角落著色的空間熵以及最小週期生成

國

立

交

通

大

學

應用數學系

碩

士

論

文

角落著色的空間熵以及最小週期生成

Spatial Entropy and Minimal Cycle of Corner

Coloring

研 究 生：張瑛峰

指導教授：林松山 教授

角落著色的空間熵以及最小週期生成

Spatial Entropy and Minimal Cycle of Corner

Coloring

研 究 生：張瑛峰 Student：Ying-Feng Chang

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

Hsinchu, Taiwan, Republic of China

i

角落著色的空間熵以及最小週期生成

學生：張瑛峰

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

這篇研究在角落著色的平面方塊的複雜性。在平面上對角落著色，角

落有 p 種顏色選擇的單位方塊並肩排著，相鄰的邊必須要有一樣的顏色 在

[12] 王浩猜測任意可以拼成全平面的磁磚集合就可以週期性的拼成全平

面。

P B

( )

是所有可以由

B

生成的週期花樣。如果

P B

( )

≠

φ

那麼

B

就有一個最

小週期生成的子集

B

'

，使得

P B

( ')

≠

φ

並且

P B

( '')

=

φ

對於

B

''

⊂

B

'

及

B

''

≠

B

'

。所

有最小週期生成的集合 C(2) 包含 17 個元素。所有最大非週期生成的集合

N(2):如果

B

∈

N

(2)

那麼

P B

( )

=

φ

並且

B

⊃

B

及

B

≠

B

意味著

P B

(

)

≠

φ

胡文貴學長和林松山老師證明過，在兩個顏色的角落著色時，王浩

的猜測，藉由表現出對於任意的

B

∈

N

(2)

，

∑

( )

B

=

φ

。更精確的說，

∑

( )

B

≠

φ

充

要

B

有一個最小週期生成的子集

我主要的工作是決定兩個顏色時任意最小週期生成的聯集的熵是否大

於零，剩下七個未能決定的情況。

ii

Spatial Entropy and Minimal Cycle of

Corner Coloring

student：

Ying-Feng Chang

Advisors：Dr.

Song-Sun Lin

Department﹙Institute﹚of

Applied Mathematics

National Chiao Tung University

ABSTRACT

This investigation studies the complexity problems of plane square

tiling with colored corners . In the corner coloring of a plane, unit

squares with colored corners of p colors are arranged side by side

such that the corners of the adjacent sides have the same colors.In [12],Wang

conjectured that any set of tiles that can tile a plane can tile the plane

periodically.i.e.,

if

∑

( )

B

≠

φ

then ( )

P B

≠

φ

.

( )

P B

is the set of all periodic patterns on



that can be generated by

B

.

If

P B

( )

≠

φ

, then

B

has a subset

B

'

of minimal cycle generator such that

( ')

P B

≠

φ

and

P B

( '')

=

φ

for

B

''

⊂

B

'

.and

B

''

≠

B

'

The set of all minimal cycle

generators C(2) contains 17 elements.N(2) is the set of all maximal

non-cycle generators: if

B

∈

N

(2)

, then

P B

( )

=

φ

and

B

⊃

B

and

B

≠

B

implies

P B

(

)

≠

φ

.

W.G. Hu and S.S.Lin proved that Wang's conjecture in corner coloring holds

when p=2 by showing that

∑

( )

B

=

φ

for any

B

∈

N

(2)

More precisely,

∑

( )

B

≠

φ

if and only if

B

has a subset of minimal cycle generator.

My primary work is to decide whether the spatial entropy of any union of

minimal cycle generators equals to zero or not when p=2 ,except 7 undetermined

cases.

iii

致

謝

首先要感謝我的指導教授,林松山教授,協助我完成本篇論文。在老師的

教導之下，學到了不同思考問題的方向，瞭解到如何當一個好老師，以及

做人處事之道。

接著感謝胡文貴學長。當我有疑惑時，學長就會義不容辭的幫助我解決

問題，讓我有信心繼續完成我的論文。

還有處理邊著色問題的陳晉育同學，他獨到的想法，讓我在角落著色上

也可以仿照他的方法處理問題。

最後，也是最重要的就是我的家人，包容我不能常常回家陪他們，是我

精神上的支柱。

iv

目

錄

中文提要

………

i

英文提要

………

ii

誌謝

………

iii

目錄

………

iv

一、

Introduction ………

1

二、

Minimal cycle generators………

2

三、

Spatial entropy………

5

1. Introduction

The coloring of unit squares onZ2_{has been studied for many years [6]. In 1961,}

in studying proving theorem by pattern recognition, Wang [12] started to study the square tiling of a plane. The unit squares with colored corners of p colors are arranged side by side such that the corners of the adjacent sides have the same colors ; the tiles cannot be rotated or reflected.

The 2 × 2 unit squares is denoted byZ2×2. Let Sp be a set of p (≥ 1) colors.

