六角形邊著兩色的決定性問題

國

立

交

通

大

學

應用數學系

碩

士

論

文

六角形邊著兩色的決定性問題

Decidability Problems of Hexagonal Edge-coloring

with Two Colors

研 究 生：賴德展

六角形邊著兩色的決定性問題

Decidability Problems of Hexagonal Edge-coloring

with Two Colors

研 究 生：賴德展

Student：De-Jan Lai

指導教授：林松山

Advisor：Song-Sun Lin

國 立 交 通 大 學

應 用 數 學 系

碩 士 論 文

A Thesis

Submitted to Department of Applied Mathematics College of Science

National Chiao Tung University in partial Fulfillment of the Requirements

for the Degree of Master

in

Applied Mathematics June 2010

六角形邊著兩色的決定性問題

學生：賴德展

指導教授

：

林松山教授

國立交通大學應用數學學系﹙研究所﹚碩士班

摘 要

這個研究是關於用邊著色的正六角形拼湊整個平面。如果對每個相交處

都有相同的顏色，則正六角形可以放在相鄰的位置。在這篇論文，我們考

慮邊上只著兩色的正六角形。我們研究的問題為：任意可佈滿整個平面的

六角形集合是否可以週期性地覆蓋整個平面。我們使用演算法來研究這個

問題，然後藉由電腦計算得到結果。最後，這篇論文的主要結果為：如果

整個平面可以被邊著兩色的正六角形拼滿，則整個平面就被週期性的區塊

圖案覆蓋，反之亦然。

Decidability Problems of Hexagonal

Edge-coloring with Two Colors

student：

De-Jan Lai

Advisors：Dr.

Song-Sun Lin

Department﹙Institute﹚of

Applied Mathematics

National Chiao Tung University

ABSTRACT

This investigation is about tiling the whole plane with regular hexagons

which have colors on edges. Regular hexagons can be placed side by side if each

of the intersections has the same color. In this paper, we consider regular

hexagons with only two colors on edges. The problem we studied is that: any set

of hexagons that can fill with the whole plane whether it can cover the whole

plane periodically. We use an algorithm to do the problem and get the result by

computers. Finally, the main result of this paper is that the whole plane can be

tiling by regular hexagons with two colors if and only if the whole plane is

covered by the local pattern periodically.

致

謝

本篇論文的完成，首先要感謝我的指導教授 ─ 林松山教授。在老師的

鞭策與教導之下，除了學業上的成長，待人接物的禮節與人品更是老師特

別著重的部分。老師常說：「做人要正直。」學生謹記在心。

其次要感謝的是胡文貴學長。學長的指導與細心解說，使我能跟上老師

的步伐。學長個性溫和，耐心無與倫比，是我遇到問題尋求幫助的第一人

選。每當遇到瓶頸與學長討論完後，總會有新的方向可以繼續前進。謝謝

學長的照顧。

其次要感謝的是我的夥伴陳泓勳，研究室位置就在我的隔壁，遇到問題

能快速回答我的疑惑，打球時也很可靠，總會帶東西回來給我吃。接下來

要感謝研究所的朋友們，這兩年來的相處，製造了不少笑點，在一起非常

快樂。其中要特別感謝林志嘉，瞎鬧中我交了第一個女朋友，改變了我的

人生。

最後要感謝我的家人，能長到這麼大都是你們的功勞。謝謝所有幫助過

我的貴人，謝謝。

目

錄

中文提要

………

i

英文提要

………

ii

誌謝

………

iii

目錄

………

iv

一、

Introduction ………

1

二、

Ordering Matrix ………

3

2.1

_Construct

T n

………

8

2.2

_Construct

Yn

………

8

三、

Symmetry of Hexagons ………

11

四、

Algorithm ………

14

五、

Conclusion ………

16

Reference ………

33

1

Introduction

There are three possible cases such that the plane is covered by one kind of regular polygons: regular triangles or regular squares or regular hexagons. See the Figure 1.

Figure 1.

Squares with colors on edges for tiling the plane is well-known as Wang tile. In 1961, Wang [12] started to study the square tiling of a plane. Wang gave a conjecture: any set of tiles that can fill with the whole plane can cover the whole plane periodically. This conjecture was false proved by Berger [4]. Berger presented a set of 20426 Wang tiles that could only tile the plane aperiodically. Later, Berger reduced the number of tiles to 104. Thereafter, smaller sets of Wang tiles were found by Knuth, L¨auchli, Robinson, Penrose, Ammann, Culik and Kari [5, 6, 7, 10, 11]. Currently, the smallest number of tiles that can tile the plane aperiodically is 13, with five colors[5]. Nevertheless, Wang’s conjecture was true proved by Hu and Lin [13] with two colors on edges. The question we want to know is that Wang’s conjecture is true or not if we use regular hexagons with two colors.

Now consider the plane covered by regular hexagons with two different colors, denoted by the numbers 0 or 1, on edges. Two hexagons can be arranged side by side if all intersections have the same color. There are 26 = 64 such hexagons. For convenience, we encode these hexagons to numbers from 1 to 64 with the following rule, where α0, α1, α2, α3, α4, α5 ∈

{0, 1}: see Figure 2.

Definition 1.1. The set {1, 2, 3, . . . , 64}, denoted by T (2), is called the total

set of hexagons with two colors. Each element of T (2) represents a specific

hexagon.

Definition 1.2. The set contained by T (2) is called a cycle generator if it can cover the plane periodically. Let C(2) be the set contains all cycle generators.

Definition 1.3. The set, B, contained by T (2) is called a minimal cycle

α0 α1 α2 α3 α4 α5 ⇐⇒W = 1 + α020+ α121+ α222+ α323+ α424+ α525 Figure 2. 1. B ∈ C(2). 2. ∀B0 $B, B0 <C(2).

Let Cm(2) be the set contains all minimal cycle generators.

Definition 1.4. The set, B, contained by T (2) is called a non-cycle generator if B < C(2). Let N(2) be the set contains all non-cycle generator.

Definition 1.5. The set, B, contained by T (2) is called a maximal non-cycle

generator if B satisfies the following two conditions:

1. B ∈ N(2).

2. ∀B0 %B, B0 ∈ C(2).

Let NM(2) be the set contains all maximal non-cycle generators.

