1. Self-Assembly with Geometric Tiles ICALP 2012 Bin FuUniversity of Texas – Pan American Matt PatitzUniversity of Arkansas Robert Schweller (Speaker)University of Texas – Pan American Robert ShelineUniversity 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. 3 Tile Assembly Model (Rothemund, Winfree, Adleman) T = G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 Tile Set: Glue Function: Temperature: x ed cba

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

19. Geometric Tile Model

20. Geometric Tiles Geometry Region

21. Geometric Tiles Geometry Region

22. Geometric Tiles Compatible Geometries

23. Geometric Tiles

24. Incompatible Geometries

25. Geometric Tiles Incompatible Geometries

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

27. n x n Squares, root(log n) tiles log n 01011

28. Assembly of n x n Squares n log n 01100 11111 11110 01011

29. Assembly of n x n Squares log n 01011

30. 2 0100000111111111 Assembly of n x n Squares -Build thicker 2 x log n seed row

31. 3 3 2 3 1 3 0 3 3 2 2 2 1 2 0 2 3 1 2 1 1 1 0 1 3 0 2 0 1 00 2 log n 0100000111111111 Assembly of n x n Squares -Build thicker 2 x log n seed row -But… cant encode general binary strings: 0 -All the same

32. 3 3 2 3 1 3 0 3 3 2 2 2 1 2 0 2 3 1 2 1 1 1 0 1 3 0 2 0 1 00 2 log n Assembly of n x n Squares 0 B3B2B1B0 A3A2A1A0 Key Idea: Geometry Decoding Tiles

33. 3 3 2 3 1 3 0 3 3 2 2 2 1 2 0 2 3 1 2 1 1 1 0 1 3 0 2 0 1 00 2 log n Assembly of n x n Squares 0 0100000111111111 B0A0A1B1A2B2A3B3A0B1B2A3B0A1A2B3B0B1A2B3B0B1B2A3

34. 3 3 2 3 1 3 0 3 3 2 2 2 1 2 0 2 3 1 2 1 1 1 0 1 3 0 2 0 1 00 2 log n Assembly of n x n Squares 0 0100000111111111 B0A0A1B1A2B2A3B3A0B1B2A3B0A1A2B3B0B1A2B3B0B1B2A3

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

36. 3 3 2 3 1 3 0 3 3 2 2 2 1 2 0 2 3 1 2 1 1 1 0 1 3 0 2 0 1 00 2 log n Assembly of n x n Squares 0 00001111 B0A0A1B1A2B2A3B3A0B1B2A3B0A1A2 1 2 0 2 0 B3A3

37. 3 3 2 3 1 3 0 3 3 2 2 2 1 2 0 2 3 1 2 1 1 1 0 1 3 0 2 0 1 00 2 log n Assembly of n x n Squares 0 0100000111111111 B0A0A1B1A2B2A3B3A0B1B2A3B0A1A2B3B0B1A2B3B0B1B2A3

38. 3 3 2 3 1 3 0 3 3 2 2 2 1 2 0 2 3 1 2 1 1 1 0 1 3 0 2 0 1 00 2 log n Assembly of n x n Squares 0 0100000111111111 B0A0A1B1A2B2A3B3A0B1B2A3B0A1A2B3B0B1A2B3B0B1B2A3 build 2 x log n block: Decode geometry into log n bit string

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

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

44. Attachment not permitted in the planar model

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

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

47. 10100110 log n Planar Geometric Tile Assembly

48. 1 1 1 1 1 0101010 0 1 0 0 0 0 1 0 1 01 0 loglog n Build log n columns with loglog n tile types Planar Geometric Tile Assembly

49. 1 1 1 1 1 0101010 0 1 0 0 0 0 1 0 1 01 0 loglog n Build log n columns with loglog n tile types Planar Geometric Tile Assembly

50. 1 1 1 1 1 0101010 0 1 0 0 0 0 1 0 1 01 0 loglog n Build log n columns with loglog n tile types Columns must assemble in proper order Planar Geometric Tile Assembly