The total set of all tiles is denoted by Σ2×2(p) ≡ SZ

2×2

p .A set B of corner-colored

tiles, such that B ⊂ Σ2×2(p), is called a basic set. Let Σ(B) be the set of all

global patterns on Z2 _{that can be constructed from the corner-colored tiles in B}

and P(B) be the set of all periodic patterns on Z2 _{that can be constructed from}

the corner-colored tiles in B. Clearly, P(B) ⊆ Σ(B). The nonemptiness problem is to determine whether or not Σ(B) 6= ∅. In [12], Wang conjectured that any set of tiles that can tile a plane can tile the plane periodically, i.e.,

(1.1) if Σ(B) 6= ∅ then P(B) 6= ∅.

First,the minimal cycle generator is introduced in [13]. B ⊂ Σw

2×2(p) is called a

minimal cycle generator if P(B) 6= ∅ and P(B0

) = ∅ whenever B0

$ B. Given p ≥ 2, denote the set of all minimal cycle generators by C(p) .

In this study, for p = 2, C(2) can be listed explicitly, C(2) has 17 members . Furthermore, under the symmetry group D4 of Z2×2 and the permutation group

Sp of colors of corners , C(2) can be classified into seven classes .

In corner coloring, the basic set of 44 tiles with six colors that can tile the plane aperiodically without any periodic patterns has been established [9]. And then (1.1) fails for p = 6. The method used to study Wang tiles (edge coloring) can also be applied to study corner coloring. For p = 2, no allowable pattern onZ5×4

can be generated from any maximal non-cycle generator. Hence, the nonemptiness problem is decidable for corner coloring.

For recent results on Wang tiles (colored edges) and colored corners with their applications to computer graphics, see Lagae and Dutr´e [8] and references therein. We consider p = 2 and the rest of this paper is arranged as follows. Section 2 introduces notations ,minimal cycle generators of corner coloring and their serial numbers . Section 3 states the theorems and methods which is used to get the bounds of the spatial entropy .

2. Minimal cycle generators

Some notations must be introduced first. In this section, Z2×2 represents the

square lattice with vertices (0, 0), (0, 1), (1, 0) and (1, 1). Furthermore, for any (i, j) ∈Z2_{, define Z}

2×2(i, j) = {(i, j), (i, j + 1), (i + 1, j), (i + 1, j + 1)}.

For given positive integers m and n, the rectangular latticeZm×n is defined by

Zm×n= {(i, j)|0 ≤ i ≤ m − 1 and 0 ≤ j ≤ n − 1} .

Denote the set of p colors by Sp= {0, 1, · · · , p− 1}. The set of all global patterns

onZ2_{with colors in S} p is denoted by Σ2 p= SZ 2 p =U |U : Z2→ Sp .

The set of all local patterns onZm×n is defined by

Σm×n(p) = {U |Zm×n: U ∈ Σ

2 p}.

Now, for any given B ⊂ Σ2×2(p), B is called a basic set. The setΣm×n(B) of all

patterns onZm×n generated by B is defined by

Σm×n(B) =U ∈ Σm×n(p) : U |Z2×2(i,j)∈ B for 0 ≤ i ≤ m − 2, 0 ≤ j ≤ n − 2 ,

and the set Σ(B) of all global patterns onZ2 _{generated by B is defined by}

Σ(B) =U ∈ Σ2

p: U |Z2×2(i,j)∈ B for i, j ∈Z .

Then, the set P(B) of all periodic patterns generated by B is defined by

P(B) = {U = (ui,j) ∈ Σ(B) | ui,j= ui+n,j= ui,j+kfor all i, j ∈Z and for some n, k ≥ 1} .

For simplicity, a periodic pattern is also called a cycle.

Now, the symmetry of the unit squareZ2×2is introduced. The symmetry group

of the rectangle Z2×2 is D4, the dihedral group of order eight. The group D4 is

generated by the rotation ρ, through π

2, and the reflection m about the y-axis.

Denote by D4= {I, ρ, ρ2, ρ3, m, mρ, mρ2, mρ3}.

m mρ3 mρ2

mρ

3

Therefore, given a basic set B ⊂ Σ2×2(p) and any element τ ∈ D4, another basic

set (B)τ can be obtained by transforming the local patterns in B by τ .

Additionally, consider the permutation group Sp on Sp. If η ∈ Sp and η(0) =

i0,η(1) = i1, · · · , η(p−1) = ip−1, we write η =  0 1 · · · p− 1 i0 i1 · · · ip−1  . For η ∈ Sp

and B ⊂ Σ2×2(p), another basic set (B)η can be obtained.

D4 and Sp can be combined to define the equivalentclasses of basic sets, as

follows: given B ⊂ Σ2×2(p), define the class [B] of B by

[B] = {B0

⊂ Σ2×2(p) : B0= ((B)τ)η, τ ∈ D4, η∈ Sp} .

