b c

  8. Tile Assembly Model (Rothemund, Winfree, Adleman) a b c x d e G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 T = e d b c

  9. Tile Assembly Model (Rothemund, Winfree, Adleman) a b c x d e G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 T = e d a b c

  10. Tile Assembly Model (Rothemund, Winfree, Adleman) a b c x d e G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 T = e d a b c

  11. Tile Assembly Model (Rothemund, Winfree, Adleman) a b c x d e G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 T = e d a b c

  12. Tile Assembly Model (Rothemund, Winfree, Adleman) a b c x d e G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 T = e d a b c

  13. Tile Assembly Model (Rothemund, Winfree, Adleman) a b c x d e G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 T = e d a b c

  14. Tile Assembly Model (Rothemund, Winfree, Adleman) a b c x d e G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 T = e x d a b c

  15. Tile Assembly Model (Rothemund, Winfree, Adleman) a b c x d e e x d a b c G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 T =

  16. Tile Assembly Model (Rothemund, Winfree, Adleman) a b c x d e e x x d a b c G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 T =

  17. Tile Assembly Model (Rothemund, Winfree, Adleman) a b c x d e x e x x d a b c G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 T =

  18. Tile Assembly Model (Rothemund, Winfree, Adleman) a b c x d e x x e x x d a b c G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 T =

  19. Geometric Tile Model

  20. Geometric Tiles Geometry Region

  21. Geometric Tiles Geometry Region

  22. Geometric Tiles Compatible Geometries

  23. Geometric Tiles

  24. Geometric Tiles Incompatible Geometries

  25. Geometric Tiles Incompatible Geometries

  26. n x n Results Tile Complexity Upper bound Lower bound Normal Tiles* Geometric Tiles Planar Geometric Tiles [*Winfree, Rothemund, Adleman, Cheng, Goel,Huang STOC 2000, 2001]

  27. n x n Squares, root(log n) tiles 0 1 0 1 1 log n

  28. Assembly of n x n Squares 1 1 1 1 1 1 1 1 1 0 n 0 1 1 0 0 0 1 0 1 1 log n

  29. Assembly of n x n Squares log n 0 1 0 1 1

  30. Assembly of n x n Squares -Build thicker 2 x log n seed row 0 1 1 1 1 0 1 1 1 0 0 1 0 1 1 0 2 log n

  31. Assembly of n x n Squares -Build thicker 2 x log n seed row -But… can’t encode general binary strings: 0 1 1 1 1 0 1 1 1 0 0 1 0 1 1 0 -All the same 0 0 0 0 1 1 1 1 2 2 2 2 3 3 3 3 2 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 log n

  32. Assembly of n x n Squares Key Idea: Geometry Decoding Tiles A3 A2 A1 A0 B3 B2 B1 B0 0 0 0 0 1 1 1 1 2 2 2 2 3 3 3 3 2 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 log n

  33. Assembly of n x n Squares A3 A3 A3 A2 A2 A2 A1 A1 A0 A0 B3 B3 B3 B2 B2 B2 B1 B1 B1 B1 B0 B0 B0 B0 0 1 1 1 1 0 1 1 1 0 0 1 0 1 1 0 0 0 0 0 1 1 1 1 2 2 2 2 3 3 3 3 2 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 log n

  34. Assembly of n x n Squares A3 A3 A3 A2 A2 A2 A1 A1 A0 A0 B3 B3 B3 B2 B2 B2 B1 B1 B1 B1 B0 B0 B0 B0 0 1 1 1 1 0 1 1 1 0 0 1 0 1 1 0 0 0 0 0 1 1 1 1 2 2 2 2 3 3 3 3 2 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 log n

  35. Assembly of n x n Squares A3 A2 B3 0 2 2 0 1

  36. Assembly of n x n Squares A3 A3 A3 A2 A2 A2 A1 A1 A0 A0 B3 B3 B2 B2 B1 B1 B0 B0 1 0 0 0 1 0 1 1 0 0 0 0 0 1 1 1 1 2 2 2 2 2 2 3 3 3 3 2 1 2 3 0 1 2 3 0 0 1 1 2 3 0 1 2 3 0 log n

  37. Assembly of n x n Squares A3 A3 A3 A2 A2 A2 A1 A1 A0 A0 B3 B3 B3 B2 B2 B2 B1 B1 B1 B1 B0 B0 B0 B0 0 1 1 1 1 0 1 1 1 0 0 1 0 1 1 0 0 0 0 0 1 1 1 1 2 2 2 2 3 3 3 3 2 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 log n

  38. Assembly of n x n Squares • build 2 x log n block: • Decode geometry into log n bit string A3 A3 A3 A2 A2 A2 A1 A1 A0 A0 B3 B3 B3 B2 B2 B2 B1 B1 B1 B1 B0 B0 B0 B0 0 1 1 1 1 0 1 1 1 0 0 1 0 1 1 0 0 0 0 0 1 1 1 1 2 2 2 2 3 3 3 3 2 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 log n

  39. n x n Results Tile Complexity Upper bound Lower bound Normal Tiles* Geometric Tiles Planar Geometric Tiles [*Winfree, Rothemund, Adleman, Cheng, Goel,Huang STOC 2000, 2001]

  40. Planar Geometric Tile Assembly Attachment requires a collision free path within the plane

  41. Planar Geometric Tile Assembly Attachment requires a collision free path within the plane Attachment not permitted in the planar model

  42. Planar Geometric Tile Assembly

  43. Planar Geometric Tile Assembly

  44. Planar Geometric Tile Assembly Attachment not permitted in the planar model

  45. n x n Results Tile Complexity Upper bound Lower bound Normal Tiles* Geometric Tiles Planar Geometric Tiles ? [*Winfree, Rothemund, Adleman, Cheng, Goel,Huang STOC 2000, 2001]

  46. n x n Results Tile Complexity Upper bound Lower bound Normal Tiles* Geometric Tiles Planar Geometric Tiles O( loglog n ) ? [*Winfree, Rothemund, Adleman, Cheng, Goel,Huang STOC 2000, 2001]

  47. Planar Geometric Tile Assembly log n 1 0 1 0 0 1 1 0

  48. Planar Geometric Tile Assembly • Build log n columns with loglog n tile types 0 0 0 0 1 1 1 1 0 0 1 1 0 0 1 1 loglog n 0 1 0 1 0 1 0 1

  49. Planar Geometric Tile Assembly • Build log n columns with loglog n tile types 0 0 0 0 1 1 1 1 0 0 1 1 0 0 1 1 loglog n 0 1 0 1 0 1 0 1

  50. Planar Geometric Tile Assembly • Build log n columns with loglog n tile types • Columns must assemble in proper order 0 0 0 0 1 1 1 1 0 0 1 1 0 0 1 1 loglog n 0 1 0 1 0 1 0 1