Tessellating Polyominoes for the Plane. Advisor : Chih -Hung Yen Student : Chia -Ying Wu. 2014/11/13. 1. 1. Outline. 1. Fundamental Concepts 1.1 Polyominoes, Enumerations, and Tessellations 1.2 Skewing Schemes and Data Templates 1.3 Surroundings 1.4 Labelings

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Tessellating Polyominoes for the Plane

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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. 2014/11/13 5 5

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. 2014/11/13 9 9

2014/11/13 • Another important problem on the subject of polyomino is the following: 『Given a polyomino P, can P tessellate the plane?』 2014/11/13 11 11 11 11

(6, 3) v =(vx,vy) = (5,1) (1, 2) (8,2) (6,2) (7,2) (3,1) (1,1) (2,1) y Q P x 2014/11/13 12 12

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. 2014/11/13 15

2014/11/13 We say that a polyomino Ptiles 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. 16 16 16

2014/11/13 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. 18 18 18

2014/11/13 If a polyomino Ptessellates the plane, then P also tiles the plane. Conversely, if Ptiles the plane, then we cannot guarantee that Ptessellates the plane. 20 20 20

2014/11/13 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. 21 21 21

2014/11/13 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. 22 22 22

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. 2014/11/13 24

control unit arithmetic processors A1 A2 At memory-processor connection network memory modules M1 M2 MN 2014/11/13 25

M0 M1 M2 MN1 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 • 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. 2014/11/13 27

M0 M1 M2 MN1 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 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. 2014/11/13 28

M0 M1 M2 MN1 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 • 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. 2014/11/13 29

Problem 1.2.1 (Shapiro; 1978) Given an M Mmatrix, 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? 2014/11/13 30

… COLUMN0 COLUMN1 COLUMN2 COLUMN3 C O L U M N N1 N1 3 ROW 0 0 1 2 2 3 ROW 1 1 3 2 ROW 2 3 ROW 3 ROW N1 N1 2014/11/13 32

For any two positive integers M and N, an (M, N)-skewing schemeis 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. 2014/11/13 33

A data template Tis 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 ij. An instance of a data template T through a vectorv 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) }. 2014/11/13 34

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. 2014/11/13 37

2014/11/13 • 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? 38 38 38

2014/11/13 • 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? 39 39 39

2014/11/13 • 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. 40 40 40

2014/11/13 • 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. 43 43 43

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. 7 4 5 6 2 3 0 1 2014/11/13 45 45 45

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 f1 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. 2014/11/13 46 46 46

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. (0,2) (0,1) (1,1) (1,2) (1,3) (1,0) (0,0) (0,3) 2014/11/13 48

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. (0,2) (0,1) (1,1) (1,2) (1,3) (1,0) (0,0) (0,3) 7 4 5 6 2 3 0 1 49