112/04/19 1

Advisor : Chih-Hung YenStudent : Chia-Ying Wu

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112/04/19 2

1. Fundamental Concepts

1.1 Polyominoes, Enumerations, and Tessellations

1.2 Skewing Schemes and Data Templates

1.3 Surroundings

1.4 Labelings

2. Main Results

2.1 Some Results

2.2 N2-Skewing Schemes

3. Conclusion

112/04/19 2

OutlineOutline

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112/04/19 5

Let the Euclidean plane (or the plane) be divided into unit

squares, that is, the four corners of a square have

coordinates (x,y), (x+1,y), (x,y+1), (x+1,y+1) for some

integers x and y. And, for each unit square, we use the

coordinate of the lower left corner to name itself.

112/04/19 55

112/04/19 6

For example, we use unit square (1,1) to represent the

unit square whose four corners have coordinates (1,1),

(2,1), (1,2) and (2,2).

112/04/19 66

(1,1) (2,1)

(1, 2) (2, 2)

y

x

112/04/19 7

A polyomino is defined as a finite, nonempty, and connected set of unit

squares.

A configuration generalizes the notation of polyomino by dropping the

requirement “being connected”.

112/04/19 777

y

x

112/04/19 8

Consider a polyomino P (or a configuration C) of size N,

namely, P (or C) consists of N unit squares.

A polyomino of size N is also called an N-omino.

112/04/19 8

112/04/19 9

In 1907, the notion of “polyomino” appeared in a puzzle

involving 5 unit squares was posed in the book 「 Canterbury

Puzzles 」 .

Between the years 1937 to 1957, many results with the

polyominoes of size 1 to 6 were first published in 「 Fairy

Chess Review 」 , a puzzle journal issued in British, under the

name “dissection problems”.

In 1953, the name polyomino was invented by Golomb and

popularized by Gardner.

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112/04/19 10

Polyominoes are sources of combinatorial problems.

The most basic one is to enumerate distinct

polyominoes of size N for a given positive integer N.

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112/04/19 1111112/04/19 11

Another important problem on the subject of polyomino is

the following:

『 Given a polyomino P, can P tessellate the plane? 』

1111112/04/19

112/04/19 12112/04/19 1212

x

(1,1) (2,1)

(1, 2)

(3,1)

(6,2) (7,2)

(6, 3)

(8,2)

v =(vx,vy) = (5,1)

Q

P

y

112/04/19 13

y

x

P

Q

112/04/19 14

y

x

Q

P

112/04/19 15

Consider two polyominoes P and Q in the plane. If Q satisfies

one of the following conditions, then Q is said to be

equivalent to P.

1. Q is a translation, a rotation, or a reflection of P.

2. Q is a rotation or a reflection of some translation of P.

3. Q is a reflection of some rotation of P.

4. A translation of Q is a reflection of some rotation of P.

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112/04/19 16112/04/19 16

We say that a polyomino P tiles the plane, or there exists a

tiling of the plane using a polyomino P, if the plane can be

composed of polyominoes that are equivalent to P and do

not overlap except along their sides.

1616

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We say that a polyomino P tessellates the plane, or there

exists a tessellation of the plane using a polyomino P, if the

plane can be composed of polyominoes that are translations

of P and do not overlap except along their sides.

1818

112/04/19 19112/04/19 19112/04/19 19

x

y

O

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112/04/19 20112/04/19 20

If a polyomino P tessellates the plane, then P also tiles the plane.

Conversely, if P tiles the plane, then we cannot guarantee that P

tessellates the plane.

2020

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It is known that every N-omino for 1 N 6 tiles the

plane and only four of all 7-ominoes cannot tile the

plane.

2121

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Golomb proved that the problem of determining whether an

arbitrary finite set of polyomino tiles the plane is undecidable;

that is, there is no NP-algorithm for this problem. On the other

hand, there does exist several methods for determining

whether a (single) polyomino tessellates the plane.

2222

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A single instruction-stream, multiple data-stream (SIMD)

computer contains one control unit, t arithmetic processors,

and one memory unit of N (independent) memory modules. All

arithmetic processor receive the same instruction from the

control unit, but operate on different items of data stored in

different memory modules.

