strict self-assembly of discrete sierpinski triangles james i. lathrop, jack h. lutz, and scott m....
TRANSCRIPT
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 2007. All rights reserved
DNA Tile Self-Assembly Seeman, starting in 1980s
DNA tile, oversimplified:
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 Self-Assembly Seeman, starting in 1980s
DNA tile, oversimplified:
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
“Sticky ends” bind with their Watson-Crick complements,so that a regular array self-assembles.
A
A
T
A T
G
CG
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 Seeman, starting in 1980s
DNA tile, oversimplified:
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
“Sticky ends” bind with their Watson-Crick complements,so that a regular array self-assembles.
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
Choice of sticky endsallows one to programthe pattern of the array.
DNA Tile Self-AssemblyWinfree, Ph.D. thesis, 1998
DNA Tile Self-AssemblyWinfree, Ph.D. thesis, 1998
Extension of Wang tiling, 1961Refined in Paul Rothemund’s Ph.D. thesis, 2001
Tile = unit square
DNA Tile Self-AssemblyWinfree, Ph.D. thesis, 1998
Extension of Wang tiling, 1961Refined in Paul Rothemund’s Ph.D. thesis, 2001
XX Y
Z
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-AssemblyWinfree, Ph.D. thesis, 1998
Extension of Wang tiling, 1961Refined in Paul Rothemund’s 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 glue’s strength.
XX Y
Z
Strength 0
Strength 1
Strength 2
DNA Tile Self-AssemblyWinfree, Ph.D. thesis, 1998
Extension of Wang tiling, 1961Refined in Paul Rothemund’s 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 glue’s strength. Tiles may have labels.
Strength 0
Strength 1
Strength 2
XX Y
ZR
DNA Tile Self-AssemblyWinfree, Ph.D. thesis, 1998
Extension of Wang tiling, 1961Refined in Paul Rothemund’s 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 glue’s strength. Tiles may have labels.Tiles cannot be rotated.
Strength 0
Strength 1
Strength 2
DNA Tile Self-AssemblyWinfree, Ph.D. thesis, 1998
Extension of Wang tiling, 1961Refined in Paul Rothemund’s 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 glue’s strength. Tiles may have labels.Tiles cannot be rotated.
Strength 0
Strength 1
Strength 2
XX Y
ZR Finitely many tile types
DNA Tile Self-AssemblyWinfree, Ph.D. thesis, 1998
Extension of Wang tiling, 1961Refined in Paul Rothemund’s 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 glue’s strength. Tiles may have labels.Tiles cannot be rotated.
Strength 0
Strength 1
Strength 2
XX Y
ZR Finitely many tile types Infinitely many of each type available
DNA Tile Self-AssemblyWinfree, Ph.D. thesis, 1998
Extension of Wang tiling, 1961Refined in Paul Rothemund’s 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 glue’s strength. Tiles may have labels.Tiles cannot be rotated.
Strength 0
Strength 1
Strength 2
XX Y
ZR Finitely many tile types Infinitely many of each type available Assembly starts from a seed tile (or seed assembly).
DNA Tile Self-AssemblyWinfree, Ph.D. thesis, 1998
Extension of Wang tiling, 1961Refined in Paul Rothemund’s 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 glue’s strength. Tiles may have labels.Tiles cannot be rotated.