51. 10100110 log n 1 1 1 1 1 0101010 0 1 0 0 0 0 1 0 1 01 0 loglog n Build log n columns with loglog n tile types Columns must assemble in proper order Somehow cap each column with specified 0 or 1 tile type. Planar Geometric Tile Assembly

52. Build log n columns with loglog n tile types Columns must assemble in proper order Somehow cap each column with specified 0 or 1 tile type. 10 0 01 1

53. Build log n columns with loglog n tile types Columns must assemble in proper order Somehow cap each column with specified 0 or 1 tile type. 10 0 01 1 010011010111

54. Build log n columns with loglog n tile types Columns must assemble in proper order Somehow cap each column with specified 0 or 1 tile type. 10 0 01 1 010011010111

55. 1 0 0 0 1 0 Planar Geometric Tile Assembly

56. 1 0 0 0 1 0 1 0 0 0 1 1

57. 1 0 0 0 1 0 1 0 0 0 1 1 10100110 log n 1 1 1 1 1 0101010 0 1 0 0 0 0 1 0 1 01 0 loglog n Build log n columns with loglog n tile types Columns must assemble in proper order Somehow cap each column with specified 0 or 1 tile type.

58. 10100110 log n 1 1 1 1 1 0101010 0 1 0 0 0 0 1 0 1 01 0 loglog n Build log n columns with loglog n tile types Columns must assemble in proper order Somehow cap each column with specified 0 or 1 tile type.

59. 1 0 0 0 1 0 Planar Geometric Tile Assembly

60. 1 0 0 0 1 0

61. 1 0 0 0 1 0

64. 1 0 0

65. 1 0 0

66. 1 0 0

67. 1 0 0

68. 1 0 0 0 1 0

69. 1 0 0 0 1 0

70. 1 0 0 0 1 0

71. 1 0 0 0 1 0

72. 10100110 log n 1 1 1 1 1 0101010 0 1 0 0 0 0 1 0 1 01 0 loglog n Build log n columns with loglog n tile types Columns must assemble in proper order Somehow cap each column with specified 0 or 1 tile type. O( loglog n ) tile types

73. n – log n log n X Y Complexity:

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

75. Outline Basic Tile Assembly Model – Rectangles – n x n squares Geometric Tile Assembly Model – More efficient n x n squares Planar Geometric Tile Assembly Model – Even MORE efficient n x n squares (A strange game.. planarity restriction helps you…) Future Directions and Other Results

76. Other Results Simulation of temperature-2 systems with temperature-1 geometric tile systems. Simulation of many glue systems with single glue geometric tile systems. Compact Geometry Design Problem – Algorithms, lower bounds

77. Future Directions Lower bound for the planar model? – Is O(1) tile complexity possible in the planar model? – If not, what about log*(n)? What can be done with just 1 tile type? – Stay tuned for: One Tile to Rule Them All: Simulating Any Turing Machine, Tile Assembly System, or Tiling System with One Rotatable Puzzle Piece by: Erik Demaine, Martin Demaine, Sandor Fekete, Matthew Patitz, Robert Schweller, Andrew Winslow, Damien Woods. What about no rotation, but relative translation placement: – Check out One Tile... -EXTENDED VERSION! SPOILER ALERT: There is totally 1 universal tile that can do anything that can be done.

78. People Bin Fu Matt Patitz Robbie Schweller Bobby Sheline

79. 79 Barish, Shulman, Rothemund, Winfree, 2009

80. DNA Origami Tiles [Masayuki Endo, Tsutomu Sugita, Yousuke Katsuda, Kumi Hidaka, and Hiroshi Sugiyama, 2010]

81. More DNA Origami Shapes [Paul Rothemund, Nature 2006]

82. Alphabet of Shapes, Built with DNA Tiles [Bryan Wei, Mingjie Dai, Peng Yin, Nature 2012]

83. 83 n x n squares with Geometric Tiles Tile Complexity: n - k k k x

84. Assembly of n x n Squares n - k k Complexity:

85. Assembly of n x n Squares n – log n log n Complexity:

86. Assembly of n x n Squares n – log n log n Complexity: seed row

87. log n 0100000111111111 Assembly of n x n Squares -Build thicker 2 x log n seed row

88. n – log n log n

89. n – log n log n X Y Complexity: