1. Strict Self-Assembly of Discrete Sierpinski Triangles
Scott M. Summers
Iowa State University

2. DNA Tile Self-Assembly: Ned Seeman, starting in 1980s
Four single DNA strands bound by Watson-Crick pairing (A-T, C-G).
DNA tile, oversimplified:

3. DNA Tile Self-Assembly: Ned Seeman, starting in 1980s
DNA tile, oversimplified:

4. DNA Tile Self-Assembly: Ned Seeman, starting in 1980s
DNA tile, oversimplified:
“Sticky ends” bind with their Watson-Crick complements, so that a regular array self- assembles.

5. DNA Tile Self-Assembly: Ned Seeman, starting in 1980s
DNA tile, oversimplified:
“Sticky ends” bind with their Watson-Crick complements, so that a regular array self- assembles.

6. DNA Tile Self-Assembly: Ned Seeman, starting in 1980s
DNA tile, oversimplified:
Choice of sticky ends allows one to program the pattern of the array.
“Sticky ends” bind with their Watson-Crick complements, so that a regular array self- assembles.

7. Theoretical Tile Self-Assembly: Erik Winfree, 1998

8. Theoretical Tile Self-Assembly: Erik Winfree, 1998
Tile = unit square

9. Theoretical Tile Self-Assembly: Erik Winfree, 1998
Tile = unit square
Each side has a glue label and strength (0, 1, or 2 notches) Y X Z

10. Theoretical Tile Self-Assembly: Erik Winfree, 1998
Tile = unit square
Each side has a glue label and strength (0, 1, or 2 notches)
If tiles abut with matching glue label and strength, then they bind with this glue’s strength Y X Z

11. Theoretical Tile Self-Assembly: Erik Winfree, 1998
Tile = unit square
Each side has a glue label and strength (0, 1, or 2 notches)
If tiles abut with matching glue label and strength, then they bind with this glue’s strength
Tiles may have labels R Y X Z

12. Theoretical Tile Self-Assembly: Erik Winfree, 1998
Tile = unit square
Each side has a glue label and strength (0, 1, or 2 notches)
If tiles abut with matching glue label and strength, then they bind with this glue’s strength
Tiles may have labels
Tiles cannot be rotated Z R Y X

13. Theoretical Tile Self-Assembly: Erik Winfree, 1998
Tile = unit square
Each side has a glue label and strength (0, 1, or 2 notches)
If tiles abut with matching glue label and strength, then they bind with this glue’s strength
Tiles may have labels
Tiles cannot be rotated
Finitely many tile types R Y X Z

14. Theoretical Tile Self-Assembly: Erik Winfree, 1998
Tile = unit square
Each side has a glue label and strength (0, 1, or 2 notches)
If tiles abut with matching glue label and strength, then they bind with this glue’s strength
Tiles may have labels
Tiles cannot be rotated
Finitely many tile types
Infinitely many of each type available R Y X Z

15. Theoretical Tile Self-Assembly: Erik Winfree, 1998
Tile = unit square
Each side has a glue label and strength (0, 1, or 2 notches)
If tiles abut with matching glue label and strength, then they bind with this glue’s strength
Tiles may have labels
Tiles cannot be rotated
Finitely many tile types
Infinitely many of each type available
Assembly starts from a seed tile R Y X Z

16. Theoretical Tile Self-Assembly: Erik Winfree, 1998
Tile = unit square
Each side has a glue label and strength (0, 1, or 2 notches)
If tiles abut with matching glue label and strength, then they bind with this glue’s strength
Tiles may have labels
Tiles cannot be rotated
Finitely many tile types
Infinitely many of each type available
Assembly starts from a seed tile
Self-assembly proceeds in a random fashion with tiles attaching one at a time R Y X Z

