Periodic Packings of d-Dimensional Polycubes
description
Transcript of Periodic Packings of d-Dimensional Polycubes
Periodic Packings of d-Dimensional Polycubes
A.V.Maleev, V.G.Rau, A.V.ShutovVladimir State Humanitarian University,
Vladimir, Russia
Definitions• Polycube is a finite union of elementary cells
of d-dimentional simple cubic lattice L with connected interior.
• Centers of elementary cells from polycube are called polycube points.
• The polycubes packing is called normal if all polycube points from the packing belong to L.
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The polycube
Definitions• The polycubes packing is called periodic if its
automorphism group contains some d-dimensional lattice Γ.
• If the fundamental domain of Γ contains only single polycube we have a translation polycube packing.
• If packing density k = 1 we have a polycube tiling.
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Problem statement• Search of all possible variants of periodic
polycubes packing for the given finite set of d-dimensional polycubes with the given packing density.
• Search of all possible variants of d-dimensional periodic polycubes tiling with the given volume of translation fundamental domain.
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Using Polycubes in Crystal Structure Prediction
• Crystal structure prediction is the prediction of the equilibrium structure of a crystal on the basis of the known structure of molecules.
• One of approaches of this problem is based on the close packing principle (A.I.Kitaigorodskii) for geometric models of molecules.
• Approximation of molecules by polycubes reduces a crystal structure prediction problem to searching of periodic polycubes packings.
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Approximation of molecule by polycube
MoleculeGeometric modelLattice pointsPolycube
Historical Background• Polyominoes are shapes made by connecting
certain numbers of equal-sized squares, each joined together with at least one other square along an edge (Golomb, 1953).
• Translation criterion and Conway’s criterion of existence of a periodic tiling of the plane for given polyomino (Conway).
• Using this criterions Rhoads (2005) enumerates and classifies the tilings for small polyominoes (n≤9).
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Packing SpaceA packing space is the pair (L;w), where L is a lattice, and w is a function w : L → {0; 1;…n-1} such that all sets w-1(i), i = 0; 1;…; n-1 are equivalent by translation to some sublattice Γ in lattice L. For any lattice point x w(x) is called the weight of this point. The number n is called the order of packing space. It is obvious that n=[L:Γ].
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1,1 1,2 1,3 1,
2,2 2,3 2,
3,3 3,
,
, , ,
The packing space is described by a :
...
0 ...
0 0 ... ,where
... ... ... ... ...
0 0 0 ...
0; 0 ( 1,2,..., ; 1, 2,
d
d
d
d d
i i i j i i
a a a a
a a a
a aY
a
a a a i d j i i
packing space matrix
)d
Packing Space
Packing Space
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1 1
( , )1 1 ,
1 1
The weight of the point ( , ,..., ) can be calculated
by the formula: ( , ,..., ) ,
where ( , ) 0 if and ( , ) 1 if .
d
ddk i j
d i j ji j
x x x
w x x x x a
k i j i j k i j i j
1 2 3
1 2 31 2 3 1
...