Given B ⊆ T (2). Let Σ(B) be the set of all global patterns on the plane that can be constructed by B and P(B) be the set of all periodic patterns on the plane that can be generated by B. Now our question can be written by if Σ(B) , ∅ whether P(B) , ∅ or not. (1.1) All we have to do is to investigate all subset of T (2) whether (1.1) is true or not. Though there are 264_{such cases. So in the following sections,}

some methods are introduced to reduce cases. In section 2, we introduce two useful matrices T and Y such that if given B ⊆ T (2), we can decide whether B ∈ C(2) with T and whether B ∈ N(2) with X. In section 3, a method called classification can reduce possible cases. In section 4, we use an algorithm to deal with the problem. Finally, we derive a conclusion in section 5.

2

Ordering Matrix

Since the plane has two dimensions, we first can construct the pattern along horizontal direction and then second along vertical direction. In the following we show how to obtain the horizontal patterns. Let X∗

1 be the

matrix of the following form( see Figure 3 ) ,where β ∈ {0, 1}. More general form of X∗

1can be written as Figure 4 where α1, α2, α

0 1, α 0 2, β ∈ {0, 1}. 0 0 0 0 0 0 0 1 1 0 0 0 1 0 0 1 0 0 1 0 0 0 1 1 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 1 1 0 0 1 1 0 1 0 1 1 0 0 1 1 1 1 1 1 0 1 1 1 1 0 1 1 0 1 1 0 0 0 0 0 1 1 0 1 1 β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β β Figure 3. X∗ 1 α0₁ α0₁ α0₁ α0₁ α0₂ α0₂ α0₂ α0₂ α1 α2 α1 α2 β β

Figure 4. the general form of X∗ 1

Let X∗_1;1 be the form of Figure 5(a). The complete matrix X∗_1;1 can be easily obtained by replacing Figure 3’ ββ from left to right of the hexagon

α0₁ α0₁ αα0₂0₂ α1 α2 0 0 (a) X∗ 1;1 α0₁ α0₁ αα0₂0₂ α1 α2 0 1 (b) X∗ 1;2 α0₁ α0₁ αα0₂0₂ α1 α2 1 0 (c) X∗_1;3 α0₁ α0₁ αα0₂0₂ α1 α2 1 1 (d) X∗_1;4 Figure 5. with 0 and 0. Each element of X∗

1;1 is denoted by x ∗ 1;1;i, j. Similarly, X ∗ 1;2, X∗ 1;3, X ∗

1;4can be defined, and their matrix forms are shown in Figure 5(b),

Figure 5(c), Figure 5(d). We can find X∗₁ = X

i=1,4

X∗_1;i (2.1)

Now consider the general form X∗

2 as Figure 6. We can define X ∗ 2;1 by

replacing Figure 6’s ββ from left to right of the hexagon with 0 and 0. X∗ 2;1

can derive from X∗

1in the following way, see equations (2.2).

α1 α2 α3 α4 α1 α2 α3 α4 α0₁ α0₁ α0₂ α0₂ α0₃ α0₃ α0₄ α0₄ β β

Figure 6. the general form of X∗ 2 Similarly, X∗ 2;2, X ∗ 2;3, X ∗

2;4 can be derive from equations (2.3), (2.4), (2.5)

X∗_2;1 =                                 x1;1;1,1X∗_1;1+ x1;2;1,1X∗_1;3 x1;1;1,2X∗_1;1+ x1;2;1,2X∗_1;3 x1;1;1,3X∗_1;1+ x1;2;1,3X∗_1;3 x1;1;1,4X∗_1;1+ x1;2;1,4X∗_1;3 x1;1;2,1X∗_1;1+ x1;2;2,1X∗_1;3 x1;1;2,2X∗_1;1+ x1;2;2,2X∗_1;3 x1;1;2,3X∗_1;1+ x1;2;2,3X∗_1;3 x1;1;2,4X∗_1;1+ x1;2;2,4X∗_1;3 x1;1;3,1X∗_1;1+ x1;2;3,1X∗_1;3 x1;1;3,2X∗_1;1+ x1;2;3,2X∗_1;3 x1;1;3,3X∗_1;1+ x1;2;3,3X∗_1;3 x1;1;3,4X∗_1;1+ x1;2;3,4X∗_1;3 x1;1;4,1X∗_1;1+ x1;2;4,1X∗_1;3 x1;1;4,2X∗_1;1+ x1;2;4,2X∗_1;3 x1;1;4,3X∗_1;1+ x1;2;4,3X∗_1;3 x1;1;4,4X∗_1;1+ x1;2;4,4X∗_1;3                                 (2.2) X∗_2;2 =                                 x1;1;1,1X∗_1;2+ x1;2;1,1X∗_1;4 x1;1;1,2X∗_1;2+ x1;2;1,2X∗_1;4 x1;1;1,3X∗_1;2+ x1;2;1,3X∗_1;4 x1;1;1,4X∗_1;2+ x1;2;1,4X∗_1;4 x1;1;2,1X∗_1;2+ x1;2;2,1X∗_1;4 x1;1;2,2X∗_1;2+ x1;2;2,2X∗_1;4 x1;1;2,3X∗_1;2+ x1;2;2,3X∗_1;4 x1;1;2,4X∗_1;2+ x1;2;2,4X∗_1;4 x1;1;3,1X∗_1;2+ x1;2;3,1X∗_1;4 x1;1;3,2X∗_1;2+ x1;2;3,2X∗_1;4 x1;1;3,3X∗_1;2+ x1;2;3,3X∗_1;4 x1;1;3,4X∗_1;2+ x1;2;3,4X∗_1;4 x1;1;4,1X∗_1;2+ x1;2;4,1X∗_1;4 x1;1;4,2X∗_1;2+ x1;2;4,2X∗_1;4 x1;1;4,3X∗_1;2+ x1;2;4,3X∗_1;4 x1;1;4,4X∗_1;2+ x1;2;4,4X∗_1;4                                 (2.3) X∗_2;3 =                                 x1;3;1,1X∗_1;1+ x1;4;1,1X∗_1;3 x1;3;1,2X∗_1;1+ x1;4;1,2X∗_1;3 x1;3;1,3X∗_1;1+ x1;4;1,3X∗_1;3 x1;3;1,4X∗_1;1+ x1;4;1,4X∗_1;3 x1;3;2,1X∗_1;1+ x1;4;2,1X∗_1;3 x1;3;2,2X∗_1;1+ x1;4;2,2X∗_1;3 x1;3;2,3X∗_1;1+ x1;4;2,3X∗_1;3 x1;3;2,4X∗_1;1+ x1;4;2,4X∗_1;3 x1;3;3,1X∗_1;1+ x1;4;3,1X∗_1;3 x1;3;3,2X∗_1;1+ x1;4;3,2X∗_1;3 x1;3;3,3X∗_1;1+ x1;4;3,3X∗_1;3 x1;3;3,4X∗_1;1+ x1;4;3,4X∗_1;3 x1;3;4,1X∗_1;1+ x1;4;4,1X∗_1;3 x1;3;4,2X∗_1;1+ x1;4;4,2X∗_1;3 x1;3;4,3X∗_1;1+ x1;4;4,3X∗_1;3 x1;3;4,4X∗_1;1+ x1;4;4,4X∗_1;3                                 (2.4)

