notes on self-assembly of discrete self-similar fractals
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Notes on self-assembly of discrete self-similar fractals
Days 32 and 33 of Comp Sci 480
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
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
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…
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)
An example
X0
An example
X0 X1
Start with any (valid) generator
An example
X0 X1 X2
An example
X0 X1 X2 X3
Not an example
X0 X1
Not a valid generator
Not an example
X0 X1
Not a valid generator
Not an example
X0 X1
Would fill the first quadrant
A famous fractal
• Let X0 = { (0,0) }
• Let V = { (0,1), (1,0) }
X0
A famous fractal
• Let X0 = { (0,0) }
• Let V = { (0,1), (1,0) }
X0 X1
A famous fractal
• Let X0 = { (0,0) }
• Let V = { (0,1), (1,0) }
X0 X1 X2
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
Self-assembly of the discrete Sierpinski triangle
• Can we self-assemble the discrete Sierpinski triangle?
• There are two ways we can do this…
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
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
Weak example
Y
Y
1
1
XXSY
X
00
0
00
10
1
11 01
0
01
11
1
10
f(x,y)
g(x,y)
h(x,y)y
x
Can you guess the pattern of inputs and outputs?
The “black” tiles
x^y
x^y
x^yy
x
f(x,y) = x XOR y = “x ^ y”g(x,y) = f(x,y)h(x,y) = f(x,y)
Y
Y
1
1
XXSY
X
00
0
00
10
1
11 01
0
01
11
1
10
The “black” tiles
Y
Y
1
1
XXSY
X
00
0
00
10
1
11 01
0
01
11
1
10
x^y
x^y
x^yy
x
f(x,y) = x XOR y = “x ^ y”g(x,y) = f(x,y)h(x,y) = f(x,y)
Y
Y
1
1
XXSY
X
00
0
00
10
1
11
01
0
01 11
1
10
Y
Y
1
Y
Y
1
Y
Y
1
Y
Y
1
Y
Y
1
1
XX
1
XX
1
XX
1
XX
1
XX
1
XX
01
0
01 11
1
10 01
0
01 11
1
10 01
0
01
10
1
11
01
0
01
10
1
11
01
0
01
00
0
0001
0
01
00
0
00
10
1
11 10
1
11
11
1
10
11
1
10
01
0
01
10
1
11 01
0
01 00
0
00
01
0
01 00
0
00 00
0
00
00
0
00
11
1
10 01
0
01
01
0
01 00
0
00
00
0
00 00
0
00
00
0
0000
0
00
00
0
00
00
0
00
00
0
00
00
0
00
00
0
00
00
0
00
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!
Scaled Sierpinski triangle
S
S?
Scaled Sierpinski triangle
S
S2
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)
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!!!
Review the Window movie lemma…
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
Window movie lemma (setup)
A
AL AR
BL BR
w w’
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)
Window movie lemma high-level example
A
AL AR
BL BR
w w’
Window movie lemma high-level example
AL
B’R = BR - (x,y)
ALB’R = AL U (BR - (x,y))
A better example
SS
w w’ = w + (3,0)
125
436
798
131215
101114
Ma,w = Ma,w’ + (-3,0) Window movie lemma says that we can do this…
Example
SS
w w’ = w + (3,0)
125
436
798
131215
101114
This must also be producible (via Window movie lemma)…
…using a different assembly sequence
Proof omitted!
The big question
• Does any discrete self-similar fractal strictly self-assemble?