Strength 0
Strength 1
Strength 2
XX Y
ZR 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
S
1
0
c c00
1
n c1
1
1
n n10
0
n n0
c R
0
L
0
E d g e b in d in g s tre n g th s :
1
2
Tile Assembly Example 1
S
S
1
0
c c00
1
n c1
1
1
n n10
0
n n0
c R
0
L
0
E d g e b in d in g s tre n g th s :
1
2
Tile Assembly Example 1
S0
L
S
1
0
c c00
1
n c1
1
1
n n10
0
n n0
c R
0
L
0
E d g e b in d in g s tre n g th s :
1
2
Tile Assembly Example 1
S0
L
c R
S
1
0
c c00
1
n c1
1
1
n n10
0
n n0
c R
0
L
0
E d g e b in d in g s tre n g th s :
1
2
Tile Assembly Example 1
S0
L
c R
c R
S
1
0
c c00
1
n c1
1
1
n n10
0
n n0
c R
0
L
0
E d g e b in d in g s tre n g th s :
1
2
Tile Assembly Example 1
S0
L
c R
c R
0
L
S
1
0
c c00
1
n c1
1
1
n n10
0
n n0
c R
0
L
0
E d g e b in d in g s tre n g th s :
1
2
Tile Assembly Example 1
S0
L
c R
c R
0
L
c RS
1
0
c c00
1
n c1
1
1
n n10
0
n n0
c R
0
L
0
E d g e b in d in g s tre n g th s :
1
2
Tile Assembly Example 1
S0
L
c R
c R
0
L
c R
0
1
n c1
S
1
0
c c00
1
n c1
1
1
n n10
0
n n0
c R
0
L
0
E d g e b in d in g s tre n g th s :
1
2
Tile Assembly Example 1
S0
L
c R
c R
0
L
c R
0
1
n c1
0
L
c R0
1
n c1
S
1
0
c c00
1
n c1
1
1
n n10
0
n n0
c R
0
L
0
E d g e b in d in g s tre n g th s :
1
2
Tile Assembly Example 1
S0
L
c R
c R
0
L
c R
0
1
n c1
0
L
c R0
1
n c1
Cooperation is key to computing with tile-assembly model.
S
1
0
c c00
1
n c1
1
1
n n10
0
n n0
c R
0
L
0
E d g e b in d in g s tre n g th s :
1
2
Tile Assembly Example 1
S0
L
c R
c R
0
L
c R
0
1
n c1
S
1
0
c c00
1
n c1
1
1
n n10
0
n n0
c R
0
L
0
E d g e b in d in g s tre n g th s :
1
2
0
0
n n0
Tile Assembly Example 1
S0
L
c R
c R
0
L
c R
0
1
n c1
S
1
0
c c00
1
n c1
1
1
n n10
0
n n0
c R
0
L
0
E d g e b in d in g s tre n g th s :
1
2
0
0
n n0
0
L
Tile Assembly Example 1
S0
L
c R
c R
0
L
c R
0
1
n c1
S
1
0
c c00
1
n c1
1
1
n n10
0
n n0
c R
0
L
0
E d g e b in d in g s tre n g th s :
1
2
0
0
n n0
0
L
c R
Tile Assembly Example 1
S0
L
c R
c R
0
L
c R
0
1
n c1
S
1
0
c c00
1
n c1
1
1
n n10
0
n n0
c R
0
L
0
E d g e b in d in g s tre n g th s :
1
2
0
0
n n0
0
L
c R
1
0
c c0
Tile Assembly Example 1
S0
L
c R
c R
0
L
c R
0
1
n c1
S
1
0
c c00
1
n c1
1
1
n n10
0
n n0
c R
0
L
0
E d g e b in d in g s tre n g th s :
1
2
0
0
n n0
0
L
c R
1
0
c c0
c R
Tile Assembly Example 1
S0
L
c R
c R
0
L
c R
0
1
n c1
S
1
0
c c00
1
n c1
1
1
n n10
0
n n0
c R
0
L
0
E d g e b in d in g s tre n g th s :
1
2
0
0
n n0
0
L
c R
1
0
c c0
c R
0
1
n c1
Tile Assembly Example 1
