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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3

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

11

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

12

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

30

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

30

41

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

52

Packing Space

13

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

15

Using this theorem we obtain the algorithm which

generates all translation packings of a given polycube

with a given packing density.

6

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

21

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

22

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

23

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

25

The numbers tZ(n) of periodic tilings of the plane on Z polyominoes with volume of the fundamental domain equals to n

26

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

27

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

1. Maleev A.V., Rau V.G., Potekhin K.A., Parkhomov L.G., Rau T.F., Stepanov S.V., Struchkov Yu.T. Discrete modeling method in molecular crystals. // Dokl. Akad. Nauk SSSR, 1990, 315, 1382-1385.

2. Potekhin K.A., Maleev A.V. Molecular cells in organic crystals. // Dokl. Akad. Nauk SSSR, 1991, 318, 1170-1173.

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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