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112/04/19 25

control unit

A1 A2 At

memory-processor connection network

M1 M2 MN

arithmetic processors

memory modules

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0,0 0,1 0,2 0, 1

1,0 1,1 1,2 1, 1

2,0 2,1 2,2 2, 1

1,0 1,1 1,2 1, 1

N

N

N

N N N N N N N

A A A A

A A A A

A A A AA

A A A A

We consider to store an N N matrix A into the N memory

modules of an SIMD computer, where A = [Ai, j] for i = 0,1,…,N －

1 and j = 0,1,…,N － 1.

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If the element Ai, j is stored in memory module j for all i and j,

then it will be possible to simultaneously fetch all the elements

of any row of A, since distinct elements of a row of A lie in

distinct memory modules.

27112/04/19

A0,0

A1,0

A2,0

.

.

.AN1,0

A0,1

A1,1

A2,1

.

.

.AN1,1

A0,2

A1,2

A2,2

.

.

.AN1,2

A0, N1

A1, N1

A2, N1

.

.

.AN1, N1

M0 M1 M2 MN1

112/04/19 28

Fetching all the elements of any column of A, however, will

result in delays, since more than one element of the

column of A (in fact all) will lie in the same memory

module.

28112/04/19

A0,0

A1,0

A2,0

.

.

.AN1,0

A0,1

A1,1

A2,1

.

.

.AN1,1

A0,2

A1,2

A2,2

.

.

.AN1,2

A0, N1

A1, N1

A2, N1

.

.

.AN1, N1

M0 M1 M2 MN1

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A0,0

A1, N1

A2, N2

.

.

.AN1,1

A0,1

A1,0

A2, N1

.

.

.AN1,2

A0,2

A1,1

A2,0

.

.

.AN1,3

A0, N1

A1, N2

A2, N3

.

.

.AN1, 0

M0 M1 M2 MN1

112/04/19

If we adopt a different storage strategy instead, where the

element Ai, j is stored in memory module i + j (mod N), then all

the elements of any row or any column of A will lie in distinct

memory modules and can be fetched simultaneously.

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Problem 1.2.1 (Shapiro; 1978)

Given an M M matrix, a collection of desirable matrix

subparts such as rows, columns, or square blocks, and an

SIMD computer with N memory modules, how do we store the

matrix so that all the elements comprising any desirable matrix

subpart are stored in different memory modules?

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112/04/19 3131112/04/19

A0,0

A1, N1

A2, N2

.

.

.AN1,1

A0,1

A1,0

A2, N1

.

.

.AN1,2

A0,2

A1,1

A2,0

.

.

.AN1,3

A0, N1

A1, N2

A2, N3

.

.

.AN1, 0

M0 M1 M2 MN1

112/04/19 32

COLUMN0

COLUMN1

ROW 0

ROW 1

32

ROW 2

0 1 2

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COLUMN2

COLUMN3

COLUMN

N1

3

N1

1

2

ROW 3

ROW N1

3

N1

2 3

3

…

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For any two positive integers M and N, an (M, N)-skewing

scheme is defined as a 2-dimentional funtion S : ℤM × ℤM

ℤN, namely, for each (i, j ) ℤM × ℤM , there exists a k ℤN

such that S(i, j ) = k.

An N-skewing scheme is a 2-dimentional funtion S : × ℤ ℤ

ℤN, namely, for each k ℤN such that S(i, j ) = k.

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A data template T is a set of ordered pairs of nonnegative

integers in which no two components identical, namely, T =

{ (x1, y1), (x2, y2), ..., (xt, yt) }, where xi 0, yi 0, and (xi, yi)

(xj, yj) for any i j.

An instance of a data template T through a vector v × ℤ ℤ,

denoted by T+v, is a set of ordered pairs of integers which

formed by componentwise addition of v to T. For example, if T

= {(x1, y1), (x2, y2), ..., (xt, yt) } and v = (vx,vy), then T+v =

{ (x1+vx, y1+vy), (x2+vx, y2+vy), ..., (xt +vx, yt +vy) }.

