Notes on self-assembly of discrete self-similar fractals

1. Notes on self-assembly of discrete self-similar fractals Days 32 and 33 of Comp Sci 480

2. I lied • Welcome back to the aTAM! • But just briefly • We forgot to discuss the self-assembly of discrete fractals • Examples of self-assembly of infinite shapes

3. Why fractals? • Mathematically interesting • Oh really? • For engineering purposes • Google “cell phone fractal antenna” • Think about why a heat sink is made the way it is • Maybe fractals will teach us about the limits of self-assembly

4. Discrete self-similar fractals • What is a discrete self-similar fractal? • Google “discrete self-similar fractal” and look at the first few images • Exactly what content shows up may depend on certain factors • A discrete self-similar fractal is an infinite set of grid points • It’s a little more complicated than this…

5. Formal definition • Let c > 1 be a natural number and X be an infinite set of grid points (all contained in the first quadrant -- but a strict subset of the first quadrant) • We say X is a c-discrete self-similar fractal if there exists a connected subset of {0, …, c – 1} x {0, …, c – 1}, say V, such that X can be written as X = U0≤i≤∞ Xi, where Xi is the ith stage of X, defined as Xi+1 = Xi U (Xi + ciV) • X0 = { (0,0) } (always) • V cannot be a “line” • V must contain at least one point from every row and every column of the square defined by {0, …, c – 1} x {0, …, c – 1} • In this case, we say that V generates X, a.k.a., V is the generator of X • Usually, we just call a c-discrete self-similar fractal a discrete self-similar fractal (c is always clear from the context)

6. An example X0

7. An example Start with any (valid) generator X0 X1

8. An example X0 X1 X2

9. An example X0 X1 X2 X3

10. Not an example Not a valid generator X0 X1

11. Not an example Not a valid generator X0 X1

12. Not an example Would fill the first quadrant X0 X1

13. A famous fractal • Let X0 = { (0,0) } • Let V = { (0,1), (1,0) } X0

14. A famous fractal • Let X0 = { (0,0) } • Let V = { (0,1), (1,0) } X0 X1

15. A famous fractal • Let X0 = { (0,0) } • Let V = { (0,1), (1,0) } X0 X1 X2

16. A famous fractal • Let X0 = { (0,0) } • Let V = { (0,1), (1,0) } • X is known as the discrete Sierpinski triangle… X0 X1 X2 X3 X4

17. Self-assembly of the discrete Sierpinski triangle • Can we self-assemble the discrete Sierpinski triangle? • There are two ways we can do this…

18. Strict self-assembly • Let X be a set of grid points (possibly infinite) • X strictly self-assembles if there exists a tile set, say T, that places tiles on – and only on – points in X • Most of our examples have focused on strict self-assembly (squares, rectangles, etc.) • T does not need to uniquely produce an assembly • The shape of every terminal assembly must be X

19. Weak self-assembly • As before, let X be a set of grid points (possibly infinite) • X weakly self-assembles if there exists a tile set, say T and a subset of “black tiles” of T, say B, and T places black tiles on – and only on – points in the set X • Can place non-black tiles on points not in X • T need not uniquely produce an assembly • T need not uniquely produce a shape • The pattern of black tiles must be unique

20. Weak example f(x,y) g(x,y) y h(x,y) x 1 1 0 0 Can you guess the pattern of inputs and outputs? 1 1 1 0 0 1 Y 0 0 1 1 1 0 0 0 1 Y 0 1 S Y 1 X X X

21. The “black” tiles x^y x^y y x^y x 1 1 0 0 f(x,y) = x XOR y = “x ^ y” g(x,y) = f(x,y) h(x,y) = f(x,y) 1 1 1 0 0 1 Y 0 0 1 1 1 0 0 0 1 Y 0 1 S Y 1 X X X

22. The “black” tiles x^y x^y y x^y x 1 1 0 0 f(x,y) = x XOR y = “x ^ y” g(x,y) = f(x,y) h(x,y) = f(x,y) 1 1 1 0 0 1 Y 0 0 1 1 1 0 0 0 1 Y 0 1 S Y 1 X X X

23. 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 Y Y Y Y Y Y 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 Y Y Y Y Y Y S Y 1 1 1 1 1 1 1 X X X X X X X X X X X X X X X

24. What do you think? • Do you think it is possible to strictly self-assemble the discrete Sierpinski triangle? • Not possible • My MS thesis at ISU • What if we scale the discrete Sierpinski triangle by a factor of c? • Replace each point in the discrete Sierpinski triangle with a cxc block of points • Still not possible • Proof never before seen… until now!

25. Scaled Sierpinski triangle S? S

26. Scaled Sierpinski triangle S2 S

27. Goal Prove that… There is no tile set in which the discrete Sierpinski triangle strictly self-assembles (at any scale factor c > 0) (many of our examples will focus on S2)

28. Proof by contradiction • Denote, as Sc, the standard discrete Sierpinski triangle scaled up by a factor of c > 0 • Each point in the discrete Sierpinski triangle is replaced by a c x c block of points • Assume T is a tile set in which Sc strictly self-assembles • We will use the Window movie lemma to get a contradiction!!!