S0
L
c R
c R
0
L
c R
0
1
n c1
S
1
0
c c00
1
n c1
1
1
n n10
0
n n0
c R
0
L
0
E d g e b in d in g s tre n g th s :
1
2
0
0
n n0
0
L
c R
1
0
c c0
c R
0
1
n c1
0
L
Tile Assembly Example 1
S0
L
c R
c R
0
L
c R
0
1
n c1
S
1
0
c c00
1
n c1
1
1
n n10
0
n n0
c R
0
L
0
E d g e b in d in g s tre n g th s :
1
2
0
0
n n0
0
L
c R
1
0
c c0
c R
0
1
n c1
0
L
0
0
n n0
Tile Assembly Example 1
S0
L
c R
c R
0
L
c R
0
1
n c1
S
1
0
c c00
1
n c1
1
1
n n10
0
n n0
c R
0
L
0
E d g e b in d in g s tre n g th s :
1
2
0
0
n n0
0
L
c R
1
0
c c0
c R
0
1
n c1
0
L
0
0
n n00
0
n n0
Tile Assembly Example 1
S0
L
c R
c R
0
L
c R
0
1
n c1
S
1
0
c c00
1
n c1
1
1
n n10
0
n n0
c R
0
L
0
E d g e b in d in g s tre n g th s :
1
2
0
0
n n0
0
L
c R
1
0
c c0
c R
0
1
n c1
0
L
0
0
n n00
0
n n0
0
1
n c1
Tile Assembly Example 1
S0
L
c R
c R
0
L
c R
0
1
n c1
S
1
0
c c00
1
n c1
1
1
n n10
0
n n0
c R
0
L
0
E d g e b in d in g s tre n g th s :
1
2
0
0
n n0
0
L
c R
1
0
c c0
c R
0
1
n c1
0
L
0
0
n n00
0
n n0
0
1
n c1
1
0
c c0
Tile Assembly Example 1
S0
L
c R
c R
0
L
c R
0
1
n c1
S
1
0
c c00
1
n c1
1
1
n n10
0
n n0
c R
0
L
0
E d g e b in d in g s tre n g th s :
1
2
0
0
n n0
0
L
c R
1
0
c c0
c R
0
1
n c1
0
L
0
0
n n00
0
n n0
0
1
n c1
1
0
c c0
0
0
n n0
Tile Assembly Example 1
S0
L
c R
c R
0
L
c R
0
1
n c1
S
1
0
c c00
1
n c1
1
1
n n10
0
n n0
c R
0
L
0
E d g e b in d in g s tre n g th s :
1
2
0
0
n n0
0
L
c R
1
0
c c0
c R
0
1
n c1
0
L
0
0
n n00
0
n n0
0
1
n c1
1
0
c c0
0
0
n n0
c R
Tile Assembly Example 1
S0
L
c R
c R
0
L
c R
0
1
n c1
S
1
0
c c00
1
n c1
1
1
n n10
0
n n0
c R
0
L
0
E d g e b in d in g s tre n g th s :
1
2
0
0
n n0
0
L
c R
1
0
c c0
c R
0
1
n c1
0
L
0
0
n n00
0
n n0
0
1
n c1
1
0
c c0
0
0
n n0
c R
0
L
Tile Assembly Example 1
S0
L
c R
c R
0
L
c R
0
1
n c1
S
1
0
c c00
1
n c1
1
1
n n10
0
n n0
c R
0
L
0
E d g e b in d in g s tre n g th s :
1
2
0
0
n n0
0
L
c R
1
0
c c0
c R
0
1
n c1
0
L
0
0
n n00
0
n n0
0
1
n c1
1
0
c c0
0
0
n n0
c R
0
L
1
1
n n1
Tile Assembly Example 1
S0
L
c R
c R
0
L
c R
0
1
n c1
S
1
0
c c00
1
n c1
1
1
n n10
0
n n0
c R
0
L
0
E d g e b in d in g s tre n g th s :
1
2
0
0
n n0
0
L
c R
1
0
c c0
c R
0
1
n c1
0
L
0
0
n n00
0
n n0
0
1
n c1
1
0
c c0
0
0
n n0
c R
0
L
1
1
n n1
0
0
n n0
Tile Assembly Example 1
S0
L
c R
c R
0
L
c R
0
1
n c1
S
1
0
c c00
1
n c1
1
1
n n10
0
n n0
c R
0
L
0
E d g e b in d in g s tre n g th s :
1
2
0
0
n n0
0
L
c R
1
0
c c0
c R
0
1
n c1
0
L
0
0
n n00
0
n n0
0
1
n c1
1
0
c c0
0
0
n n0
c R
0