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In fact, for any data template T or any instance T+v of T,

there exists uniquely a polyomino or a configuration

which is corresponding to T or T+v, and vice versa.

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y

x(0,0) (1,0)

(0,1)

(2,0)

For example, a data template T = { (0,0), (1,0), (2,0), (0,1) }.

P

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We say that an (M, N)-skewing scheme or an N-skewing

scheme, denoted by S, is valid for a collection of data

templates, denoted by T1, T2, ..., Tr , if and only if, for any two

ordered pairs (i1, j1) and (i2, j2) satisfying S(i1, j1) = S(i2, j2), there

exists no l {1, 2, ..., r } such that an instance of Tl contains

both (i1, j1) and (i2, j2) as components.

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Problem 1.2.2

Consider a collection of data templates, how do we determine

if there is a valid (M, N)-skewing scheme for this collection of

data templates, and if a valid (M, N)-skewing scheme exists,

how do we determine what it is?

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Problem 1.2.3

Consider a data template T of size N satisfying that a

polyomino corresponds to T, how do we determine whether

there is a valid N-skewing scheme for T, if a valid N-skewing

scheme for T exists, how do we determine what it is?

Problem 1.2.4

Consider a data template T of size N satisfying that a

configuration corresponds to T, how do we determine whether

there is a valid N-skewing scheme for T, if a valid N-skewing

scheme for T exists, how do we determine what it is?

3939

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Theroem 1.2.5 (Shapiro; 1978) (Wijshoff and van Leeuwen;

1984)

Consider a data template T of size N. Then there exists a valid N-

skewing scheme for T if and only if the polyomino or the

configuration corresponding to T tessellates the plane.

4040

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4242112/04/19

P0

P1

P2

P3

P4

P5

P

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Theorem 1.3.1 (Beauquier and Nivat; 1991)

Consider a polyomino P. Then the plane can be tessellated by

P if and only if there exists a surrounding of P.

4343

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Consider a polyomino P (or a configuration C) of size N in the

plane. Then P (or C) has a 1-linear labeling if we can label the

unit squares of P (or C) by using the elements of ℤN exactly

once in such a way that the labels of unit squares in each row

is an arithmetic sequence with skip parameter a and the labels

of unit squares in each column is an arithmetic sequence with

skip parameter b, where a, b ℤN and a = b is allowed.

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00 11

22

44

33

55 66

77

112/04/19 46

Consider a polyomino P (or a configuration C) of size N in the

plane. Then P (or C) has a 2-linear labeling if we can label the

unit squares of P (or C) by using the elements of ℤf × ℤN/f

exactly once for some f 1 and f | N (i.e., f divides N) in such a

way that the labels of unit squares in each row is an arithmetic

sequence with skip parameter A=(a1,a2) and the labels of unit

squares in each column is an arithmetic sequence with skip

parameter B=(b1,b2), where A, B ℤf × ℤN/f and A = B is

allowed.

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(0,2)(0,2)

(1,1)(1,1)

(0,0)(0,0)

(1,2)(1,2) (1,3)(1,3) (1,0)(1,0)

(0,1)(0,1)

(0,3)(0,3)

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If a polyomino P (or a configuration C) of size N has a 1-linear

labeling with skip parameter a and b in ℤN , then P (or C) also

has a 2-linear labeling with skip parameters A = (0,a) and B =

(b, 0) ℤ1 × ℤN . But the converse is not.

112/04/19 48

(0,2)(0,2)

(1,1)(1,1)

(0,0)(0,0)

(1,2)(1,2) (1,3)(1,3) (1,0)(1,0)

(0,1)(0,1)

(0,3)(0,3)

Theorem 1.4.1 (Chen, Hwang and Yen; 2006)

A polyomino P of size N tessellates the plane if and only if P has a

2-linear labeling.

4949112/04/19

00 11

22

44

33

55 66

77 (0,2)(0,2)

(1,1)(1,1)

(0,0)(0,0)

(1,2)(1,2) (1,3)(1,3) (1,0)(1,0)

(0,1)(0,1)

(0,3)(0,3)

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Lemma 2.1.1

An N-skewing scheme S is N-periodic if and only if S(i, j) = S(i + N, j) = S(i,

j + N) for any (i, j ) ℤ × ℤ.

