strict self-assembly of discrete sierpinski triangles
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Strict Self-Assembly of Discrete Sierpinski Triangles
Scott M. Summers
Iowa State University
DNA Tile Self-Assembly:Ned Seeman, starting in 1980s
Four single DNA strandsbound by Watson-Crickpairing (A-T, C-G).
A
A
AA
A
A
T
T
T T
T
T
AT
AT
AT
AT
A
T
A
T
A
T
A
T
A T
A T
A T
A T
A
T
A
T
A
T
A
T
C
C
G
G
CG
CG
CG
C
G
C
G
C
G C
G
C G
C G
C G
C
G
C
G
DNA tile, oversimplified:
A
A
AA
A
A
T
T
T T
T
T
AT
AT
AT
AT
A
T
A
T
A
T
A
T
A T
A T
A T
A T
A
T
A
T
A
T
A
T
C
C
G
G
CG
CG
CG
C
G
C
G
C
G C
G
C G
C G
C G
C
G
C
G
DNA Tile Self-Assembly:Ned Seeman, starting in 1980s
DNA tile, oversimplified:
A
A
AA
A
A
T
T
T T
T
T
AT
AT
AT
AT
A
T
A
T
A
T
A
T
A T
A T
A T
A T
A
T
A
T
A
T
A
T
C
C
G
G
CG
CG
CG
C
G
C
G
C
G C
G
C G
C G
C G
C
G
C
G
A
A
T
A T
G
CG
C G
DNA Tile Self-Assembly:Ned Seeman, starting in 1980s
“Sticky ends” bind with their Watson-Crick complements,so that a regular array self-assembles.
DNA tile, oversimplified:
A
A
AA
A
A
T
T
T T
T
T
AT
AT
AT
AT
A
T
A
T
A
T
A
T
A T
A T
A T
A T
A
T
A
T
A
T
A
T
C
C
G
G
CG
CG
CG
C
G
C
G
C
G C
G
C G
C G
C G
C
G
C
G
“Sticky ends” bind with their Watson-Crick complements,so that a regular array self-assembles.
DNA Tile Self-Assembly:Ned Seeman, starting in 1980s
DNA tile, oversimplified:
A
A
AA
A
A
T
T
T T
T
T
AT
AT
AT
AT
A
T
A
T
A
T
A
T
A T
A T
A T
A T
A
T
A
T
A
T
A
T
C
C
G
G
CG
CG
CG
C
G
C
G
C
G C
G
C G
C G
C G
C
G
C
G
A
A
AA
A
A
T
T
T T
T
T
AT
AT
AT
AT
A
T
A
T
A
T
A
T
A T
A T
A T
A T
A
T
A
T
A
T
A
T
C
C
G
G
CG
CG
CG
C
G
C
G
C
G C
G
C G
C G
C G
C
G
C
G
A
A
AA
A
A
T
T
T T
T
T
AT
AT
AT
AT
A
T
A
T
A
T
A
T
A T
A T
A T
A T
A
T
A
T
A
T
A
T
C
C
G
G
CG
CG
CG
C
G
C
G
C
G C
G
C G
C G
C G
C
G
C
G
A
A
AA
A
A
T
T
T T
T
T
AT
AT
AT
AT
A
T
A
T
A
T
A
T
A T
A T
A T
A T
A
T
A
T
A
T
A
T
C
C
G
G
CG
CG
CG
C
G
C
G
C
G C
G
C G