The number of -dimentional packing spaces of the order
coincides with the number of sublattices with the index :
( ) ... (Delone, 1940).d
d d dd d
n
d n
n
I n
Packing Space
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07n
1 2 3 4 5 6 0 1 2 3 4 5 6 0
7 3
0 1Y
0 1 2 3 4 5 6 0 1 2 3 45 641 2 3 4 5 6 0 1 2 3 4 5 6 0 1
0 1 2 3 4 5 6 0 1 2 3 45 6 52 3 4 5 6 0 1 2 3 4 5 6 0 1 2
0 1 2 3 4 5 6 0 1 2 3 46 5 63 4 5 6 0 1 2 3 4 5 6 0 1 20 1 2 3 4 5 6 0 1 2 3 4 5 6
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4 5 6 0 1 2 3 4 5 6 0 1 21 2 3 4 5 6 0 1 2 3 4 5 6
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5 6 0 1 2 3 4 5 6 0 1 22 3 4 5 6 0 1 2 3 4 5 6
30
41
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Packing Space
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2The number of 2-dimentional packing spaces of the order 7: (7) 7 1 8.I 0 1 2 3 4 5 6
0 1 2 3 4 5 6
0 1 2 3 4 5 6
0 1 2 3 4 5 6
0 1 2 3 4 5 6
0 1 2 3 4 5 6
0 1 2 3 4 5 6
0 1 2 3 4 5 6
6 0 1 2 3 4 5
5 6 0 1 2 3 4
4 5 6 0 1 2 3
3 4 5 6 0 1 2
2 3 4 5 6 0 1
1 2 3 4 5 6 0
0 1 2 3 4 5 6
5 6 0 1 2 3 4
3 4 5 6 0 1 2
1 2 3 4 5 6 0
6 0 1 2 3 4 5
4 5 6 0 1 2 3
2 3 4 5 6 0 1
0 1 2 3 4 5 6
4 5 6 0 1 2 3
1 2 3 4 5 6 0
5 6 0 1 2 3 4
2 3 4 5 6 0 1
6 0 1 2 3 4 5
3 4 5 6 0 1 2
0 1 2 3 4 5 6
3 4 5 6 0 1 2
6 0 1 2 3 4 5
2 3 4 5 6 0 1
5 6 0 1 2 3 4
1 2 3 4 5 6 0
4 5 6 0 1 2 3
0 1 2 3 4 5 6
2 3 4 5 6 0 1
4 5 6 0 1 2 3
6 0 1 2 3 4 5
1 2 3 4 5 6 0
3 4 5 6 0 1 2
5 6 0 1 2 3 4
0 1 2 3 4 5 6
1 2 3 4 5 6 0
2 3 4 5 6 0 1
3 4 5 6 0 1 2
4 5 6 0 1 2 3
5 6 0 1 2 3 4
6 0 1 2 3 4 5
0 0 0 0 0 0 0
1 1 1 1 1 1 1
2 2 2 2 2 2 2
3 3 3 3 3 3 3
4 4 4 4 4 4 4
5 5 5 5 5 5 5
6 6 6 6 6 6 6
7 0
0 1
7 1
0 1
7 2
0 1
7 3
0 1
7 4
0 1
7 5
0 1
7 6
0 1
1 0
0 7
Theorem 1
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1
Let be -dimensional polycube. Then
the following conditions are equivalent:
1) There exists the polycube packing of with
packing density .
2) There exists the packing space ( ; ) of ord
i i rP d
P
rk
nL w
0
0 1
er
such that for any vector weights of the points
+ are pairwise different.i i r
n
x
x
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Using this theorem we obtain the algorithm which
generates all translation packings of a given polycube
with a given packing density.
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7k
For example: we will find all possible packings
6hexamino "4" with packing density .
7k
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0 1 2 3 4 5 6
0 1 2 3 4 5 6
0 1 2 3 4 5 6
0 1 2 3 4 5 6
0 1 2 3 4 5 6
0 1 2 3 4 5 6
0 1 2 3 4 5 6
0 1 2 3 4 5 6
6 0 1 2 3 4 5
5 6 0 1 2 3 4
4 5 6 0 1 2 3
3 4 5 6 0 1 2
2 3 4 5 6 0 1
1 2 3 4 5 6 0
0 1 2 3 4 5 6
5 6 0 1 2 3 4
3 4 5 6 0 1 2
1 2 3 4 5 6 0
6 0 1 2 3 4 5
4 5 6 0 1 2 3
2 3 4 5 6 0 1
0 1 2 3 4 5 6
4 5 6 0 1 2 3
1 2 3 4 5 6 0
5 6 0 1 2 3 4
2 3 4 5 6 0 1
6 0 1 2 3 4 5
3 4 5 6 0 1 2
0 1 2 3 4 5 6
3 4 5 6 0 1 2
6 0 1 2 3 4 5
2 3 4 5 6 0 1
5 6 0 1 2 3 4
1 2 3 4 5 6 0
4 5 6 0 1 2 3
0 1 2 3 4 5 6
2 3 4 5 6 0 1
4 5 6 0 1 2 3
6 0 1 2 3 4 5
1 2 3 4 5 6 0
3 4 5 6 0 1 2
5 6 0 1 2 3 4
0 1 2 3 4 5 6
1 2 3 4 5 6 0
2 3 4 5 6 0 1
3 4 5 6 0 1 2
4 5 6 0 1 2 3
5 6 0 1 2 3 4
6 0 1 2 3 4 5
0 0 0 0 0 0 0
1 1 1 1 1 1 1
2 2 2 2 2 2 2
3 3 3 3 3 3 3
4 4 4 4 4 4 4
5 5 5 5 5 5 5
6 6 6 6 6 6 6
7 0
0 1
7 1
0 1
7 2
0 1
7 3
0 1
7 4
0 1
7 5
0 1
7 6
0 1
1 0
0 7
Algorithm 1
There are two variants of packing.