X∗_2;4 =                                 x1;3;1,1X∗_1;2+ x1;4;1,1X∗_1;4 x1;3;1,2X∗_1;2+ x1;4;1,2X∗_1;4 x1;3;1,3X∗_1;2+ x1;4;1,3X∗_1;4 x1;3;1,4X∗_1;2+ x1;4;1,4X∗_1;4 x1;3;2,1X∗_1;2+ x1;4;2,1X∗_1;4 x1;3;2,2X∗_1;2+ x1;4;2,2X∗_1;4 x1;3;2,3X∗_1;2+ x1;4;2,3X∗_1;4 x1;3;2,4X∗_1;2+ x1;4;2,4X∗_1;4 x1;3;3,1X∗_1;2+ x1;4;3,1X∗_1;4 x1;3;3,2X∗_1;2+ x1;4;3,2X∗_1;4 x1;3;3,3X∗_1;2+ x1;4;3,3X∗_1;4 x1;3;3,4X∗_1;2+ x1;4;3,4X∗_1;4 x1;3;4,1X∗_1;2+ x1;4;4,1X∗_1;4 x1;3;4,2X∗_1;2+ x1;4;4,2X∗_1;4 x1;3;4,3X∗_1;2+ x1;4;4,3X∗_1;4 x1;3;4,4X∗_1;2+ x1;4;4,4X∗_1;4                                 (2.5) X∗_n;1=                                 x1;1;1,1X∗_n−1;1+ x1;2;1,1X∗_n−1;3 x1;1;1,2X∗_n−1;1+ x1;2;1,2X∗_n−1;3 x1;1;1,3X∗_n−1;1+ x1;2;1,3X∗_n−1;3 x1;1;1,4X∗_n−1;1+ x1;2;1,4X∗_n−1;3 x1;1;2,1X∗_n−1;1+ x1;2;2,1X∗_n−1;3 x1;1;2,2X∗_n−1;1+ x1;2;2,2X∗_n−1;3 x1;1;2,3X∗_n−1;1+ x1;2;2,3X∗_n−1;3 x1;1;2,4X∗_n−1;1+ x1;2;2,4X∗_n−1;3 x1;1;3,1X∗_n−1;1+ x1;2;3,1X∗_n−1;3 x1;1;3,2X∗_n−1;1+ x1;2;3,2X∗_n−1;3 x1;1;3,3X∗_n−1;1+ x1;2;3,3X∗_n−1;3 x1;1;3,4X∗_n−1;1+ x1;2;3,4X∗_n−1;3 x1;1;4,1X∗_n−1;1+ x1;2;4,1X∗_n−1;3 x1;1;4,2X∗_n−1;1+ x1;2;4,2X∗_n−1;3 x1;1;4,3X∗_n−1;1+ x1;2;4,3X∗_n−1;3 x1;1;4,4X∗_n−1;1+ x1;2;4,4X∗_n−1;3                                 (2.8) So, X∗₂ = X i=1,4 X∗_2;i (2.6)

By observation, the recursive formula of X∗ nis

X∗_n= X

i=1,4

X∗_n;i (2.7)

where X∗_n;1, X∗_n;2, X∗_n;3, X∗_n;4are derived from equations (2.8), (2.9), (2.10), (2.11) alternatively.

Remark 2.1. Now we get a recursive formula about X∗

n;i where i = 1, 2, 3, 4

which are useful to get matrices we want, i.e. Tnand Yn.

Definition 2.2. A function ϕ is called a counting function if ϕ(α1α2· · · αn) = α12n−1+ α22n−2+· · · + αn20+ 1