C G
C G
C
G
C
G
A
A
AA
A
A
T
T
T T
T
T
AT
AT
AT
AT
A
T
A
T
A
T
A
T
A T
A T
A T
A T
A
T
A
T
A
T
A
T
C
C
G
G
CG
CG
CG
C
G
C
G
C
G C
G
C G
C G
C G
C
G
C
G
A
A
AA
A
A
T
T
T T
T
T
AT
AT
AT
AT
A
T
A
T
A
T
A
T
A T
A T
A T
A T
A
T
A
T
A
T
A
T
C
C
G
G
CG
CG
CG
C
G
C
G
C
G C
G
C G
C G
C G
C
G
C
G
A
A
AA
A
A
T
T
T T
T
T
AT
AT
AT
AT
A
T
A
T
A
T
A
T
A T
A T
A T
A T
A
T
A
T
A
T
A
T
C
C
G
G
CG
CG
CG
C
G
C
G
C
G C
G
C G
C G
C G
C
G
C
G
A
A
AA
A
A
T
T
T T
T
T
AT
AT
AT
AT
A
T
A
T
A
T
A
T
A T
A T
A T
A T
A
T
A
T
A
T
A
T
C
C
G
G
CG
CG
CG
C
G
C
G
C
G C
G
C G
C G
C G
C
G
C
G
A
A
AA
A
A
T
T
T T
T
T
AT
AT
AT
AT
A
T
A
T
A
T
A
T
A T
A T
A T
A T
A
T
A
T
A
T
A
T
C
C
G
G
CG
CG
CG
C
G
C
G
C
G C
G
C G
C G
C G
C
G
C
G
A
A
AA
A
A
T
T
T T
T
T
AT
AT
AT
AT
A
T
A
T
A
T
A
T
A T
A T
A T
A T
A
T
A
T
A
T
A
T
C
C
G
G
CG
CG
CG
C
G
C
G
C
G C
G
C G
C G
C G
C
G
C
G
A
A
AA
A
A
T
T
T T
T
T
AT
AT
AT
AT
A
T
A
T
A
T
A
T
A T
A T
A T
A T
A
T
A
T
A
T
A
T
C
C
G
G
CG
CG
CG
C
G
C
G
C
G C
G
C G
C G
C G
C
G
C
G
A
A
AA
A
A
T
T
T T
T
T
AT
AT
AT
AT
A
T
A
T
A
T
A
T
A T
A T
A T
A T
A
T
A
T
A
T
A
T
C
C
G
G
CG
CG
CG
C
G
C
G
C
G C
G
C G
C G
C G
C
G
C
G
A
A
AA
A
A
T
T
T T
T
T
AT
AT
AT
AT
A
T
A
T
A
T
A
T
A T
A T
A T
A T
A
T
A
T
A
T
A
T
C
C
G
G
CG
CG
CG
C
G
C
G
C
G C
G
C G
C G
C G
C
G
C
G
A
A
AA
A
A
T
T
T T
T
T
AT
AT
AT
AT
A
T
A
T
A
T
A
T
A T
A T
A T
A T
A
T
A
T
A
T
A
T
C
C
G
G
CG
CG
CG
C
G
C
G
C
G C
G
C G
C G
C G
C
G
C
G
A
A
AA
A
A
T
T
T T
T
T
AT
AT
AT
AT
A
T
A
T
A
T
A
T
A T
A T
A T
A T
A
T
A
T
A
T
A
T
C
C
G
G
CG
CG
CG
C
G
C
G
C
G C
G
C G
C G
C G
C
G
C
G
A
A
AA
A
A
T
T
T T
T
T
AT
AT
AT
AT
A
T
A
T
A
T
A
T
A T
A T
A T
A T
A
T
A
T
A
T
A
T
C
C
G
G
CG
CG
CG
C
G
C
G
C
G C
G
C G
C G
C G
C
G
C
G
DNA Tile Self-Assembly:Ned Seeman, starting in 1980s
Choice of sticky endsallows one to programthe pattern of the array.
“Sticky ends” bind with their Watson-Crick complements,so that a regular array self-assembles.