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0 1 2 3 4 5 6
4 5 6 0 1 2 3
1 2 3 4 5 6 0
5 6 0 1 2 3 4
2 3 4 5 6 0 1
6 0 1 2 3 4 5
3 4 5 6 0 1 2
0 1 2 3 4 5 6
3 4 5 6 0 1 2
6 0 1 2 3 4 5
2 3 4 5 6 0 1
5 6 0 1 2 3 4
1 2 3 4 5 6 0
4 5 6 0 1 2 3
7 3
0 1
7 4
0 1
Algorithm 1
0 1 2 3 4 5 6
3 4 5 6 0 1 2
6 0 1 2 3 4 5
2 3 4 5 6 0 1
5 6 0 1 2 3 4
1 2 3 4 5 6 0
4 5 6 0 1 2 3
0 1 2 3 4 5 6
3 4 5 6 0 1 2
6 0 1 2 3 4 5
2 3 4 5 6 0 1
5 6 0 1 2 3 4
1 2 3 4 5 6 0
4 5 6 0 1 2 3
0 1 2 3 4 5 6
3 4 5 6 0 1 2
6 0 1 2 3 4 5
2 3 4 5 6 0 1
5 6 0 1 2 3 4
1 2 3 4 5 6 0
4 5 6 0 1 2 3
0 1 2 3 4 5 6
4 5 6 0 1 2 3
1 2 3 4 5 6 0
5 6 0 1 2 3 4
2 3 4 5 6 0 1
6 0 1 2 3 4 5
3 4 5 6 0 1 2
0 1 2 3 4 5 6
4 5 6 0 1 2 3
1 2 3 4 5 6 0
5 6 0 1 2 3 4
2 3 4 5 6 0 1
6 0 1 2 3 4 5
3 4 5 6 0 1 2
0 1 2 3 4 5 6
4 5 6 0 1 2 3
1 2 3 4 5 6 0
5 6 0 1 2 3 4
2 3 4 5 6 0 1
6 0 1 2 3 4 5
3 4 5 6 0 1 2
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Complexity of Algorithm 1
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Let ( ) be a computational complexity of this
algorithm.
In two-dimensional case we have
( ) ( ln ln )
In -dimensional case we have
( ) ( ( )),
where ( ) is a number of sublattices
d
d d
d
C n
C n O n n
d
C n O nI n
I n
Theorem 2
dof
with the index .n
Z
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Algorithm 2
1 1
The theorem 1 can be generalized to finite sets of polycubes.
Consider a finite sets of polycubes , where
The following conditions are equivalent:
1) There exists the polyc
jj j ijj M i rP P
Theorem 3
M
1
0j 01 j M 1 ; 1
ube packing of the set with
packing density , .
2) There exists the packing space ( ; ) of order and the set
of the vectors x such that the points
are pairj
j
jj
ij j i r j M
P
Rk R r
nL w n
x
wise different and have pairwise different weights.
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Algorithm 2
0 1 2 3 4 5 6 7 8 9 10 11 12 13
11 12 13 0 1 2 3 4 5 6 7 8 9 10
8 9 10 11 12 13 0 1 2 3 4 5 6 7
5 6 7 8 9 10 11 12 13 0 1 2 3 4
2 3 4 5 6 7 8 9 10 11 12 13 0 1
13 0 1 2 3 4 5 6 7 8 9 10 11 12
10 11 12 13 0 1 2 3 4 5 6 7 8 9
7 8 9 10 11 12 13 0 1 2 3 4 5 6
4 5 6 7 8 9 10 11 12 13 0 1 2 3
1 2 3 4 5 6 7 8 9 10 11 12 13 0
12 13 0 1 2 3 4 5 6 7 8 9 10 11
9 10 11 12 13 0 1 2 3 4 5 6 7 8
6 7 8 9 10 11 12 13 0 1 2 3 4 5
3 4 5 6 7 8 9 10 11 12 13 0 1 2
0 1 2 3 4 5 6 7 8 9 10 11 12 13
1k
1 -P
14 3
0 1Y
2 -P
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Algorithm 2
0 1 2 3 4 5 6 7 8 9 10 11 12 13 0 1 2 3 4 5 6 7 8 9 10 11 12 13
11 12 13 0 1 2 3 4 5 6 7 8 9 10 11 12 13 0 1 2 3 4 5 6 7 8 9 10
8 9 10 11 12 13 0 1 2 3 4 5 6 7 8 9 10 11 12 13 0 1 2 3 4 5 6 7
5 6 7 8 9 10 11 12 13 0 1 2 3 4 5 6 7 8 9 10 11 12 13 0 1 2 3 4
2 3 4 5 6 7 8 9 10 11 12 13 0 1 2 3 4 5 6 7 8 9 10 11 12 13 0 1
13 0 1 2 3 4 5 6 7 8 9 10 11 12 13 0 1 2 3 4 5 6 7 8 9 10 11 12
10 11 12 13 0 1 2 3 4 5 6 7 8 9 10 11 12 13 0 1 2 3 4 5 6 7 8 9
7 8 9 10 11 12 13 0 1 2 3 4 5 6 7 8 9 10 11 12 13 0 1 2 3 4 5 6
4 5 6 7 8 9 10 11 12 13 0 1 2 3 4 5 6 7 8 9 10 11 12 13 0 1 2 3
1 2 3 4 5 6 7 8 9 10 11 12 13 0 1 2 3 4 5 6 7 8 9 10 11 12 13 0
12 13 0 1 2 3 4 5 6 7 8 9 10 11 12 13 0 1 2 3 4 5 6 7 8 9 10 11
9 10 11 12 13 0 1 2 3 4 5 6 7 8 9 10 11 12 13 0 1 2 3 4 5 6 7 8
6 7 8 9 10 11 12 13 0 1 2 3 4 5 6 7 8 9 10 11 12 13 0 1 2 3 4 5
3 4 5 6 7 8 9 10 11 12 13 0 1 2 3 4 5 6 7 8 9 10 11 12 13 0 1 2
0 1 2 3 4 5 6 7 8 9 10 11 12 13 0 1 2 3 4 5 6 7 8 9 10 11 12 13
11 12 13 0 1 2 3 4 5 6 7 8 9 10 11 12 13 0 1 2 3 4 5 6 7 8 9 10
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Packing Code
1