X∗_n;2=                                 x1;1;1,1X∗_n−1;2+ x1;2;1,1X∗_n−1;4 x1;1;1,2X∗_n−1;2+ x1;2;1,2X∗_n−1;4 x1;1;1,3X∗_n−1;2+ x1;2;1,3X∗_n−1;4 x1;1;1,4X∗_n−1;2+ x1;2;1,4X∗_n−1;4 x1;1;2,1X∗_n−1;2+ x1;2;2,1X∗_n−1;4 x1;1;2,2X∗_n−1;2+ x1;2;2,2X∗_n−1;4 x1;1;2,3X∗_n−1;2+ x1;2;2,3X∗_n−1;4 x1;1;2,4X∗_n−1;2+ x1;2;2,4X∗_n−1;4 x1;1;3,1X∗_n−1;2+ x1;2;3,1X∗_n−1;4 x1;1;3,2X∗_n−1;2+ x1;2;3,2X∗_n−1;4 x1;1;3,3X∗_n−1;2+ x1;2;3,3X∗_n−1;4 x1;1;3,4X∗_n−1;2+ x1;2;3,4X∗_n−1;4 x1;1;4,1X∗_n−1;2+ x1;2;4,1X∗_n−1;4 x1;1;4,2X∗_n−1;2+ x1;2;4,2X∗_n−1;4 x1;1;4,3X∗_n−1;2+ x1;2;4,3X∗_n−1;4 x1;1;4,4X∗_n−1;2+ x1;2;4,4X∗_n−1;4                                 (2.9) X∗_n;3 =                                 x1;3;1,1X∗_n−1;1+ x1;4;1,1X∗_n−1;3 x1;3;1,2X∗_n−1;1+ x1;4;1,2X∗_n−1;3 x1;3;1,3X∗_n−1;1+ x1;4;1,3X∗_n−1;3 x1;3;1,4X∗_n−1;1+ x1;4;1,4X∗_n−1;3 x1;3;2,1X∗_n−1;1+ x1;4;2,1X∗_n−1;3 x1;3;2,2X∗_n−1;1+ x1;4;2,2X∗_n−1;3 x1;3;2,3X∗_n−1;1+ x1;4;2,3X∗_n−1;3 x1;3;2,4X∗_n−1;1+ x1;4;2,4X∗_n−1;3 x1;3;3,1X∗_n−1;1+ x1;4;3,1X∗_n−1;3 x1;3;3,2X∗_n−1;1+ x1;4;3,2X∗_n−1;3 x1;3;3,3X∗_n−1;1+ x1;4;3,3X∗_n−1;3 x1;3;3,4X∗_n−1;1+ x1;4;3,4X∗_n−1;3 x1;3;4,1X∗_n−1;1+ x1;4;4,1X∗_n−1;3 x1;3;4,2X∗_n−1;1+ x1;4;4,2X∗_n−1;3 x1;3;4,3X∗_n−1;1+ x1;4;4,3X∗_n−1;3 x1;3;4,4X∗_n−1;1+ x1;4;4,4X∗_n−1;3                                 (2.10) X∗_n;4 =                                 x1;3;1,1X∗_n−1;2+ x1;4;1,1X∗_n−1;4 x1;3;1,2X∗_n−1;2+ x1;4;1,2X∗_n−1;4 x1;3;1,3X∗_n−1;2+ x1;4;1,3X∗_n−1;4 x1;3;1,4X∗_n−1;2+ x1;4;1,4X∗_n−1;4 x1;3;2,1X∗_n−1;2+ x1;4;2,1X∗_n−1;4 x1;3;2,2X∗_n−1;2+ x1;4;2,2X∗_n−1;4 x1;3;2,3X∗_n−1;2+ x1;4;2,3X∗_n−1;4 x1;3;2,4X∗_n−1;2+ x1;4;2,4X∗_n−1;4 x1;3;3,1X∗_n−1;2+ x1;4;3,1X∗_n−1;4 x1;3;3,2X∗_n−1;2+ x1;4;3,2X∗_n−1;4 x1;3;3,3X∗_n−1;2+ x1;4;3,3X∗_n−1;4 x1;3;3,4X∗_n−1;2+ x1;4;3,4X∗_n−1;4 x1;3;4,1X∗_n−1;2+ x1;4;4,1X∗_n−1;4 x1;3;4,2X∗_n−1;2+ x1;4;4,2X∗_n−1;4 x1;3;4,3X∗_n−1;2+ x1;4;4,3X∗_n−1;4 x1;3;4,4X∗_n−1;2+ x1;4;4,4X∗_n−1;4                                 (2.11)

· · · · · · · · · α1 α1 α2 α2 α3 α4 α2n−3 α2n−2 α2n−1 α2n α2n α0_2n α0_2n α0₁ α0₁ α0₂ α0₂ α0₃ α0_2n−4 α0_2n−3 α0_2n−2 α0_2n−1 β β

Figure 7. the general form of Tn

2.1

Construct T

n

The general form of Tnis shown in Figure 7. Notice that the upper colors

are circularly shifting one position such that the patterns can tile above the original patterns if we multiply Tnand Tn.

Define a permutation matrix Pn.

Pn=

h

P_{n;i, j} i

22n_×22n (2.12)

Given 1 ≤ j ≤ 22n_{, ∃α}

1, · · · , α2nsuch that j = ϕ(α1· · · α2n), then

( Pn;i, j= 1 , if i = ϕ(α2nα1· · · α2n−1)

P_{n;i, j}= 0 , otherwise (2.13)

By changing variables, we get more simple form of Pnwhere

( Pn;i,2i−1 = 1 and Pn;i+22n−1_,2i = 1 , if 1 ≤ i ≤ 22n−1

Pn;i, j = 0 , otherwise (2.14)

Thus Tn = X∗nPn.

2.2

Construct Y

n

The general form of Yn =

h

Y_{n;i, j} i

22n−1_×22n−1 is shown in Figure 8. Notice

the difference between Ynand Xn.

Let Y∗_n= 4 X i=1 X∗_n;i. (2.15)

· · · · · · · · · α1 α1 α2 α2 α2n−1 α2n−1 α3 α4 α0₂ α0₂ α0₃ α0₃ α0₄ α2n−3 α2n−2 α0_2n−3 α0_2n−2 α0_2n−1 α0_2n α0_2n

Figure 8. the general form of Yn

· · · · · · · · · α1 α1 α2 α2 α3 α4 α2n−3 α2n−2 α2n−1 α2n α2n α0₁ α0₁ α0₂ α0₂ α0₃ α0₄ α0_2n−3 α0_2n−2 α0_2n−1 α0_2n α0_2n

Figure 9. the general form of Y∗ n

Now Y∗

nhas the matrix form in Figure 9.

We construct Yn from Y∗n. Given 1 ≤ i ≤ 22n−1, then there exist

α1, α2, · · · , α2n−1 such that i = ϕ(α1α2· · · α2n−1). Define the set , Cn;1(i), such

that

C_n;1(i) = {ϕ(α1α2· · · α2n−1α2n) : α2n= 0, 1}

= {2i − 1 + 0, 2i − 1 + 1} = {2i − 1, 2i}

Given 1 ≤ j ≤ 22n−1, then there exist α0₂, α0₃, · · · , α0_2n such that j = ϕ(α0₂α0₃· · · α0_2n). Define the set, Cn;2(j), such that

C_n;2(j) = {ϕ(α0₁α0₂· · · α0_2n−1α0_2n) : α0₁ = 0, 1} = { j, j + 22n−1} Then Yn = h Y_{n;i, j} i 22n−1_×22n−1 where Yn;i, j = X p∈Cn;1(i) q∈Cn;2( j) Y∗_n;p,q

3

Symmetry of Hexagons

The symmetry group of the hexagon is D6, the dihedral group where

| D6 |= 12. The group D6 = {I, ρ, ρ2, ρ3, ρ4, ρ5, m, mρ, mρ2, mρ3, mρ4, mρ5}

where ρ is rotating π

3 counterclockwise, m is reflecting about the y-axis.