17. Theoretical Tile Self-Assembly: Erik Winfree, 1998
Tile = unit square
Each side has a glue label and strength (0, 1, or 2 notches)
If tiles abut with matching glue label and strength, then they bind with this glue’s strength
Tiles may have labels
Tiles cannot be rotated
Finitely many tile types
Infinitely many of each type available
Assembly starts from a seed tile
Self-assembly proceeds in a random fashion with tiles attaching one at a time
A tile can attach to the existing assembly if it binds with total strength at least the “temperature” R Y X Z

18. Tile Assembly Example c 1 X L S R Y n

19. Tile Assembly Example Temperature = 2 1 1 Y X S 1 Y Y Y X X S n n c 1n 1 n c 1 1 c n Y c R X S R L L L
Temperature = 2 1 c n Y c R Y c R Y c R X L X L S R L

20. Cooperation is the key to computing with the Tile Assembly Model.
Tile Assembly Example n 1 n X L Y c R 1 n c 1 1 c n Y c R X S R L L L
Temperature = 2 Y c R R Y c R
Cooperation is the key to computing with the Tile Assembly Model. 1 c n Y c R X L X L S R L

21. Tile Assembly Example Temperature = 2 1 Y 1 1 1 1 Y 1 Y 1 1 Y X S 1 1n 1 c n c 1 c 1 c 1 Y c R n 1 n n n 1 n 1 n 1 c n Y c R c 1 1 c n Y c R n n n 1 n 1 c n c 1 Y c R X S n n n 1 n n 1 c n Y c R R L L L n n n n 1 c n c 1 c 1 Y c R
Temperature = 2 R n n n n n n 1 n 1 c n Y c R R n n n n n n 1 c n c 1 Y c R R n n n n n n n 1 c n Y c R R X L X L X L X L X L X L X X S R L L L L L

22. Tile Assembly Example Temperature = 2 1 Y 1 1 1 1 Y 1 Y 1 1 Y X S 1 1n 1 c n c 1 c 1 c 1 Y c n 1 n R R n n 1 n 1 n 1 c n Y c R c 1 1 c n Y c R R n n n 1 n 1 c n c 1 Y c R R X S n n n 1 n n 1 c n Y c R R L L L R n n n n 1 c n c 1 c 1 Y c R
Temperature = 2 R n n n n n n 1 n 1 c n Y c R R n n n n n n 1 c n c 1 Y c R R n n n n n n n 1 c n Y c R R X X X X X X X X S R L L L L L L L L L L L L L L L L L

23. Another Tile Assembly Example1

24. Another Tile Assembly Example1 1
Temperature = 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
The “discrete Sierpinski triangle” 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

25. Experimental Self-Assembly: Rothemund, Papadakis and Winfree, 2004
From: Algorithmic Self-Assembly of DNA Sierpinski Triangles Rothemund PWK, Papadakis N, Winfree E PLoS Biology Vol. 2, No. 12, e424 doi: /journal.pbio

26. Typical test bed for new research on fractals: Sierpinski triangles
Objective
Study the self-assembly of discrete fractal structures in the Tile Assembly Model.
Typical test bed for new research on fractals: Sierpinski triangles

27. Self-Assembly of Sierpinski Triangles

28. Self-Assembly of Sierpinski Triangles
We have already seen theoretical and molecular self-assemblies of Sierpinski triangles.
But these are really just self-assemblies of entire two-dimensional surfaces onto which a picture of the Sierpinski triangle is “painted.”
But I want to study the more difficult problem of the self-assembly of shapes and nothing else, i.e., strict self-assembly.

29. Some Formal Definitions

30. Some Formal Definitions
Let X be a set of grid points.
The set X weakly self-assembles if there exists a finite set of tile types T that places “black” tiles on—and only on—every point that belongs to X.
The set X strictly self-assembles if there exists a finite set of tile types T that places tiles on—and only on—every point that belongs to X.

31. Today’s Objective
Study the strict self-assembly of discrete Sierpinski triangles in the Tile Assembly Model.

32. Impossibility of the Strict Self-Assembly of the Discrete Sierpinski Triangle
THEOREM (Summers, Lathrop and Lutz, 2007). The discrete Sierpinski triangle does NOT strictly self-assemble in the Tile Assembly Model.
Why?