Every polycube packing can be associated with some -tuple
( ;...; ) with 0 2 -1 for 1 .
This -tuple is called a .
We use this code to recognize packings equivalent by some
moveme
dn i
n
c c c i d
n
packing code
nt of dimensional space.
We also use this coding to obtain an algorithm for generation
of all periodic polycube tilings with a given volume of
fundamental domain and a given number of polycubes.
d
3
2
1
0
Packing Code
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0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1
4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5
1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2
5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6
2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3
6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0
3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4
0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1
4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5
3 2 0 2 1 3 1 24
Packing Code
0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1
4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5
1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2
5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6
2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3
6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0
3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4
0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1
4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5
3 2 0 2 1 3 10 1 2 3 4 5 6 7 3
Packing space matrix 0 1
Packing code:
Packing Code
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The numbers tZ(n) of periodic tilings of the plane on Z polyominoes with volume of the fundamental domain equals to n
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nZ
3 4 5 6 7 8 9 10 11 12 13 14 15
1 3 10 12 43 48 171 253 632 815 3205 3236 9304 17434
2 2 8 15 73 100 445 851 2472 3573 18091 18858 64986 142940
3 1 5 10 54 91 521 1160 4064 6685 41092 47022 188354 472222
4 1 3 26 47 341 894 3773 7111 51716 66314 305075 860394
5 1 8 19 147 452 2241 4898 41724 60677 320231 1010519
6 1 4 45 145 941 2326 23515 38889 236807 834188
7 1 10 44 278 816 9537 18279 128673 508920
8 1 7 68 202 2936 6380 52994 235652
9 1 11 47 654 1728 16575 84466
10 1 6 132 341 4070 23360
11 1 18 66 749 5140
12 1 7 128 837
13 1 14 138
14 1 13
15 1
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1( )t n
n
1 2
1 1 2
4
There exist constants , such that
2 ( ) 2.7n n
c c
c t n c
Theorem
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3. Maleev A.V. n-Dimensional pacing space. // Crystallography Reports, 1995, 40.4. Maleev A.V., Lysov A.E., Potekhin K.A. Symmetry n -Dimensional pacing space. //
Crystallography Reports 1998, 43, 721-727.5. Maleev A.V. An Algorithm and Program of Exhaustive Search for Possible Tiling of a Plane
with Polyominoes. // Crystallography Reports, 2001, 46, 154-156.6. Maleev A.V. Generation of Molecular Bravais Structures by the Method of Discrete
Modeling of Packings. Crystallography Reports, 46, 2001, 13–18.7. Maleev A.V. Generation of the Structures of Molecular Crystals with Two Molecules
Related by the Center of Inversion in a Primitive Unit Cell. // Crystallography Reports, 47, 2002, 731–735.
8. Maleev A.V. Generation of Structures of Molecular Crystals with Two Molecules Related by a Twofold Axis or a Plane of Symmetry in a Primitive Unit Cell. // Crystallography Reports, 2006, 51, 559–563.
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