See Figure 10. m ρ _ρ S2;0 S2;π₃ S_2;2π 3 Figure 10.

For example, see Figure 11.

α0 α0 α1 α1 α2 α2 α3 α3 α3 α4 α4 α5 α5 =⇒ mρ2 Figure 11.

There are three permutation groups S2 in three directions, denoted by

S2;0, S2;π

3, S2;2π3 , on edge coloring of hexagons as Figure 10. According to

the symmetry group and the permutation group, we define an equivalent class of a set B ⊆ T (2), denoted by [B], as follows:

Let A = {τηoηπ

3η2π3 | τ ∈ D6, η0 ∈ S2;0, η π

obviously | A |= (2 × 6) × 2 × 2 × 2 = 96. Given B ⊆ T (2), then [B] = {ξ(B) | ξ ∈ A}.

Some observations about an equivalent class: 1. if P(B) = ∅ (, ∅), then P(B0) = ∅ (, ∅) ∀B0 ∈ [B] 2. if Σ(B) = ∅ (, ∅), then Σ(B0) = ∅ (, ∅) ∀B0 ∈ [B]

In addition, T (2) can be decomposed into four types, denoted by K0,

K_I, KII, KIII, according to equivalent classes as Figure 12.

We know

1. if t ∈ KI, then ξ(t) ∈ KI∀ξ ∈ A

2. if t ∈ KII, then ξ(t) ∈ KII ∀ξ ∈ A

3. if t ∈ KIII, then ξ(t) ∈ KIII ∀ξ ∈ A

So we construct two groups GIand GII. Let GI=KIIand GII =KI∪ KIII,

then

1. if t ∈ GI, then ξ(t) ∈ GI∀ξ ∈ A

2. if t ∈ GII, then ξ(t) ∈ GII ∀ξ ∈ A

Lemma 3.1. Given B = BI∪ BII, where BI ⊆ GI, BII ⊆ GII. For any BII ∈ [BII],

there exists B0_I ⊆ G_Isuch that B0_I∪ BII ∈ [B]

Proof. Since BII ∈ [BII], there exists ξ ∈ A such that BII = ξ(BII). We know

ξ(B) = ξ(BI) ∪ ξ(BII) ∈ [B]

where ξ(BI) ⊆ GI and ξ(BII) ⊆ GII. Therefore,

ξ(BI) ∪ ξ(BII) = ξ(BI) ∪ BII ∈ [B]

where ξ(BI) ⊆ GI 

Remark 3.2. This lemma tells us that we can classify GII with A to reduce

0 0 0 0 0 0 1 0 1 0 1 0 0 10 0 0 1 0 0 1 19 0 1 1 1 0 1 28 1 0 0 0 1 0 37 1 1 0 1 1 0 46 1 0 1 0 1 1 55 1 1 1 1 1 1 64 (a) K0 0 0 0 1 0 0 2 0 0 1 1 0 1 20 1 0 0 1 1 0 38 1 0 1 1 1 1 56 0 1 0 0 0 0 9 0 1 1 0 0 1 27 1 1 0 0 1 0 45 1 1 1 0 1 1 63 1 0 0 0 0 0 5 1 1 0 1 0 0 14 1 0 1 0 0 1 23 1 1 1 1 0 1 32 0 0 0 0 1 0 33 0 1 0 1 1 0 42 0 0 1 0 1 1 51 0 1 1 1 1 1 60 0 0 1 0 0 0 3 0 1 1 1 0 0 12 1 0 1 0 1 0 39 1 1 1 1 1 0 48 0 0 0 0 0 1 17 0 1 0 1 0 1 26 1 0 0 0 1 1 53 1 1 0 1 1 1 62 (b) KI 1 0 1 0 0 0 7 1 1 1 1 0 0 16 0 0 0 0 1 1 49 0 1 0 1 1 1 58 0 1 1 0 0 0 11 1 0 1 0 1 1 47 0 0 0 1 0 1 18 1 0 0 1 1 1 54 1 1 0 0 0 0 13 1 1 1 0 0 1 31 0 0 0 1 1 0 34 0 0 1 1 1 1 52 1 0 0 0 0 1 21 1 1 0 1 0 1 30 0 0 1 0 1 0 35 0 1 1 1 1 0 44 0 0 1 1 0 0 4 1 0 1 1 1 0 40 0 1 0 0 0 1 25 1 1 0 0 1 1 61 1 0 0 1 0 0 6 1 0 1 1 0 1 24 0 1 0 0 1 0 41 0 1 1 0 1 1 59 (c) KII 0 1 1 0 1 0 43 0 1 0 0 1 1 57 1 0 1 1 0 0 8 1 0 0 1 0 1 22 0 0 1 1 1 0 36 0 0 0 1 1 1 50 1 1 1 0 0 0 15 1 1 0 0 0 1 29 (d) KIII Figure 12.

4

Algorithm

According to the above discussions, we now have T and Y to check whether B ∈ C(2) or B ∈ N(2) or not for B ⊆ T (2) and a method called classification to reduce computations. Using the following algorithm to achieve our goal.