L
1
1
n n1
0
0
n n0
0
0
n n0
Tile Assembly Example 1
S0
L
c R
c R
0
L
c R
0
1
n c1
S
1
0
c c00
1
n c1
1
1
n n10
0
n n0
c R
0
L
0
E d g e b in d in g s tre n g th s :
1
2
0
0
n n0
0
L
c R
1
0
c c0
c R
0
1
n c1
0
L
0
0
n n00
0
n n0
0
1
n c1
1
0
c c0
0
0
n n0
c R
0
L
1
1
n n1
0
0
n n0
0
1
n c1
0
0
n n0
0
0
n n0
Tile Assembly Example 1
S0
L
c R
c R
0
L
c R
0
1
n c1
S
1
0
c c00
1
n c1
1
1
n n10
0
n n0
c R
0
L
0
E d g e b in d in g s tre n g th s :
1
2
0
0
n n0
0
L
c R
1
0
c c0
c R
0
1
n c1
0
L
0
0
n n00
0
n n0
0
1
n c1
1
0
c c0
0
0
n n0
c R
0
L
1
1
n n1
0
0
n n0
0
1
n c1
0
0
n n0
0
0
n n0
c R
Tile Assembly Example 1
S0
L
c R
c R
0
L
c R
0
1
n c1
S
1
0
c c00
1
n c1
1
1
n n10
0
n n0
c R
0
L
0
E d g e b in d in g s tre n g th s :
1
2
0
0
n n0
0
L
c R
1
0
c c0
c R
0
1
n c1
0
L
0
0
n n00
0
n n0
0
1
n c1
1
0
c c0
0
0
n n0
c R
0
L
1
1
n n1
0
0
n n0
0
1
n c1
0
0
n n0
0
0
n n0
c R
0
L
Tile Assembly Example 1
S0
L
c R
c R
0
L
c R
0
1
n c1
S
1
0
c c00
1
n c1
1
1
n n10
0
n n0
c R
0
L
0
E d g e b in d in g s tre n g th s :
1
2
0
0
n n0
0
L
c R
1
0
c c0
c R
0
1
n c1
0
L
0
0
n n00
0
n n0
0
1
n c1
1
0
c c0
0
0
n n0
c R
0
L
1
1
n n1
0
0
n n0
0
1
n c1
0
0
n n0
0
0
n n0
c R
0
L0
L
Tile Assembly Example 1
S0
L
c R
c R
0
L
c R
0
1
n c1
S
1
0
c c00
1
n c1
1
1
n n10
0
n n0
c R
0
L
0
E d g e b in d in g s tre n g th s :
1
2
0
0
n n0
0
L
c R
1
0
c c0
c R
0
1
n c1
0
L
0
0
n n00
0
n n0
0
1
n c1
1
0
c c0
0
0
n n0
c R
0
L
1
1
n n1
0
0
n n0
0
1
n c1
0
0
n n0
0
0
n n0
c R
0
L0
L
0
0
n n0
Tile Assembly Example 1
S0
L
c R
c R
0
L
c R
0
1
n c1
S
1
0
c c00
1
n c1
1
1
n n10
0
n n0
c R
0
L
0
E d g e b in d in g s tre n g th s :
1
2
0
0
n n0
0
L
c R
1
0
c c0
c R
0
1
n c1
0
L
0
0
n n00
0
n n0
0
1
n c1
1
0
c c0
0
0
n n0
c R
0
L
1
1
n n1
0
0
n n0
0
1
n c1
0
0
n n0
0
0
n n0
c R
0
L0
L
0
0
n n00
0
n n0
Tile Assembly Example 1
S0
L
c R
c R
0
L
c R
0
1
n c1
S
1
0
c c00
1
n c1
1
1
n n10
0
n n0
c R
0
L
0
E d g e b in d in g s tre n g th s :
1
2
0
0
n n0
0
L
c R
1
0
c c0
c R
0
1
n c1
0
L
0
0
n n00
0
n n0
0
1
n c1
1
0
c c0
0
0
n n0
c R
0
L
1
1
n n1
0
0
n n0
0
1
n c1
0
0
n n0
0
0
n n0
c R
0
L0
L
0
0
n n00
0
n n0
0
0
n n0
Tile Assembly Example 1
S0
L
c R
c R
0
L
c R
0
1
n c1
S
1
0
c c00
1
n c1
1
1
n n10
0
n n0
c R
0
L
0
E d g e b in d in g s tre n g th s :
1
2
0
0
n n0
0
L
c R
1
0
c c0
c R
0
1
n c1
0
L
0
0
n n00
0
n n0
0
1
n c1
1
0
c c0
0
0
n n0
c R
0
L
1
1
n n1
0
0
n n0
0
1
n c1
0
0
n n0
0
0
n n0
c R
0
L0
L
0
0
n n00
0
n n0
0
0
n n0
