# Self-Assembly with Geometric Tiles ICALP 2012 Bin FuUniversity of Texas – Pan American Matt PatitzUniversity of Arkansas Robert Schweller (Speaker)University.

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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

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) 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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)

Geometric Tile Model

Geometric Tiles Geometry Region

Geometric Tiles Geometry Region

Geometric Tiles Compatible Geometries

Geometric Tiles

Incompatible Geometries

Geometric Tiles Incompatible Geometries

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]

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

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

Assembly of n x n Squares log n 01011

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

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

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

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

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

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

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

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

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

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

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

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

Planar Geometric Tile Assembly

Attachment not permitted in the planar model

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]

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] ?

10100110 log n Planar Geometric Tile Assembly

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

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

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

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

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

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

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

1 0 0 0 1 0 Planar Geometric Tile Assembly

1 0 0 0 1 0 1 0 0 0 1 1

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.

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.

1 0 0 0 1 0 Planar Geometric Tile Assembly

1 0 0 0 1 0

1 0 0 0 1 0

1 0 0

1 0 0

1 0 0

1 0 0

1 0 0 0 1 0

1 0 0 0 1 0

1 0 0 0 1 0

1 0 0 0 1 0

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

n – log n log n X Y Complexity:

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]

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

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

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.

People Bin Fu Matt Patitz Robbie Schweller Bobby Sheline

79 Barish, Shulman, Rothemund, Winfree, 2009

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

More DNA Origami Shapes [Paul Rothemund, Nature 2006]

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

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

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

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

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

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

n – log n log n

n – log n log n X Y Complexity:

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