Definition 4.1. P(" n l

0 k #

) is the set of sets, i.e. each element is a set. Each element contains tiles that can tile the local pattern such that the width of the local pattern is n hexagons, the height of the local pattern is k hexagons and the shift of the local pattern is l. See Figure ??, where i is from 1 to 2n and αi = ( α0_{2(l+1)+(i−1)} ,if 2(l + 1) + (i − 1) ≤ 2 α0_{2(l+1)+(i−1)−2n} ,otherwise n k α1 α2 α2n α0₁ α0₂ α0_2n

Define two sets C∗_{and N}∗_{which mean cycle and non-cycle alternatively}

such that C∗_{(m) = {B : B ∈ P(}" n l 0 k # ) where m = nk, 0 ≤ l ≤ n − 1, n, k, l ∈ N}

and a recursive definition of N∗

( N∗_{(0) = {B}

1∪ B2 : B1 ⊆ GI, B2 ⊆ GII}

N∗_{(m) = {B : B ∈ N}∗_{(m − 1) and c * B for all c ∈ C}∗_(m)}

Since the condition Σ(B) = ∅ for all B ∈ N∗_{(m) is hard to check and the}

Algorithm 1Algorithm m ← 0 repeat m ← m + 1 C∗(m) N∗_(m)

until Σ(B) = ∅ for all B ∈ N∗_(m)

Algorithm 2Modified Algorithm

C(2) ← ∅ // cycle generator N(2) ← ∅ // Non-cycle generator U? _{← ∅ // Unknown} m ← 0 // Loop index repeat m ← m + 1 C∗_(m) C(2) ← C(2) ∪ C∗_(m) N∗_(m) untilm == 12 repeat pick B ∈ N∗₍₁₂₎ N∗_{(12) ← N}∗_{(12) r {B}}

if time is acceptable and P(B) , ∅ then

C(2) ← C(2) ∪ {B}

else if time is acceptable and Σ(B) = ∅ then

N(2) ← N(2) ∪ {B}

else

U? _{← U}? _{∪ {B}}

end if

5

Conclusion

According to the modified algorithm, if | U? _{|= 0 and Σ(B) = ∅ ∀B ∈ N(2),}

then Wang’s conjecture is true for hexagons with two colors, i.e. if Σ(B) , ∅ then P(B) , ∅.

Unfortunately, there are three cases we can’t decide. 1. 2 4 15 20 21 56 59

2. 2 15 20 25 35 38 3. 2 15 20 25 35 56

Since Tn and Yn is too large to compute for n is sufficient large ( 6 ),

another method for these three cases is introduced in the following. Method:

1. Find all possible situations that can tile like the graph above, and then record the informations of colored edges.

2. Use the informations to tile the plane with the width is 8 tile.

3. If all possible cases is checked and can’t tile the plane with some height, then the set Σ(B) = ∅.

The last three cases are shown. 1. 2 4 15 20 21 56 59 ( 8, 17 ) 2. 2 15 20 25 35 38 ( 8, 15 ) 3. 2 15 20 25 35 56 ( 8, 13 )

Theorem 5.1. Given B ⊆ T (2).

if Σ(B) , ∅ then P(B) , ∅.

The other cases are decided and show in Table 1. Only show [Cm(2)] in

A

Table1

TileNum [Cm(2)] 3tile { 2 5 41 } { 2 3 25 } { 2 5 59 } { 2 29 35 } { 2 11 53 } { 4 13 49 } { 2 11 25 } { 4 29 57 } { 2 29 43 } 4tile { 2 3 5 57 } { 2 4 13 57 } { 2 5 11 49 } { 2 5 11 58 } { 4 11 21 49 } { _{2 11 22 59 }} { 2 11 20 61 } { 2 4 29 59 } { _{2 3 21 59 }} { 2 4 29 41 } { 4 11 22 57 } { 4 11 21 58 } { 2 5 60 63 } { 2 22 43 63 } { 2 11 56 61 } { 4 15 54 57 }

TileNum [Cm(2)] { 2 20 47 61 } { _{8 15 50 57 }} { 2 5 11 57 } { 2 11 30 41 } { _{2 3 21 57 }} { 2 7 25 36 } { 2 5 43 58 } { _{2 11 21 59 }} { 2 21 36 63 } { 2 5 43 57 } { _{2 7 21 41 }} { 2 11 29 41 } { 2 11 22 57 } { _{2 7 25 44 }} { 2 15 29 60 } { 2 7 21 59 } { 2 5 17 43 } { 2 11 38 61 } { 8 29 43 50 } { 2 11 24 57 } { 2 11 31 41 } { 2 11 22 41 } { 2 8 25 43 } { 2 7 25 51 } { 2 7 41 62 } { 2 7 25 60 } Continued. . .

TileNum [Cm(2)] { 2 7 41 53 } { 2 5 25 43 } { 4 15 21 58 } { 2 7 41 54 } { _{2 7 25 52 }} { 2 8 21 41 } { 4 6 31 57 } { _{2 22 31 43 }} { 2 7 43 53 } { 2 15 21 60 } { _{2 5 44 57 }} { 2 11 29 60 } { 2 7 29 60 } { 2 5 25 44 } { 2 3 21 41 } { 4 6 11 57 } { 4 13 50 57 } { 2 11 21 62 } { 2 11 22 61 } { 4 11 21 50 } { 2 7 44 61 } { 4 15 21 57 } { 2 11 21 57 } { 4 21 47 58 } 5tile { 2 3 29 39 58 } { 2 7 25 38 59 } { 2 7 25 35 62 } Continued. . .

TileNum [Cm(2)] { 2 4 29 60 63 } { _{2 5 15 52 57 }} { 2 7 29 36 57 } { 2 21 31 36 59 } { _{4 6 15 49 57 }} { 2 15 22 35 57 } { 4 13 22 43 58 } { _{2 11 22 43 62 }} { 2 4 21 59 63 } { 2 11 13 20 57 } { _{2 7 25 41 50 }} { 2 5 12 15 49 } { 2 7 21 43 57 } { _{2 7 29 44 57 }} { 2 3 23 29 42 } { 2 4 21 60 63 } { 2 8 11 41 62 } { 2 3 21 43 61 } { 2 7 25 43 62 } { 2 15 22 43 57 } { 4 7 18 21 41 } { 2 7 25 50 59 } { 4 6 11 31 49 } { 2 11 31 38 57 } { 2 7 41 50 61 } { 2 7 52 57 61 } Continued. . .

TileNum [Cm(2)] { 2 11 29 38 59 } { 2 3 21 43 60 } { 2 22 31 40 59 } { 2 8 20 43 61 } { _{2 5 22 43 62 }} { 2 22 31 35 61 } { 2 3 8 60 61 } { _{2 21 24 43 62 }} { 2 11 20 29 57 } { 2 4 15 17 42 } { _{2 4 11 31 57 }} { 2 3 11 29 49 } { 2 8 15 25 51 } { 2 4 15 41 62 } { 2 5 15 49 57 } { 2 7 22 41 57 } { 2 7 25 43 50 } { 2 3 11 21 61 } { 2 3 7 41 61 } { 2 8 20 41 61 } { 2 4 5 15 58 } { 2 3 5 15 58 } { 2 4 21 42 43 } { 2 7 13 51 57 } { 2 4 13 47 58 } { 2 8 11 57 61 } { 2 3 7 22 41 } Continued. . .