1
0
c c0
Tile Assembly Example 1
S0
L
c R
c R
0
L
c R
0
1
n c1
S
1
0
c c00
1
n c1
1
1
n n10
0
n n0
c R
0
L
0
E d g e b in d in g s tre n g th s :
1
2
0
0
n n0
0
L
c R
1
0
c c0
c R
0
1
n c1
0
L
0
0
n n00
0
n n0
0
1
n c1
1
0
c c0
0
0
n n0
c R
0
L
1
1
n n1
0
0
n n0
0
1
n c1
0
0
n n0
0
0
n n0
c R
0
L0
L
0
0
n n00
0
n n0
0
0
n n0
1
0
c c0
0
0
n n0
Tile Assembly Example 1
S0
L
c R
c R
0
L
c R
0
1
n c1
S
1
0
c c00
1
n c1
1
1
n n10
0
n n0
c R
0
L
0
E d g e b in d in g s tre n g th s :
1
2
0
0
n n0
0
L
c R
1
0
c c0
c R
0
1
n c1
0
L
0
0
n n00
0
n n0
0
1
n c1
1
0
c c0
0
0
n n0
c R
0
L
1
1
n n1
0
0
n n0
0
1
n c1
0
0
n n0
0
0
n n0
c R
0
L0
L
0
0
n n00
0
n n0
0
0
n n0
1
0
c c0
0
0
n n0
0
0
n n0
Tile Assembly Example 1
S0
L
c R
c R
0
L
c R
0
1
n c1
S
1
0
c c00
1
n c1
1
1
n n10
0
n n0
c R
0
L
0
E d g e b in d in g s tre n g th s :
1
2
0
0
n n0
0
L
c R
1
0
c c0
c R
0
1
n c1
0
L
0
0
n n00
0
n n0
0
1
n c1
1
0
c c0
0
0
n n0
c R
0
L
1
1
n n1
0
0
n n0
0
1
n c1
0
0
n n0
0
0
n n0
c R
0
L0
L
0
0
n n00
0
n n0
0
0
n n0
1
0
c c0
0
0
n n0
0
0
n n0
1
0
c c0
Tile Assembly Example 1
S0
L
c R
c R
0
L
c R
0
1
n c1
S
1
0
c c00
1
n c1
1
1
n n10
0
n n0
c R
0
L
0
E d g e b in d in g s tre n g th s :
1
2
0
0
n n0
0
L
c R
1
0
c c0
c R
0
1
n c1
0
L
0
0
n n00
0
n n0
0
1
n c1
1
0
c c0
0
0
n n0
c R
0
L
1
1
n n1
0
0
n n0
0
1
n c1
0
0
n n0
0
0
n n0
c R
0
L0
L
0
0
n n00
0
n n0
0
0
n n0
1
0
c c0
0
0
n n0
0
0
n n00
0
n n0
1
0
c c0
Tile Assembly Example 1
S0
L
c R
c R
0
L
c R
0
1
n c1
S
1
0
c c00
1
n c1
1
1
n n10
0
n n0
c R
0
L
0
E d g e b in d in g s tre n g th s :
1
2
0
0
n n0
0
L
c R
1
0
c c0
c R
0
1
n c1
0
L
0
0
n n00
0
n n0
0
1
n c1
1
0
c c0
0
0
n n0
c R
0
L
1
1
n n1
0
0
n n0
0
1
n c1
0
0
n n0
0
0
n n0
c R
0
L0
L
0
0
n n00
0
n n0
0
0
n n0
1
0
c c0
0
0
n n0
0
0
n n00
0
n n0
0
1
n c1
1
0
c c0
Tile Assembly Example 1
S0
L
c R
c R
0
L
c R
0
1
n c1
S
1
0
c c00
1
n c1
1
1
n n10
0
n n0
c R
0
L
0
E d g e b in d in g s tre n g th s :
1
2
0
0
n n0
0
L
c R
1
0
c c0
c R
0
1
n c1
0
L
0
0
n n00
0
n n0
0
1
n c1
1
0
c c0
0
0
n n0
c R
0
L
1
1
n n1
0
0
n n0
0
1
n c1
0
0
n n0
0
0
n n0
c R
0
L0
L
0
0
n n00
0
n n0
0
0
n n0
1
0
c c0
0
0
n n0
0
0
n n00
0
n n0
0
1
n c1
0
0
n n0
1
0
c c0
Tile Assembly Example 1
S0
L
c R
c R
0
L
c R
0
1
n c1
S
1
0
c c00
1
n c1
1
1
n n10
0
n n0
c R
0
L
0
E d g e b in d in g s tre n g th s :
1
2
0
0
n n0
0
L
c R
1
0
c c0
c R
0
1
n c1
0
L
0
0
n n00
0
n n0
0
1
n c1
1
0
c c0
0
0
n n0
c R
0
L
1
1
n n1
0
0
n n0
0
1
n c1
0