33. Impossibility of the Strict Self-Assembly of the Discrete Sierpinski Triangle

34. Denote the discrete Sierpinski triangle as S
Impossibility of the Strict Self-Assembly of the Discrete Sierpinski Triangle
Assume (for the sake of contradiction) that S strictly self-assembles in the tile set denoted as T
Denote the discrete Sierpinski triangle as S
Now look at the points rk = (2k + 1,2k) for all natural numbers k = 0, 1, 2, …

35. Impossibility of the Strict Self-Assembly of the Discrete Sierpinski Triangle
Assume (for the sake of contradiction) that S strictly self-assembles in the tile set denoted as T
Now look at the points rk = (2k + 1,2k) for all natural numbers k = 0, 1, 2, …
Since T is finite and S is infinite, there must be two numbers i and j such that T places the same tile type at ri and rj

36. Impossibility of the Strict Self-Assembly of the Discrete Sierpinski Triangle
Assume (for the sake of contradiction) that S strictly self-assembles in the tile set denoted as T
Now look at the points rk = (2k + 1,2k) for all natural numbers k = 0, 1, 2, …
Since T is finite and S is infinite, there must be two numbers i and j such that T places the same tile type at ri and rj
But then some structure other than S could just as easily strictly self-assemble in T—a contradiction!

37. Impossibility of the Strict Self-Assembly of the Discrete Sierpinski Triangle
Assume (for the sake of contradiction) that S strictly self-assembles in the tile set denoted as T
Now look at the points rk = (2k + 1,2k) for all natural numbers k = 0, 1, 2, …
Since T is finite and S is infinite, there must be two numbers i and j such that T places the same tile type at ri and rj
But then some structure other than S could just as easily strictly self-assemble in T—a contradiction!

38. Now What? Perhaps we could “approximately”
strictly self-assemble the discrete Sierpinski triangle

39. The Fibered Sierpinski Triangle

40. The Fibered Sierpinski Triangle
The First Stage

41. The Fibered Sierpinski Triangle

42. The Fibered Sierpinski Triangle
The Second Stage

43. The Fibered Sierpinski Triangle

44. The Fibered Sierpinski Triangle
The Third Stage

45. The Fibered Sierpinski Triangle

46. The Fibered Sierpinski Triangle
The Fourth Stage

47. Similarity Between Fibered and Standard Sierpinski Triangle
Both fractals even share the same discrete fractal dimension
(i.e., log23 ≈ 1.585)

48. Strict Self-Assembly of the Fibered Sierpinski Triangle
THEOREM (Summers, Lathrop and Lutz, 2007). The fibered Sierpinski triangle strictly self-assembles in the Tile Assembly Model.
In fact, our tile set contains only 51 unique tile types.

49. The Key Observation
The fibered Sierpinski triangle is made up of a bunch of squares and rectangles.

50. Strict Self-Assembly of the Fibered Sierpinski Triangle (sketch)
Standard fixed-width counter
Modified fixed-width counter 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

51. Strict Self-Assembly of the Fibered Sierpinski Triangle (sketch)1 # 1 1 1 S # 1 1

52. Strict Self-Assembly of the Fibered Sierpinski Triangle (sketch)1 1 1 1 1 1 # # 1

53. Strict Self-Assembly of the Fibered Sierpinski Triangle (sketch)1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 # # # 1 1

54. Further Reading
Scott M Summers (with James I. Lathrop and Jack H. Lutz). Strict Self-Assembly of Discrete Sierpinski Triangles. Theoretical Computer Science, 410:384—405, 2009.

55. Summary

56. Summary The discrete Sierpinski triangle weakly self-assembles
The discrete Sierpinski triangle does not strictly self-assemble
OPEN QUESTION: Does any (non-trivial) discrete self-similar fractal strictly self-assemble?
The fibered Sierpinski triangle strictly self-assembles

57. Any Questions?
Thank you!