TileNum [Cm(2)] { 2 7 30 57 60 } { _{2 3 29 60 63 }} { 2 4 11 31 49 } { 2 7 41 50 63 } { _{2 7 36 57 61 }} { 2 4 11 31 58 } { 2 7 25 50 63 } { _{2 7 22 57 59 }} { 2 3 29 47 58 } { 2 7 41 56 57 } { _{2 7 25 56 57 }} { 2 7 29 52 57 } { 2 3 24 57 61 } { _{2 7 44 57 62 }} { 2 5 29 48 60 } { 2 3 8 13 57 } { 2 11 29 36 57 } { 2 4 5 16 57 } { 2 15 20 43 61 } { 2 3 8 58 61 } { 2 3 8 59 61 } { 2 7 41 52 61 } { 2 7 20 25 57 } { 2 4 16 23 57 } { 2 3 16 38 57 } { 2 11 29 40 52 } Continued. . .

TileNum [Cm(2)] { 2 11 29 36 52 } { 2 7 21 43 61 } { 2 4 15 53 57 } { 2 7 21 25 43 } { _{2 7 21 47 57 }} { 2 8 21 43 57 } { 2 15 21 36 57 } { _{2 7 29 52 59 }} { 2 4 17 32 43 } { 2 15 21 43 62 } { _{2 4 15 53 60 }} { 2 3 16 29 57 } { 2 3 7 20 57 } { 2 4 29 44 49 } { 2 7 35 52 61 } { 2 5 15 57 58 } { 2 3 29 32 59 } { 2 8 11 20 57 } { 2 8 29 54 59 } { 2 15 22 54 59 } { 2 7 24 35 61 } { 2 3 6 44 61 } { 2 4 31 44 53 } { 2 4 17 31 44 } { 2 3 24 29 57 } { 2 4 21 30 41 } { 2 3 29 59 60 } Continued. . .

TileNum [Cm(2)] { 2 4 16 53 59 } { _{2 8 24 43 61 }} { 2 8 25 31 51 } { 2 4 11 21 41 } { _{2 3 8 13 58 }} { 2 3 13 38 57 } { 2 3 13 39 57 } { _{2 7 13 35 57 }} { 4 8 11 29 58 } { 2 8 11 29 57 } { _{2 4 22 41 63 }} { 2 11 21 36 41 } { 2 3 8 14 57 } { _{2 3 8 23 57 }} { 2 11 29 38 52 } { 2 4 13 50 59 } { 4 8 29 47 49 } { 2 11 30 36 59 } { 2 7 47 57 62 } { 2 3 30 60 61 } { 2 11 24 43 62 } { 2 4 31 43 62 } { 2 3 22 42 63 } { 2 8 11 52 61 } { 2 5 12 50 63 } { 2 3 16 57 61 } Continued. . .

TileNum [Cm(2)] { 2 3 7 52 61 } { 2 15 22 43 59 } { 2 4 31 38 57 } { 2 22 43 47 59 } { _{2 4 22 32 41 }} { 2 11 29 38 57 } { 4 15 22 43 57 } { _{2 11 29 36 61 }} { 2 4 13 22 59 } { 4 8 29 43 49 } { _{2 4 5 58 63 }} { 2 4 21 42 63 } { 2 4 13 35 58 } { 2 4 7 30 41 } { 2 7 11 57 62 } { 2 4 5 15 57 } { 2 7 36 38 61 } { 4 6 11 15 49 } { 2 7 20 22 41 } { 4 7 18 22 41 } { 2 11 32 36 57 } { 2 15 22 47 57 } { 2 15 21 43 57 } { 2 5 15 44 58 } { 2 4 21 31 59 } { 2 4 21 31 41 } { 2 21 31 43 59 } Continued. . .

TileNum [Cm(2)] { 2 11 21 43 61 } { _{2 11 21 43 60 }} { 2 4 29 39 42 } { 2 3 23 42 61 } { _{2 5 44 47 58 }} { 2 3 23 38 57 } { 2 5 44 58 63 } { _{2 11 29 56 57 }} { 2 11 21 41 63 } { 15 25 35 38 56 } { _{2 15 20 44 61 }} { 2 20 29 47 58 } { 2 23 35 44 61 } 6tile { _{2 7 22 25 35 41 }} { 2 3 8 29 57 60 } { 2 3 8 29 58 59 } { 2 5 12 15 50 57 } { 2 8 11 30 57 60 } { 2 8 11 13 50 57 } { 2 3 13 22 43 49 } { 2 5 15 36 58 61 } { 2 5 15 36 57 62 } { 2 5 29 44 47 50 } { 2 7 30 38 41 59 } { 4 7 18 21 43 57 } { 2 7 25 35 38 61 } Continued. . .

TileNum [Cm(2)] { 2 8 11 31 50 57 } { 2 8 29 47 50 59 } { 2 7 31 36 57 62 } { 2 15 22 36 59 61 } { _{2 3 29 38 39 57 }} { 4 6 31 47 49 58 } { 2 7 13 35 50 61 } { _{2 4 5 23 57 63 }} { 2 4 5 29 44 58 } { 2 3 5 29 44 58 } { _{2 4 13 43 58 62 }} { 2 3 11 20 30 49 } { 2 4 15 32 57 62 } { 2 4 32 47 57 62 } { 2 4 31 47 49 62 } { 2 4 21 47 59 62 } { 2 7 20 29 41 44 } { 2 8 11 50 57 63 } { 2 7 24 25 35 57 } { 2 4 15 31 49 58 } { 2 8 15 25 56 57 } { 2 7 29 36 41 61 } { 2 4 15 30 41 49 } { 2 7 22 25 41 43 } { 2 4 13 31 51 58 } { 2 3 6 29 47 49 } { 2 7 22 25 43 59 } Continued. . .