Proof.

() It is trivial.

() We want to show that S(i, j) = S(i + N, j + N) for any (i, j ) ℤ × ℤ and

any two integers and . Without loss of generality, we suppose that

0 and 0. For any (i, j ) ℤ × ,ℤ since S(i, j) = S(i + N, j ) = S(i, j + N), we

have S(i, j) = S(i + N, j ) = S(i + 2N, j ) = … = S(i + (1)N, j ) = S(i + N, j )

= S(i + N, j + N) = S(i + N, j + 2N) = … = S(i + N, j + ( 1)N) = S(i +

N, j + N). Thus S is N-periodic.112/04/19 52

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Lemma 2.1.2

A linear N-skewing scheme is also N-periodic.

Proof.

Let S be a linear N-skewing scheme. By Lemma 2.1.1, it suffices to

show that S(i, j) =S(i + N, j) = S(i, j + N) for any (i, j ) ℤ × ℤ. Since S

is linear, there exist a and b in ℤN such that S(i, j) = ia + jb (mod N) for

any (i, j ) ℤ × ℤ, where a = b is allowed. Hence, for any (i, j ) ℤ ×

ℤ, we have S(i, j) = ia + jb (mod N) = ia + Na + jb (mod N) = (i + N)a +

jb (mod N) =S(i + N, j) and S(i, j) = ia + jb (mod N) = ia + jb + Nb (mod

N) =ia + (j + N)b (mod N) = S(i, j + N). Therefore, S is an N-periodic

N-skewing scheme.112/04/19 53

112/04/19 54

Theorem 2.1.3

Problem 1.1.1, Problem 1.1.2, Problem 1.2.3, Problem 1.3.1, and

Problem 1.4.2 are pairwise equivalent.

Problem 1.1.1. For what classes of polyominoes do the existence of

tessellations of the plane?

Problem 1.1.2. For what classes of polyominoes do the existence of

periodic tessellations of the plane?

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Problem 1.2.3. Consider a data template T of size N satisfying that a

polyomino corresponds to T, how do we determine whether there is a

valid N-skewing scheme for T, if a valid N-skewing scheme for T

exists, how do we determine what it is?

Problem 1.3.1. For what classes of polyominoes do the existence of

surroundings?

Problem 1.4.2. For what classes of polyominoes do the existence of

2–linear labelings?

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Theorem 2.1.4

Problem 1.1.3, and Problem 1.2.4 are equivalent.

Problem 1.1.3.

Does the existence of tessellations of the plane using a polyomino P

imply the extence of periodic tessllations of the plane using P?

Problem 1.2.4. Consider a data template T of size N satisfying that a

configuration corresponds to T, how do we determine whether there is a

valid N-skewing scheme for T, if a valid N-skewing scheme for T exists,

how do we determine what it is?

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One feasible method for determine the existence of such a (A, B)

is to test each ordered pairs of elements in ℤf × ℤN/f for any f 1

and f | N. However, it is inefficient. Let TN denote the total number

of tests performed and fN denote the number of factors of N. Then

112/04/19 575757

21 1 .N NT N f

N 2 3 4 5 10 100

fN 2 2 3 2 4 9

TN 1 4 18 16 243 78408

112/04/19 58

On the other hand, if N is a prime number, then fN = 2 and

TN (N 1)2; otherwise, fN N and TN (N 1)3. Therefore, a

method to find (A, B) quickly is required.

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An N2-skewing scheme is a mapping S: ℤ ℤ ℤf × ℤN/f for

some f 1 and f | N, namely, for each (i, j) ℤ ℤ, there

exists a (k1, k2) ℤf × ℤN/f such that S(i, j) = (k1, k2) .

An N2-skewing scheme S is valid for a collection of data

templates, denoted by T1, T2, ..., Tr , if and only if, for any two

ordered pairs (i1, j1) and (i2, j2) satisfying S(i1, j1) = S(i2, j2), there

exists no l {1, 2, ..., r} such that an instance of Tl contains both

(i1, j1) and (i2, j2) as components.

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An N2-skewing scheme S is N-periodic if S(i, j) = S( i + N, j +

N ) for any (i, j) ℤ ℤ and any two integers and .