DNA tile, oversimplified:
A
A
AA
A
A
T
T
T T
T
T
AT
AT
AT
AT
A
T
A
T
A
T
A
T
A T
A T
A T
A T
A
T
A
T
A
T
A
T
C
C
G
G
CG
CG
CG
C
G
C
G
C
G C
G
C G
C G
C G
C
G
C
G
A
A
AA
A
A
T
T
T T
T
T
AT
AT
AT
AT
A
T
A
T
A
T
A
T
A T
A T
A T
A T
A
T
A
T
A
T
A
T
C
C
G
G
CG
CG
CG
C
G
C
G
C
G C
G
C G
C G
C G
C
G
C
G
A
A
AA
A
A
T
T
T T
T
T
AT
AT
AT
AT
A
T
A
T
A
T
A
T
A T
A T
A T
A T
A
T
A
T
A
T
A
T
C
C
G
G
CG
CG
CG
C
G
C
G
C
G C
G
C G
C G
C G
C
G
C
G
A
A
AA
A
A
T
T
T T
T
T
AT
AT
AT
AT
A
T
A
T
A
T
A
T
A T
A T
A T
A T
A
T
A
T
A
T
A
T
C
C
G
G
CG
CG
CG
C
G
C
G
C
G C
G
C G
C G
C G
C
G
C
G
A
A
AA
A
A
T
T
T T
T
T
AT
AT
AT
AT
A
T
A
T
A
T
A
T
A T
A T
A T
A T
A
T
A
T
A
T
A
T
C
C
G
G
CG
CG
CG
C
G
C
G
C
G C
G
C G
C G
C G
C
G
C
G
A
A
AA
A
A
T
T
T T
T
T
AT
AT
AT
AT
A
T
A
T
A
T
A
T
A T
A T
A T
A T
A
T
A
T
A
T
A
T
C
C
G
G
CG
CG
CG
C
G
C
G
C
G C
G
C G
C G
C G
C
G
C
G
A
A
AA
A
A
T
T
T T
T
T
AT
AT
AT
AT
A
T
A
T
A
T
A
T
A T
A T
A T
A T
A
T
A
T
A
T
A
T
C
C
G
G
CG
CG
CG
C
G
C
G
C
G C
G
C G
C G
C G
C
G
C
G
A
A
AA
A
A
T
T
T T
T
T
AT
AT
AT
AT
A
T
A
T
A
T
A
T
A T
A T
A T
A T
A
T
A
T
A
T
A
T
C
C
G
G
CG
CG
CG
C
G
C
G
C
G C
G
C G
C G
C G
C
G
C
G
A
A
AA
A
A
T
T
T T
T
T
AT
AT
AT
AT
A
T
A
T
A
T
A
T
A T
A T
A T
A T
A
T
A
T
A
T
A
T
C
C
G
G
CG
CG
CG
C
G
C
G
C
G C
G
C G
C G
C G
C
G
C
G
A
A
AA
A
A
T
T
T T
T
T
AT
AT
AT
AT
A
T
A
T
A
T
A
T
A T
A T
A T
A T
A
T
A
T
A
T
A
T
C
C
G
G
CG
CG
CG
C
G
C
G
C
G C
G
C G
C G
C G
C
G
C
G
A
A
AA
A
A
T
T
T T
T
T
AT
AT
AT
AT
A
T
A
T
A
T
A
T
A T
A T
A T
A T
A
T
A
T
A
T
A
T
C
C
G
G
CG
CG
CG
C
G
C
G
C
G C
G
C G
C G
C G
C
G
C
G
A
A
AA
A
A
T
T
T T
T
T
AT
AT
AT
AT
A
T
A
T
A
T
A
T
A T
A T
A T
A T
A
T
A
T
A
T
A
T
C
C
G
G
CG
CG
CG
C
G
C
G
C
G C
G
C G
C G
C G
C
G
C
G
A
A
AA
A
A
T
T
T T
T
T
AT
AT
AT
AT
A
T
A
T
A
T
A
T
A T
A T
A T
A T
A
T
A
T
A
T
A
T
C
C
G
G
CG
CG
CG
C
G
C
G
C
G C
G
C G
C G
C G
C
G
C
G
A
A
AA
A
A
T
T
T T
T
T
AT
AT
AT
AT
A
T
A
T
A
T
A
T
A T
A T
A T
A T
A
T
A
T
A
T
A
T
C
C
G
G
CG
CG
CG
C
G
C
G
C
G C
G
C G
C G
C G
C
G
C
G
A
A
AA
A
A
T
T
T T
T
T
AT
AT
AT
AT
A
T
A
T
A
T
A
T
A T
A T
A T
A T
A
T
A
T
A
T
A
T
C
C
G
G
CG
CG
CG
C
G
C
G
C
G C
G
C G
C G
C G
C
G
C
G
Theoretical Tile Self-Assembly: Erik Winfree, 1998
Theoretical Tile Self-Assembly: Erik Winfree, 1998
Tile = unit square
Theoretical Tile Self-Assembly: Erik Winfree, 1998
Y
ZX
Tile = unit square
Each side has a glue label and strength (0, 1, or 2 notches)
Theoretical Tile Self-Assembly: Erik Winfree, 1998
Y
ZX
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
Theoretical Tile Self-Assembly: Erik Winfree, 1998
RY
ZX
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
Theoretical Tile Self-Assembly: Erik Winfree, 1998
RY
ZX
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
Theoretical Tile Self-Assembly: Erik Winfree, 1998
RY
ZX