TileNum [Cm(2)] { 2 3 8 29 57 59 } { _{2 8 11 30 57 59 }} { 2 3 13 40 59 62 } { 2 3 8 31 41 58 } { _{2 3 22 47 60 61 }} { 4 8 13 43 54 57 } { 2 4 21 41 48 63 } { _{2 3 7 54 60 61 }} { 2 3 16 54 59 61 } { 2 3 24 47 58 61 } { _{2 4 29 31 36 49 }} { 2 4 8 21 57 59 } { 2 4 21 31 36 57 } { _{2 4 29 40 49 63 }} { 2 3 15 56 57 62 } { 2 3 8 29 58 60 } { 2 3 15 24 57 58 } { 2 3 6 47 58 61 } { 2 4 15 22 57 63 } { 2 4 29 47 49 56 } { 2 4 13 21 51 59 } { 2 4 17 21 41 48 } { 2 3 6 13 43 49 } { 2 3 8 31 49 59 } { 2 4 5 17 48 57 } { 2 3 14 21 43 62 } Continued. . .

TileNum [Cm(2)] { 2 4 30 31 41 51 } { 2 3 6 11 49 61 } { 2 3 29 40 59 62 } { 2 3 8 31 57 58 } { _{2 3 7 29 50 59 }} { 2 3 7 60 61 62 } { 4 8 13 54 57 59 } { _{2 3 5 11 34 61 }} { 2 3 13 43 54 58 } { 2 8 25 30 41 51 } { _{4 7 18 22 43 57 }} { 2 7 29 36 51 61 } { 2 3 13 22 47 58 } { 2 7 22 25 35 43 } { 2 3 5 44 61 62 } { 2 3 6 39 49 61 } { 2 4 29 31 44 50 } { 2 3 7 18 29 59 } { 2 4 13 31 58 59 } { 2 7 25 38 41 56 } { 2 4 21 31 43 57 } { 2 4 15 18 21 59 } { 2 7 29 38 41 59 } { 2 7 25 38 43 61 } { 2 22 30 40 47 59 } { 2 4 29 39 44 58 } { 2 4 15 29 39 58 } Continued. . .

TileNum [Cm(2)] { 2 3 29 39 60 61 } { _{2 4 5 23 48 57 }} { 4 8 13 31 57 58 } { 2 3 5 12 58 63 } { _{2 4 11 13 56 58 }} { 2 3 8 29 38 57 } { 2 5 12 23 48 57 } { _{4 7 22 30 41 57 }} { 2 7 29 36 41 51 } { 2 5 15 44 52 61 } { _{2 7 21 47 60 61 }} { 2 11 29 31 44 57 } { 2 3 13 24 54 57 } { _{2 7 29 40 41 51 }} { 2 8 15 21 52 57 } { 2 11 30 39 52 57 } { 2 7 11 20 30 57 } { 2 7 29 41 44 59 } { 2 7 22 25 41 56 } { 2 7 22 41 56 61 } { 2 7 20 25 56 59 } { 2 7 20 36 41 61 } { 2 7 30 38 41 51 } { 2 4 13 43 56 58 } { 2 4 13 40 53 59 } { 2 15 21 40 57 59 } Continued. . .

TileNum [Cm(2)] { 2 4 15 30 41 56 } { 2 4 15 21 38 59 } { 2 3 21 47 60 62 } { 2 11 29 44 47 57 } { _{2 7 22 29 41 56 }} { 2 5 15 17 36 57 } { 2 7 29 41 44 51 } { _{2 22 29 40 47 59 }} { 2 24 31 43 54 61 } { 2 7 20 36 43 61 } { _{2 5 20 23 44 61 }} { 2 11 24 29 44 61 } { 2 5 20 23 47 58 } { 3 8 22 34 44 61 } { 2 3 16 29 38 59 } { 15 22 25 36 49 59 } { 2 15 24 35 54 61 } { 2 8 31 44 54 61 } 7tile { 4 7 8 30 41 54 57 } { 2 7 22 30 40 41 59 } { 2 5 15 32 36 56 57 } { 2 7 22 32 40 41 59 } { 2 7 22 32 40 43 57 } { 2 3 12 22 31 48 49 } { 2 7 20 29 41 56 61 } { 4 6 24 29 40 43 50 } { 2 3 12 22 31 39 49 } Continued. . .

TileNum [Cm(2)]

{ 2 3 22 31 39 48 49 } { _{2 11 29 30 39 47 52 }} { 2 5 24 29 47 52 58 } { 2 7 16 22 29 40 59 }

References

[1] J.C. Ban and S.S. Lin, Patterns generation and transition matrices in

multi-dimensional lattice models, Discrete Contin. Dyn. Syst. 13 (2005), no. 3,

637–658.1

[2] J.C. Ban, W.G. Hu, S.S. Lin and Y.H. Lin, Zeta functions for

two-dimensional shifts of finite type , preprint (2008).

[3] J.C. Ban, S.S. Lin and Y.H. Lin, Patterns generation and spatial entropy in

two dimensional lattice models, Asian J. Math. 11 (2007), 497–534.

[4] R. Berger, The undecidability of the domino problem, Memoirs Amer. Math. Soc., 66 (1966).

[5] K. Culik II, An aperiodic set of 13 Wang tiles, Discrete Mathematics, 160 (1996), 245–251.

[6] B. Gr ¨unbaum and G. C. Shephard, Tilings and Patterns, New York: W. H. Freeman, (1986).

[7] J. Kari, A small aperiodic set of Wang tiles, Discrete Mathematics, 160 (1996), 259–264.

[8] A. Lagae and P. Dutr´e, An alternative for Wang tiles: colored edges versus

colores corners, ACM Trans. Graphics, 25 (2006), no. 4, 1442–1459.

[9] A. Lagae , J. Kari and P. Dutr´e, Aperiodic sets of square tiles with

col-ored corners, Report CW 460, Department of Computer Science, K.U.

Leuven, Leuven, Belgium. Aug 2006.

[10] R. Penrose, Bull. Inst. Math. Appl. 10 (1974), 266.

[11] R.M. Robinson, Undecidability and nonperiodicity for tilings of the plane, Inventiones Mathematicae, 12 (1971), 177–209.

[12] H. Wang, Proving theorems by pattern recognition-II, Bell System Tech. Journal, 40 (1961), 1–41.

[13] W.G. Hu and S.S Lin, Nonemptiness problems of plane square tiling with