0
n n0
0
0
n n0
c R
0
L0
L
0
0
n n00
0
n n0
0
0
n n0
1
0
c c0
0
0
n n0
0
0
n n00
0
n n0
0
1
n c1
0
0
n n0
1
0
c c00
1
n c1
Tile Assembly Example 1
S0
L
c R
c R
0
L
c R
0
1
n c1
S
1
0
c c00
1
n c1
1
1
n n10
0
n n0
c R
0
L
0
E d g e b in d in g s tre n g th s :
1
2
0
0
n n0
0
L
c R
1
0
c c0
c R
0
1
n c1
0
L
0
0
n n00
0
n n0
0
1
n c1
1
0
c c0
0
0
n n0
c R
0
L
1
1
n n1
0
0
n n0
0
1
n c1
0
0
n n0
0
0
n n0
c R
0
L0
L
0
0
n n00
0
n n0
0
0
n n0
1
0
c c0
0
0
n n0
0
0
n n00
0
n n0
0
1
n c1
0
0
n n0
0
0
n n0
1
0
c c00
1
n c1
Tile Assembly Example 1
S0
L
c R
c R
0
L
c R
0
1
n c1
S
1
0
c c00
1
n c1
1
1
n n10
0
n n0
c R
0
L
0
E d g e b in d in g s tre n g th s :
1
2
0
0
n n0
0
L
c R
1
0
c c0
c R
0
1
n c1
0
L
0
0
n n00
0
n n0
0
1
n c1
1
0
c c0
0
0
n n0
c R
0
L
1
1
n n1
0
0
n n0
0
1
n c1
0
0
n n0
0
0
n n0
c R
0
L0
L
0
0
n n00
0
n n0
0
0
n n0
1
0
c c0
0
0
n n0
0
0
n n00
0
n n0
0
1
n c1
0
0
n n0
0
0
n n0
1
1
n n1
1
0
c c00
1
n c1
Tile Assembly Example 1
S0
L
c R
c R
0
L
c R
0
1
n c1
S
1
0
c c00
1
n c1
1
1
n n10
0
n n0
c R
0
L
0
E d g e b in d in g s tre n g th s :
1
2
0
0
n n0
0
L
c R
1
0
c c0
c R
0
1
n c1
0
L
0
0
n n00
0
n n0
0
1
n c1
1
0
c c0
0
0
n n0
c R
0
L
1
1
n n1
0
0
n n0
0
1
n c1
0
0
n n0
0
0
n n0
c R
0
L0
L
0
0
n n00
0
n n0
0
0
n n0
1
0
c c0
0
0
n n0
0
0
n n00
0
n n0
0
1
n c1
0
0
n n0
0
0
n n0
1
1
n n1
0
0
n n0
1
0
c c00
1
n c1
Tile Assembly Example 1
S0
L
c R
c R
0
L
c R
0
1
n c1
S
1
0
c c00
1
n c1
1
1
n n10
0
n n0
c R
0
L
0
E d g e b in d in g s tre n g th s :
1
2
0
0
n n0
0
L
c R
1
0
c c0
c R
0
1
n c1
0
L
0
0
n n00
0
n n0
0
1
n c1
1
0
c c0
0
0
n n0
c R
0
L
1
1
n n1
0
0
n n0
0
1
n c1
0
0
n n0
0
0
n n0
c R
0
L0
L
0
0
n n00
0
n n0
0
0
n n0
1
0
c c0
0
0
n n0
0
0
n n00
0
n n0
0
1
n c1
0
0
n n0
0
0
n n0
1
1
n n1
0
0
n n0
1
0
c c00
1
n c1
0
0
n n0
Tile Assembly Example 1
S0
L
c R
c R
0
L
c R
0
1
n c1
S
1
0
c c00
1
n c1
1
1
n n10
0
n n0
c R
0
L
0
E d g e b in d in g s tre n g th s :
1
2
0
0
n n0
0
L
c R
1
0
c c0
c R
0
1
n c1
0
L
0
0
n n00
0
n n0
0
1
n c1
1
0
c c0
0
0
n n0
c R
0
L
1
1
n n1
0
0
n n0
0
1
n c1
0
0
n n0
0
0
n n0
c R
0
L0
L
0
0
n n00
0
n n0
0
0
n n0
1
0
c c0
0
0
n n0
0
0
n n00
0
n n0
0
1
n c1
0
0
n n0
0
0
n n0
1
1
n n1
0
0
n n0
1
0
c c00
1
n c1
0
0
n n0
0
0
n n0
Tile Assembly Example 1
S0
L
c R
c R
0
L
c R
0
1
n c1
S
1
0
c c00
1
n c1
1
1
n n10
0
n n0
c R
0
L
0
E d g e b in d in g s tre n g th s :
1
2
0
0
n n0
0
L
c R
1
0
c c0
c R
0
1
n c1
0
L
0
0
n n00
0
n n0
0
1
n c1
1
0
c c0
0
0