An N2-skewing scheme S is periodic if there exist u = (u1, u2)

and v = (v1, v2) in ℤ ℤ such that S(i, j) = S( i + u1 + v1, j +

u2 + v2 ) for any (i, j) ℤ ℤ and any two integers and .

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An N2-skewing scheme S is linear if there exist A = (a1, a2) and

B = (b1, b2) in ℤf × ℤN/f such that S(i, j) = iA + jB (mod (f , N/f)) =

(ia1 + jb1 (mod f ), ia2 + jb2 (mod N/f )) for any (i, j) ℤ ℤ,

where A = B is allowed.

If an N-skewing scheme S is linear, then it is easy to obtain a

linear N2-skewing scheme S : ℤ ℤ ℤ1 × ℤN by letting S(i,

j)=(0, S(i, j)) for any (i, j) ℤ ℤ.

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112/04/19 65

Lemma 2.2.1

An N2-skewing scheme S is N-periodic if and only if S(i, j) = S(i + N, j) = S(i,

j + N) for any (i, j ) ℤ × ℤ.

Proof.

() It is trivial.

() We want to show that S(i, j) = S(i + N, j + N) for any (i, j ) ℤ × ℤ and

any two integers and . Without loss of generality, we suppose that

0 and 0. For any (i, j ) ℤ × ,ℤ since S(i, j) = S(i + N, j ) = S(i, j + N), we

have S(i, j) = S(i + N, j ) = S(i + 2N, j ) = … = S(i + (1)N, j ) = S(i + N, j )

= S(i + N, j + N) = S(i + N, j + 2N) = … = S(i + N, j + ( 1)N) = S(i +

N, j + N). Thus S is N-periodic.112/04/19 65

112/04/19 66

Lemma 2.2.2

A linear N2-skewing scheme is also N-periodic.

Proof.

Let S be a linear N2-skewing scheme. By Lemma 2.2.1, it suffices to

show that S(i, j) =S(i + N, j) = S(i, j + N) for any (i, j ) ℤ × ℤ. Since S

is linear, there exist A = (a1, a2) and B = (b1, b2) in ℤf × ℤN/f such that

S(i, j) = iA + jB (mod (f , N/f)) = (ia1 + jb1 (mod f ), ia2 + jb2 (mod N/f ))

for any (i, j) ℤ ℤ, where A = B is allowed.

112/04/19 66

112/04/19 67

Hence, for any (i, j ) ℤ × ℤ, we have

S(i, j) = iA + jB (mod (f , N/f))

= (ia1 + jb1 (mod f ), ia2 + jb2 (mod N/f ))

= (ia1 + f N/f a1+ jb1 (mod f ), ia2 + f N/f a2 + jb2 (mod

N/f ))

= (ia1 + Na1+ jb1 (mod f ), ia2 + Na2 + jb2 (mod N/f ))

= ((i + N)a1+ jb1 (mod f ), (i + N)a2 + jb2 (mod N/f ))

= (i + N)A+ jB (mod (f , N/f))

= S(i + N, j)

and112/04/19 67

112/04/19 68

S(i, j) = iA + jB (mod (f , N/f))

= (ia1 + jb1 (mod f ), ia2 + jb2 (mod N/f ))

= (ia1 + jb1 + f N/f b1 (mod f ), ia2 + jb2 + f N/f b2 (mod

N/f ))

= (ia1 + jb1 + Nb1 (mod f ), ia2 + jb2 + Nb2 (mod N/f ))

= (ia1+ (j + N)b1 (mod f ), ia2 + (j + N)b2 (mod N/f ))

= iA+ (j + N)B (mod (f , N/f))

= S(i, j+ N)

Therefore, S is an N-periodic N2-skewing scheme.

112/04/19 68

112/04/19 69112/04/19 69

112/04/19 70

The study of tessellations of the plane using polyominoes has a

long history in mathematics. And the motivation of our study

comes from the labelings of polyominoes. We revise some

definitions and terminologies, and give some results on the

correlations between different methods of determining whether

a polyomino tessellates the plane. However, there are still

some unsolved problems.

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