Strict Self-Assembly of Discrete Sierpinski Triangles James I. Lathrop, Jack H. Lutz, and Scott M. Summers Iowa State University © James I. Lathrop, Jack H. Lutz, and Scott M. Summers All rights reserved

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

DNA Tile Self-Assembly Seeman, starting in 1980s DNA tile, oversimplified: Four single DNA strands bound by Watson-Crick pairing (A-T, C-G). Sticky ends bind with their Watson-Crick complements, so that a regular array self- assembles.

DNA Tile Self-Assembly Seeman, starting in 1980s DNA tile, oversimplified: Four single DNA strands bound by Watson-Crick pairing (A-T, C-G). Sticky ends bind with their Watson-Crick complements, so that a regular array self- assembles. Choice of sticky ends allows one to program the pattern of the array.

DNA Tile Self-Assembly Winfree, Ph.D. thesis, 1998

Extension of Wang tiling, 1961 Refined in Paul Rothemunds Ph.D. thesis, 2001 Tile = unit square

DNA Tile Self-Assembly Winfree, Ph.D. thesis, 1998 Extension of Wang tiling, 1961 Refined in Paul Rothemunds Ph.D. thesis, 2001 Strength 0 Strength 1 Strength 2 Tile = unit square Each side has glue of certain kind and strength (0, 1, or 2).

DNA Tile Self-Assembly Winfree, Ph.D. thesis, 1998 Extension of Wang tiling, 1961 Refined in Paul Rothemunds Ph.D. thesis, 2001 Tile = unit square Each side has glue of certain kind and strength (0, 1, or 2). If tiles abut with matching kinds of glue, then they bind with this glues strength. Strength 0 Strength 1 Strength 2

DNA Tile Self-Assembly Winfree, Ph.D. thesis, 1998 Extension of Wang tiling, 1961 Refined in Paul Rothemunds Ph.D. thesis, 2001 Tile = unit square Each side has glue of certain kind and strength (0, 1, or 2). If tiles abut with matching kinds of glue, then they bind with this glues strength. Tiles may have labels. Strength 0 Strength 1 Strength 2

DNA Tile Self-Assembly Winfree, Ph.D. thesis, 1998 Extension of Wang tiling, 1961 Refined in Paul Rothemunds Ph.D. thesis, 2001 Tile = unit square Each side has glue of certain kind and strength (0, 1, or 2). If tiles abut with matching kinds of glue, then they bind with this glues strength. Tiles may have labels. Tiles cannot be rotated. Strength 0 Strength 1 Strength 2

DNA Tile Self-Assembly Winfree, Ph.D. thesis, 1998 Extension of Wang tiling, 1961 Refined in Paul Rothemunds Ph.D. thesis, 2001 Tile = unit square Each side has glue of certain kind and strength (0, 1, or 2). If tiles abut with matching kinds of glue, then they bind with this glues strength. Tiles may have labels. Tiles cannot be rotated. Strength 0 Strength 1 Strength 2 Finitely many tile types

DNA Tile Self-Assembly Winfree, Ph.D. thesis, 1998 Extension of Wang tiling, 1961 Refined in Paul Rothemunds Ph.D. thesis, 2001 Tile = unit square Each side has glue of certain kind and strength (0, 1, or 2). If tiles abut with matching kinds of glue, then they bind with this glues strength. Tiles may have labels. Tiles cannot be rotated. Strength 0 Strength 1 Strength 2 Finitely many tile types Infinitely many of each type available

DNA Tile Self-Assembly Winfree, Ph.D. thesis, 1998 Extension of Wang tiling, 1961 Refined in Paul Rothemunds Ph.D. thesis, 2001 Tile = unit square Each side has glue of certain kind and strength (0, 1, or 2). If tiles abut with matching kinds of glue, then they bind with this glues strength. Tiles may have labels. Tiles cannot be rotated. Strength 0 Strength 1 Strength 2 Finitely many tile types Infinitely many of each type available Assembly starts from a seed tile (or seed assembly).

DNA Tile Self-Assembly Winfree, Ph.D. thesis, 1998 Extension of Wang tiling, 1961 Refined in Paul Rothemunds Ph.D. thesis, 2001 Tile = unit square Each side has glue of certain kind and strength (0, 1, or 2). If tiles abut with matching kinds of glue, then they bind with this glues strength. Tiles may have labels. Tiles cannot be rotated. Strength 0 Strength 1 Strength 2 Finitely many tile types Infinitely many of each type available Assembly starts from a seed tile (or seed assembly). A tile can attach to existing assembly if it binds with total strength at least 2 (the temperature).

Tile Assembly Example 1

Cooperation is key to computing with tile-assembly model.

Tile Assembly Example 1

Tile Assembly Example 2

Tile Assembly Example 2 Discrete Sierpinski Triangle The set S is called the discrete Sierpinski triangle.

Tile Assembly Example 2 Discrete Sierpinski Triangle The points in this subset of S are shown in black.

Tile Assembly Example 2 A simulation

Theorem (Winfree 1998) The Tile Assembly Model is computationally universal, i.e., it can simulate any Turing machine.

Theorem (Winfree 1998) The Tile Assembly Model is computationally universal, i.e., it can simulate any Turing machine. Hence nanoscale self-assembly can be algorithmically directed.