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
Theoretical Tile Self-Assembly: Erik Winfree, 1998
RY
ZX
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
Theoretical Tile Self-Assembly: Erik Winfree, 1998
RY
ZX
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
Theoretical Tile Self-Assembly: Erik Winfree, 1998
RY
ZX
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
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”
RY
ZX
Tile Assembly Example
00
cc
1
X0
LL SLR
YcR
R
11
cn
0
11
nn
100
nn
0
SX0
11c
0n0
0c
1c
11n
1n0
0n
0n
11c
0n
Temperature = 2
Tile Assembly Example
LLLR
YcR
R
YcR
R
YcR
R
SR
LX0LLX
0LL
YcR
R
11c
0n
Cooperation is the key to computing with the Tile
Assembly Model.
Tile Assembly Example
SX0
11c
0n0
0c
1c
11n
1n0
0n
0n
Temperature = 2
LLLR
YcR
R
YcR
R
YcR
R
SR
LX0LLX
0LL
YcR
RX0
LL
YcR
R
11
cn
0
S
Yc
Yc
Yc
Yc
X0
X0
11c
0n0
0n
0n0
0n
0n0
0n
0n0
0n
0n0
0n
0n0
0n
0n
00c
1c1
1c
0n0
0n
0n0
0n
0n0
0n
0n0
0n
0n0
0n
0n
11c
0n1
1n
1n0
0n
0n0
0n
0n0
0n
0n0
0n
0n0
0n
0n
00c
1c0
0c
1c1
1c
0n0
0n
0n0
0n
0n0
0n
0n0
0n
0n
11c
0n0
0n
0n1
1n
1n0
0n
0n0
0n
0n0
0n
0n
00c
1c1
1c
0n1
1n
1n0
0n
0n0
0n
0n0
0n
0n
11c
0n1
1n
1n1
1n
1n0
0n
0n0
0n
0n
00n
0n
00n
0n
00n
0n
00c
1c0
0c
1c0
0c
1c1
1c
0n0
0n
0n
Tile Assembly Example
SX0
11c
0n0
0c
1c
11n
1n0
0n
0n
Temperature = 2
LLLR
YcR
R
R
R
R
R
R
R
R
R
R
YcR
R
YcR
R
YcR
R
YcR
R
X0
L L X0
L L X0
L L X0
L L X0
L L X0
L L L L L L L
S
Yc
Yc
Yc
Yc
Yc
Yc
Yc
X0
X0
X0
X0
X0
X0
X0
11c
0n0
0n
0n0
0n
0n0
0n
0n0
0n
0n0
0n
0n0
0n
0n
00c
1c1
1c
0n0
0n
0n0
0n
0n0
0n
0n0
0n
0n0
0n
0n
11c
0n1
1n
1n0
0n
0n0
0n
0n0
0n
0n0
0n
0n0
0n
0n
00c
1c0
0c
1c1
1c
0n0
0n
0n0
0n
0n0
0n
0n0
0n
0n
11c
0n0
0n
0n1
1n
1n0
0n
0n0
0n
0n0
0n
0n
00c
1c1
1c
0n1
1n
1n0
0n
0n0
0n
0n0
0n
0n
11c
0n1
1n
1n1
1n
1n0
0n
0n0
0n
0n
X0
00n
0n
00n
0n
00n
0n
Yc00c
1c0
0c
1c0
0c
1c1
1c
0n0
0n
0n
Tile Assembly Example
SX0
11c
0n0
0c
1c
11n
1n0
0n
0n
Temperature = 2
LLLR
YcR
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
L L L L L L L L L L L L L L L L L
Another Tile Assembly Example
00
01
1
1 1
1
1 1
1
1
11
11
11
10
1
11
11
000
00
0
Another Tile Assembly Example
1
1 1
11
000
11 1
11
10
111
01 0
00
00
000
11
111
01
111
10 0
00
11 1
11
10
111
01
000
00
000
00
111
10
111
10
000
11 0
00
00
000
11
000
11
000
11
111
10 0
00
11 0
00
00
000
00
111
01 1
11
01
000
00
111
01 0
00
11
000
11 1
11
10 0
00
11 0
00
00 0
00
00
000
00
000
00
000
00
000
00 0
00
00
111
01 0
00
11 0
00
00 0
00
00 0
00
00 0
00
00
000
11 0
00
00 0
00
00 0
00
00 0
00
00
000
00
000
00
The “discrete Sierpinski triangle”
Temperature = 2
000
11
1
1
11 1 1
1 11
1
1 11
1
1 11
1
1 11
1
1 11
1
1 11
1
1 11
1
1 11
1
111 1
11 1 1
11 1 1
11 1 1
11 1 1
11 1 1
11 1 1
11 1 1
11 1
From: Algorithmic Self-Assembly of DNA Sierpinski Triangles Rothemund PWK, Papadakis N, Winfree E PLoS Biology Vol. 2, No. 12, e424 doi:10.1371/journal.pbio.0020424