29. Review the Window movie lemma…

30. Window movie lemma (setup) Let a = (a0, a1, …) be an assembly sequence with final result A Can be an infinite sequence Let w be a window that cuts A into AL and AR Let w’ = w + (x,y), for (x,y) ≠ (0,0), be a translation of w that cuts A into BL and BR Let Ma,w and Ma,w’ be window movies for w and w’, respectively Assume AL contains the seed tile

31. Window movie lemma (setup) AL AR A w w’ BL BR

32. Window movie lemma If Ma,w = Ma,w’ - (x,y), for some (x,y) ≠ (0,0), then the following two assemblies are producible: ALB’R = AL U B’R Where B’R = BR - (x,y) B’LAR = B’L U AR Where B’L = BL + (x,y)

33. Window movie lemma high-level example AL AR w A w’ BL BR

34. Window movie lemma high-level example AL ALB’R = AL U (BR - (x,y)) B’R = BR - (x,y)

35. A better example S S w w’ = w + (3,0) 5 6 8 14 15 2 3 9 11 12 1 4 7 10 13 Ma,w = Ma,w’ + (-3,0)  Window movie lemma says that we can do this…

36. Example S S w w’ = w + (3,0) 5 6 8 14 15 2 3 9 11 12 1 4 7 10 13 This must also be producible (via Window movie lemma)… …using a different assembly sequence

37. Proof omitted!

38. The big question • Does any discrete self-similar fractal strictly self-assemble? • Currently nobody knows

39. Another famous fractal… This is the generator X0 X1 X2 X3

40. Some questions • Does the discrete Sierpinski carpet strictly self-assemble? • Unknown • Does it weakly self-assemble? • Yes

41. Sierpinski carpet weakly self-assembles f(x,y) g(x,y) y h(x,y) x Y 1 Y S Y 11 X X X

42. Three neighbors y f(x,y,z) g(x,y,z) y h(x,y,z) x,z z x Y 1 Y S Y 11 X X X

43. The glues “my left neighbor and I” y y,(x+y+z)%3 (x+y+z)%3 (x+y+z)%3 “me” y x,z z x Y 1 Y S Y 11 X X X

44. The expanded tile set… 0 1 1 0 2 2 0 0 1 2 0 0 2 1 2 0 0 1 2 0 1 2 1 2 2 0 1 20 10 00 11 11 21 20 01 00 02 02 01 10 12 10 12 21 21 00 02 01 22 12 22 22 10 20 2 0 0 0 2 2 2 0 1 2 2 1 0 0 1 1 1 0 2 1 1 2 1 2 0 1 0 1 1 1 2 2 0 1 0 0 0 0 1 2 0 0 2 1 1 0 2 0 2 2 0 2 2 1 02 00 11 10 12 01 00 02 22 21 10 01 12 20 22 01 20 20 12 10 11 21 11 22 02 21 00 Y 1 Y S Y 11 X X X

45. 2 2 1 0 1 2 2 2 1 2 2 2 0 1 0 1 1 0 1 1 1 2 0 1 1 2 1 1 2 2 0 0 2 0 0 2 2 1 1 2 1 2 0 0 1 0 0 0 00 20 02 21 02 21 21 22 02 22 11 22 00 11 00 00 02 01 02 21 12 10 00 02 10 10 10 11 22 11 12 10 21 00 11 20 11 02 12 11 12 21 10 21 12 11 21 12 2 0 0 2 1 1 1 0 1 1 0 0 1 1 1 2 2 0 2 1 1 0 1 0 1 2 1 0 2 1 2 1 2 0 1 0 2 1 1 2 2 0 0 2 1 0 1 2 0 0 0 0 1 1 2 1 0 2 2 0 1 2 1 1 1 0 1 2 0 1 2 2 1 2 0 0 2 1 2 2 0 1 1 0 2 2 1 2 0 2 2 0 1 1 2 1 02 12 10 11 02 00 21 11 01 21 02 00 10 10 12 02 20 21 22 02 10 11 21 12 12 20 11 21 10 11 11 10 11 21 12 00 02 00 21 11 22 11 11 02 00 00 11 11 Y Y Y Y Y Y 1 1 1 1 1 1 Y Y Y Y Y Y S Y 11 11 11 11 11 11 11 11 X X X X X X X X X X X X X X X X X Which tiles should be “black”??

46. 1 1 0 0 2 2 2 2 0 0 2 2 2 1 1 1 0 1 2 1 2 1 0 1 0 0 2 2 1 1 0 2 0 2 1 0 1 2 2 1 0 1 2 1 1 2 0 0 22 00 02 12 22 21 10 21 12 11 20 02 10 11 21 01 12 02 21 00 11 00 00 21 11 10 22 12 10 20 21 10 11 00 02 00 12 22 02 11 10 21 02 11 21 02 11 12 0 2 2 1 1 1 1 1 1 2 1 1 2 2 2 1 1 1 2 2 2 1 1 1 2 2 2 1 0 0 0 0 1 0 0 2 1 1 0 1 0 0 0 0 0 2 1 0 0 2 1 0 2 0 0 1 0 2 2 1 0 2 0 1 2 2 2 1 2 1 1 0 0 1 2 1 2 1 0 1 2 2 1 1 2 2 1 2 0 0 1 0 0 1 2 1 10 21 02 12 11 11 11 11 21 00 02 11 12 02 02 21 00 22 10 01 12 11 20 11 00 11 21 10 21 21 20 02 02 02 10 00 00 10 11 10 11 00 11 11 12 22 12 21 Y Y Y Y Y Y 1 1 1 1 1 1 Y Y Y Y Y Y S Y 11 11 11 11 11 11 11 11 X X X X X X X X X X X X X X X X X All tiles with a label ≠ “0”

47. Approximating fractals • Strict self-assembly of discrete self-similar fractals is probably impossible • Can still strictly self-assemble fractal-like shapes • Time for some fiber…

48. The Fibered Sierpinski triangle How is it defined?

49. The Fibered Sierpinski triangle The first stage

50. The Fibered Sierpinski triangle The second stage