Nanoscale Sierpinski Triangles From DNA Tiles 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 Algorithmic Self- Assembly of DNA Sierpinski Triangles

Nanoscale Sierpinski Triangles From DNA Tiles Rothemund, Papadakis, and Winfree, Feynman Prize in nanotechnology Prize for theoretical work: Prize for experimental work:

Nanoscale Sierpinski Triangles From DNA Tiles Rothemund, Papadakis, and Winfree, Feynman Prize in nanotechnology Prize for theoretical work: Rothemund and Winfree Prize for experimental work:

Nanoscale Sierpinski Triangles From DNA Tiles Rothemund, Papadakis, and Winfree, Feynman Prize in nanotechnology Prize for theoretical work: Rothemund and Winfree Prize for experimental work: Rothemund and Winfree

Fractal Structures Mathematical examples:

Fractal Structures Mathematical examples: the Cantor set,

Fractal Structures Mathematical examples: the Cantor set, the von Koch curve,

Fractal Structures Mathematical examples: the Cantor set, the von Koch curve, the Sierpinski triangle,

Fractal Structures Mathematical examples: the Cantor set, the von Koch curve, the Sierpinski triangle, the Menger sponge,

Fractal Structures Mathematical examples: the Cantor set, the von Koch curve, the Sierpinski triangle, the Menger sponge, and many more exotic sets.

Fractal Structures Mathematical examples: the Cantor set, the von Koch curve, the Sierpinski triangle, the Menger sponge, and many more exotic sets. Each has fractal dimension that is less than the dimension of the space or surface that it occupies.

Fractal Structures Mathematical examples: the Cantor set, the von Koch curve, the Sierpinski triangle, the Menger sponge, and many more exotic sets. Each has fractal dimension that is less than the dimension of the space or surface that it occupies. Physical examples (usefully modeled as fractals):

Fractal Structures Mathematical examples: the Cantor set, the von Koch curve, the Sierpinski triangle, the Menger sponge, and many more exotic sets. Each has fractal dimension that is less than the dimension of the space or surface that it occupies. Physical examples (usefully modeled as fractals): a fern leaf Jungle fern FreeJunglePictures.com Barnsley fractal fern

Fractal Structures Mathematical examples: the Cantor set, the von Koch curve, the Sierpinski triangle, the Menger sponge, and many more exotic sets. Each has fractal dimension that is less than the dimension of the space or surface that it occupies. Physical examples (usefully modeled as fractals): a fern leaf a lung

Fractal Structures Mathematical examples: the Cantor set, the von Koch curve, the Sierpinski triangle, the Menger sponge and many more exotic sets. Each has fractal dimension that is less than the dimension of the space or surface that it occupies. Physical examples (usefully modeled as fractals): a fern leaf a lung a neuron

Fractal Structures Advantages of fractal structures Materials transport Heat exchange Information processing Robustness

Fractal Structures Advantages of fractal structures Materials transport Heat exchange Information processing Robustness OBJECTIVE Study the self-assembly of fractal structures in the Tile Assembly Model.

Fractal Structures Advantages of fractal structures Materials transport Heat exchange Information processing Robustness OBJECTIVE Study the self-assembly of fractal structures in the Tile Assembly Model. Typical test bed for new research on fractals:

Fractal Structures Advantages of fractal structures Materials transport Heat exchange Information processing Robustness OBJECTIVE Study the self-assembly of fractal structures in the Tile Assembly Model. Typical test bed for new research on fractals: Sierpinski triangles

Self-Assembly of Sierpinski Triangles We have already seen theoretical and molecular self- assemblies of Sierpinski triangles.

Self-Assembly of Sierpinski Triangles We have already seen theoretical and molecular self- assemblies of Sierpinski triangles. Observation. These are really self-assemblies of entire two-dimensional surfaces on which Sierpinski triangles are painted.

Self-Assembly of Sierpinski Triangles We have already seen theoretical and molecular self- assemblies of Sierpinski triangles. Observation. These are really self-assemblies of entire two-dimensional surfaces on which Sierpinski triangles are painted. To achieve advantages of fractal structures, we need strict self-assembly, i.e., the self-assembly of the fractals and nothing else.

Self-Assembly of Sierpinski Triangles We have already seen theoretical and molecular self- assemblies of Sierpinski triangles. Observation. These are really self-assemblies of entire two-dimensional surfaces on which Sierpinski triangles are painted. To achieve advantages of fractal structures, we need strict self-assembly, i.e., the self-assembly of the fractals and nothing else. TODAYS OBJECTIVE Study the strict self-assembly of discrete Sierpinski triangles.

Impossibility of Strict Self-Assembly of S Theorem. The set S does not strictly self- assemble in the Tile Assembly Model.

Impossibility of Strict Self-Assembly of S Observation: S is an infinite tree with arbitrarily deep finite sub-trees.

Definition of Fibered Sierpinski Triangle What is the fibered Sierpinski triangle?

Definition of Fibered Sierpinski Triangle

The Fibered Sierpinski Triangle Lemma. Dim ς (S) = Dim ς (T)

The Fibered Sierpinski Triangle Lemma. Dim ς (S) = Dim ς (T) Theorem. The set T, unlike S, strictly self- assembles in the Tile Assembly Model.

The Fibered Sierpinski Triangle Question: How do we strictly self-assemble the fibered Sierpinski triangle?

The Fibered Sierpinski Triangle Question: How do we strictly self-assemble the fibered Sierpinski triangle? Answer: We will use a modified fixed- width version of the infinite binary counter shown earlier in this talk.

The Fibered Sierpinski Triangle A simulation

We use the local determinism method of Soloveichik and Winfree (2004) to rigorously prove the correctness of this self-assembly. The Fibered Sierpinski Triangle

The standard discrete Sierpinski triangle S does not strictly self-assemble. Summary

The standard discrete Sierpinski triangle S does not strictly self-assemble. A higher-bandwidth, fibered version T of S does self-assemble. Summary

The standard discrete Sierpinski triangle S does not strictly self-assemble. A higher-bandwidth, fibered version T of S does self-assemble. Thank you! Summary