• Currently nobody knows
Another famous fractal…
X0 X1 X2 X3
This is the generator
Some questions
• Does the discrete Sierpinski carpet strictly self-assemble?– Unknown
• Does it weakly self-assemble?– Yes
Sierpinski carpet weakly self-assembles
Y
Y
1
11
XXSY
X
f(x,y)
g(x,y)
h(x,y)y
x
Three neighbors
Y
Y
1
11
XXSY
X
f(x,y,z)
g(x,y,z)
h(x,y,z)y
x,z
y
xz
The glues
Y
Y
1
11
XXSY
X
(x+y+z)%3
y,(x+y+z)%3
y
x,z
y
xz
(x+y+z)%3
“my left neighbor and I”
“me”
Y
Y
1
11
XXSY
X
000
00
00 101
01
10 202
02
20
110
01
10 211
02
20 012
00
00
220
02
20 021
00
00 122
01
10
100
11
11 201
12
21 002
10
01
210
12
21 011
10
01 012
10
01
020
10
01 121
11
11 222
12
21
200
22
22 001
20
02 102
21
12
010
20
02 111
21
12 212
22
22
120
21
12 221
22
22 022
20
02
The expanded tile set…
Y
Y
1
11
XXSY
X
Y
Y
1
Y
Y
1
Y
Y
1
Y
Y
1
Y
Y
1
11
XX
11
XX
11
XX
11
XX
11
XX
11
XX
11
XX
000
00
00 202
02
20
211
02
20
012
00
00 021
00
00
122
01
10
201
12
21
210
12
21
011
10
01
112
11
11
020
10
01
121
11
11
200
22
22
102
21
12
010
20
02
111
21
12
212
22
22
120
21
12
221
22
22
022
20
02
211
02
20011
10
01 111
21
12 211
02
20011
10
01
210
12
21 121
11
11102
21
12 210
12
21 102
21
12
121
11
11 011
10
01
011
10
01 211
02
20
210
12
21 102
21
12
112
11
11 121
11
11 210
12
21
211
02
20
000
00
00
000
00
00 000
00
00 202
02
20
200
22
22 102
21
12 121
11
11 112
11
11
Which tiles should be “black”??
Y
Y
1
11
XXSY
X
000
00
00 202
02
20
012
00
00 021
00
00
122
01
10
201
12
21
011
10
01
112
11
11
020
10
01
200
22
22
010
20
02
111
21
12
212
22
22
120
21
12
221
22
22
022
20
02
011
10
01 211
02
20011
10
01
121
11
11 011
10
01
011
10
01 211
02
20
210
12
21 102
21
12
112
11
11 121
11
11 210
12
21
211
02
20
000
00
00
000
00
00 000
00
00 202
02
20
200
22
22 102
21
12 121
11
11 112
11
11
All tiles with a label ≠ “0”
11
XX
11
XX
11
XX
11
XX
11
XX
11
XX
11
XX
Y
Y
1
Y
Y
1
Y
Y
1
Y
Y
1
Y
Y
1
210
12
21 102
21
12 121
11
11 210
12
21 102
21
12 121
11
11 210
12
21 102
21
12
211
02
20 211
02
20 111
21
12
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…
The Fibered Sierpinski triangle
How is it defined?
The first stage
The Fibered Sierpinski triangle
The second stage
The Fibered Sierpinski triangle
The third stage
The Fibered Sierpinski triangle
The fourth stage
The Fibered Sierpinski triangle
Both fractals even share the same discrete fractal dimension(i.e., log23 ≈ 1.585)
Visual similarity
Strict self-assembly of the Fibered Sierpinski triangle
• The Fibered Sierpinski triangle strictly self-assembles
• The smallest tile set I know of that does this has 51 tile types
The fibered Sierpinski triangle is made up of a bunch of squares and
rectangles
Key observation
Modified fixed-width counter
# times to count each number x:
max{1, # 0’s to the right of the rightmost 1 that x has}
A modified counter
0 0 1
10 0
110
1 00
0 01
101
11 0
111
Self-assembly of stage 0
S
#
#'
1
2
Y
X
1' 2'
Self-assembly of stage 1
10
1
1
1
#
1
0
#
01
1
11
1
01
Self-assembly of stage 1
1
110
0
0
101
1
111
0
1
1
1
0
0 1
10
0
10
0
0
0
0
11
0
00
1
00
1
10
1
10
1
11
1
11
#
#
#
10
1
11
0
11
01
110
1
1
0
1
1
1
More fibered fractals!
• Any “nice” discrete self-similar fractal has a “fibered” version
• Same fractal dimension
• The fibered version can strictly self-assemble
“Nice” discrete self-similar fractals
• Let X be a discrete self-similar fractal– Generated by V
• X is “nice” if the set ({0, …, c – 1} x {0}) U ({0} x {0, …, c – 1}) is contained in V– This is just the leftmost column and
bottommost row of the c x c square with lower left point at the origin
• Examples…
Some discrete self-similar fractals
Nice…
NOT nice…
FiberedFractalTiler.exe
Summary
• The discrete Sierpinski triangle weakly self-assembles• It does not strictly self-assemble (at any scale factor c >
0)• The fibered Sierpinski triangle strictly self-assembles• The discrete Sierpinski carpet weakly self-assembles• Does any (non-trivial) discrete self-similar fractal strictly
self-assemble?
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