n n0
c R
0
L
1
1
n n1
0
0
n n0
0
1
n c1
0
0
n n0
0
0
n n0
c R
0
L0
L
0
0
n n00
0
n n0
0
0
n n0
1
0
c c0
0
0
n n0
0
0
n n00
0
n n0
0
1
n c1
0
0
n n0
0
0
n n0
1
1
n n1
0
0
n n0
1
0
c c00
1
n c1
0
0
n n0
0
0
n n0
1
1
n n1
Tile Assembly Example 1
S0
L
c R
c R
0
L
c R
0
1
n c1
S
1
0
c c00
1
n c1
1
1
n n10
0
n n0
c R
0
L
0
E d g e b in d in g s tre n g th s :
1
2
0
0
n n0
0
L
c R
1
0
c c0
c R
0
1
n c1
0
L
0
0
n n00
0
n n0
0
1
n c1
1
0
c c0
0
0
n n0
c R
0
L
1
1
n n1
0
0
n n0
0
1
n c1
0
0
n n0
0
0
n n0
c R
0
L0
L
0
0
n n00
0
n n0
0
0
n n0
1
0
c c0
0
0
n n0
0
0
n n00
0
n n0
0
1
n c1
0
0
n n0
0
0
n n0
1
1
n n1
0
0
n n0
1
0
c c00
1
n c1
0
0
n n0
0
0
n n0
1
1
n n10
0
n n0
0
1
n c1
Tile Assembly Example 1
S0
L
c R
c R
0
L
c R
0
1
n c1
S
1
0
c c00
1
n c1
1
1
n n10
0
n n0
c R
0
L
0
E d g e b in d in g s tre n g th s :
1
2
0
0
n n0
0
L
c R
1
0
c c0
c R
0
1
n c1
0
L
0
0
n n00
0
n n0
0
1
n c1
1
0
c c0
0
0
n n0
c R
0
L
1
1
n n1
0
0
n n0
0
1
n c1
0
0
n n0
0
0
n n0
c R
0
L0
L
0
0
n n00
0
n n0
0
0
n n0
1
0
c c0
0
0
n n0
0
0
n n00
0
n n0
0
1
n c1
0
0
n n0
0
0
n n0
1
1
n n1
0
0
n n0
1
0
c c00
1
n c1
0
0
n n0
0
0
n n0
1
1
n n10
0
n n0
0
1
n c11
1
n n1
Tile Assembly Example 1
S0
L
c R
c R
0
L
c R
0
1
n c1
S
1
0
c c00
1
n c1
1
1
n n10
0
n n0
c R
0
L
0
E d g e b in d in g s tre n g th s :
1
2
0
0
n n0
0
L
c R
1
0
c c0
c R
0
1
n c1
0
L
0
0
n n00
0
n n0
0
1
n c1
1
0
c c0
0
0
n n0
c R
0
L
1
1
n n1
0
0
n n0
0
1
n c1
0
0
n n0
0
0
n n0
c R
0
L0
L
0
0
n n00
0
n n0
0
0
n n0
1
0
c c0
0
0
n n0
0
0
n n00
0
n n0
0
1
n c1
0
0
n n0
0
0
n n0
1
1
n n1
0
0
n n0
1
0
c c00
1
n c1
0
0
n n0
0
0
n n0
1
1
n n10
0
n n0
0
1
n c11
1
n n1
0
0
n n0
Tile Assembly Example 1
S0
L
c R
c R
0
L
c R
0
1
n c1
S
1
0
c c00
1
n c1
1
1
n n10
0
n n0
c R
0
L
0
E d g e b in d in g s tre n g th s :
1
2
0
0
n n0
0
L
c R
1
0
c c0
c R
0
1
n c1
0
L
0
0
n n00
0
n n0
0
1
n c1
1
0
c c0
0
0
n n0
c R
0
L
1
1
n n1
0
0
n n0
0
1
n c1
0
0
n n0
0
0
n n0
c R
0
L0
L
0
0
n n00
0
n n0
0
0
n n0
1
0
c c0
0
0
n n0
0
0
n n00
0
n n0
0
1
n c1
0
0
n n0
0
0
n n0
1
1
n n1
0
0
n n0
1
0
c c00
1
n c1
0
0
n n0
0
0
n n0
1
1
n n10
0
n n0
0
1
n c11
1
n n1
0
0
n n0
1
1
n n1
Tile Assembly Example 1
S0
L
c R
c R
0
L
c R
0
1
n c1
S
1
0
c c00
1
n c1
1
1
n n10
0
n n0
c R
0
L
0
E d g e b in d in g s tre n g th s :
1
2
0
0
n n0
0
L
c R
1
0
c c0
c R
0
1
n c1
0
L
0
0
n n00
0
n n0
0
1
n c1
1
0
c c0
0
0
n n0
c R
0
L
1
1
n n1
0
0
n n0
0
1
n c1
0
0
n n0
0
0
n n0
c R
0
L0
L
0
0
n n00
0
n n0
0
0
n n0