Experimental Self-Assembly: Rothemund, Papadakis and Winfree, 2004
Objective
Study the self-assembly of discrete fractal structures in the
Tile Assembly Model.
Typical test bed for new research on fractals: Sierpinski triangles
Self-Assembly of Sierpinski Triangles
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.
Some Formal Definitions
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.
Today’s Objective
Study the strict self-assembly of discrete Sierpinski triangles in
the Tile Assembly Model.
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?
Impossibility of the Strict Self-Assembly of the Discrete Sierpinski Triangle
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 TDenote the discrete
Sierpinski triangle as SNow look at the points rk = (2k + 1,2k) for
all natural numbers k = 0, 1, 2, …
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
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!
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!
Now What?
Perhaps we could “approximately”
strictly self-assemble the discrete Sierpinski triangle
The Fibered Sierpinski Triangle
The Fibered Sierpinski Triangle
The First Stage
The Fibered Sierpinski Triangle
The Fibered Sierpinski Triangle
The Second Stage
The Fibered Sierpinski Triangle
The Fibered Sierpinski Triangle
The Third Stage
The Fibered Sierpinski Triangle
The Fibered Sierpinski Triangle
The Fourth Stage
Similarity Between Fibered and Standard Sierpinski Triangle
Both fractals even share the same discrete fractal dimension
(i.e., log23 ≈ 1.585)
THEOREM (Summers, Lathrop and Lutz, 2007). The fibered Sierpinski triangle strictly self-assembles in the Tile Assembly Model.
Strict Self-Assembly of the Fibered Sierpinski Triangle
In fact, our tile set contains only 51 unique tile types.
The Key Observation
The fibered Sierpinski triangle is made up of a bunch of squares and
rectangles.
Strict Self-Assembly of the Fibered Sierpinski Triangle (sketch)
0
01
11
0
0
1 0
11 0
011
111
10 0
110
1 0
0
0
1 0
11 0
11 0
111
Standard fixed-width counter
Modified fixed-width counter
100 100
Strict Self-Assembly of the Fibered Sierpinski Triangle (sketch)
S
1
1
1
1
1
#
1 #
1
Strict Self-Assembly of the Fibered Sierpinski Triangle (sketch)
#
10
1 0
11
1
1
0 1
0 1
1
1
#
Strict Self-Assembly of the Fibered Sierpinski Triangle (sketch)
#
1
#
0 0
1 00
110
1 0 0
01 0
101
1 01
111
1
1
11
0
0
1 0
1
0
1
1
0 1
0
0
0
1
1
0
1
1
1
1
1
1
1
0 0
110
1 0 1
1
0 1
0 1
1
1
#
11
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.
Summary
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
Any Questions?
Thank you!
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