1
0
c c0
0
0
n n0
0
0
n n00
0
n n0
0
1
n c1
0
0
n n0
0
0
n n0
1
1
n n1
0
0
n n0
1
0
c c00
1
n c1
0
0
n n0
0
0
n n0
1
1
n n10
0
n n0
0
1
n c11
1
n n1
0
0
n n0
1
1
n n10
0
n n0
Tile Assembly Example 1
S0
L
c R
c R
0
L
c R
0
1
n c1
S
1
0
c c00
1
n c1
1
1
n n10
0
n n0
c R
0
L
0
E d g e b in d in g s tre n g th s :
1
2
0
0
n n0
0
L
c R
1
0
c c0
c R
0
1
n c1
0
L
0
0
n n00
0
n n0
0
1
n c1
1
0
c c0
0
0
n n0
c R
0
L
1
1
n n1
0
0
n n0
0
1
n c1
0
0
n n0
0
0
n n0
c R
0
L0
L
0
0
n n00
0
n n0
0
0
n n0
1
0
c c0
0
0
n n0
0
0
n n00
0
n n0
0
1
n c1
0
0
n n0
0
0
n n0
1
1
n n1
0
0
n n0
1
0
c c00
1
n c1
0
0
n n0
0
0
n n0
1
1
n n10
0
n n0
0
1
n c11
1
n n1
0
0
n n0
1
1
n n10
0
n n0
Tile Assembly Example 2
odd is, 2
i
jijiS
Tile Assembly Example 2Discrete Sierpinski Triangle
odd is, 2
i
jijiS
The set S is called the discrete Sierpinski triangle.
Tile Assembly Example 2Discrete Sierpinski Triangle
The “points” in this subset of S are shown in black.
Tile Assembly Example 2
A simulation
1 1
00
0 1
01
1
1 1 0
01
0 1
01
1
1 0 1
01
1
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 TrianglesFrom 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:10.1371/journal.pbio.0020424
Nanoscale Sierpinski TrianglesFrom DNA Tiles
Rothemund, Papadakis, and Winfree, 2004
2006 Feynman Prize in nanotechnologyPrize for theoretical work:Prize for experimental work:
Nanoscale Sierpinski TrianglesFrom DNA Tiles
Rothemund, Papadakis, and Winfree, 2004
2006 Feynman Prize in nanotechnologyPrize for theoretical work: Rothemund and WinfreePrize for experimental work:
Nanoscale Sierpinski TrianglesFrom DNA Tiles
Rothemund, Papadakis, and Winfree, 2004
2006 Feynman Prize in nanotechnologyPrize for theoretical work: Rothemund and WinfreePrize 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 fernFreeJunglePictures.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
OBJECTIVEStudy the self-assembly of fractal structures in the Tile Assembly Model.
Fractal Structures
Advantages of fractal structures Materials transport Heat exchange Information processing Robustness
OBJECTIVEStudy 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
OBJECTIVEStudy 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.
TODAY’S OBJECTIVEStudy 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. Dimς(S) = Dimς(T) ≈ 1.518.
The Fibered Sierpinski Triangle
Lemma. Dimς(S) = Dimς(T) ≈ 1.518.
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