Definition 2.1. _{For B ⊂ Σ2×2(p),}

(i) B is called a cycle generator if P(B) 6= ∅.

(ii) B is called a minimal cycle generator if P(B) 6= ∅ and P(B0

) = ∅ for all B0

$ B.

(iii) C(p) is the set of all minimal cycle generators that are subsets of Σ2×2(p).

From now on, only the case p = 2 is considered: S2= {0, 1}. In another work [1],

the horizontal ordering matrix X2×2= [xp,q]4×4for all local patterns in Σ2×2(2) is

defined by (2.1) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 .

For convenience, the name of each local pattern in Σ2×2(2) is given as follows.

Definition 2.2. _{Denote by} (2.2) X_2×2=     O e2 e4 r e1 t I e3 e3 J b e1 l e4 e2 E     .

The following theorem groups all 17 minimal cycle generators into seven classes . Table A.1 presents the symmetries of minimal cycle generators in Cc(2)

Theorem 2.3. _{(i) Cc(2) contains 17 elements and is classified into seven classes} of minimal cycle generators which are given by

(2) [{b, t}] = {{b, t}, {l, r}}, (3) [{I, J}] = {{I, J}},

(4) [{e1, e2, e3, e4}] = {{e1, e2, e3, e4}, {e1, e2, e3, e4}},

(5) [{e1, e2, e3, e4, b}] = {{e1, e2, e3, e4, b}, {e3, e4, e1, e2, t}, {e1, e3, e2, e4, r},

{e2, e4, e1, e3, l}},

(6) [{e1, e4, J}] = {{e1, e4, J}, {e2, e3, I}, {e2, e3, J}, {e1, e4, I}},

(7) [{e1, e4, e1, e4}] = {{e1, e4, e1, e4}, {e2, e3, e2, e3}}.

For convenience, the mainimal cycle generators mentioned above are numbered first by classes and then ordering.

Serial number Original minimal cycle

(1.1) {O} (1.2) {E} (2.1) {b,t} (2.2) {l,r} (3) {I,J} (4.1) {e1, e2, e3, e4} (4.2) {e1, e2, e3, e4} (5.1) {e1, e2, e3, e4, b} (5.2) {e3, e4, e1, e2, t} (5.3) {e1, e3, e2, e4, r} (5.4) {e2, e4, e1, e3, l} (6.1) {e1, e4, J} (6.2) {e2, e3, I} (6.3) {e2, e3, J} (6.4) {e1, e4, I} (7.1) {e1, e4, e1, e4} (7.2) {e2, e3, e2, e3} Table 2.1 Notations:

1 : (a)1 which means any minimal cycle generator of (a) , for 1 ≤ a ≤ 7

2 : [a]1 which means there exists one minimal cycle generator of (a) , for

5

3. Spatial entropy

Let Γm×n(B) = card(Σm×n(B)), be the number of distinct patterns in Σm×n(B).

The Spatial entropy h(B) of Σ(B) is defined as h(B) ≡ lim

m→∞,n→∞

log(Γm×n(B))

m× n .

The spatial entropy , h(B) is clearly independent of the choice of elements in [B]: for any B0 _{∈ [B].}

Definition 3.1. _{A K+1 multiple index}

BK ≡ β1β2· · · βkβ1

is called a diagonal cycle if

βK+1= β1 and βk ∈ {1, 4}.

for each 1 ≤ k ≤ K + 1

Theorem 3.2. _{Let β1β2· · · βkβ1 be a diagonal cycle . Then for any m ≥ 2,} h(A2) ≥ 1 mKlog ρ(Sm;β1β2Sm;β2β3· · · Sm;βKβ1) and h(A2) ≥ 1 mK log ρ(Wm;β1β2Wm;β2β3· · · Wm;βKβ1)

In particular, if a diagonal cycle β1β2· · · βKβ1 exists and m ≥ 2 such that

ρ(Sm;β1β2Sm;β2β3· · · Sm;βKβ1) > 1,

or

ρ(Wm;β1β2Wm;β2β3· · · Wm;βKβ1) > 1

then h(A2) > 0

This theorem is in [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.]

Theorem 3.3. _{Given a basic set B,if there are two adjacent sides of all tiles in B} ,such that the corners of the sides have different colors . Then,if given the colored corners of the adjacent sides in Σm×n(B) ,the rest colors will be determined.

Hence we can show the upper bound of Γm×n(B) is related to the (m+n)th power

,then we can get h(B) = 0 by the definition of spatial entropy.

[This theorem is first proved by J.Y. Chen in edge coloring.]

Methods: 1 :

h(B) > 0 

P1: Theorem 3.2

2 : h(B) = 0                O1: Theorem 3.3

O1.1 : If the patterns in the horizontal(or vertical) way have only one

choice,then Γm×n(B) ≤ (|B|)n(or (|B|)m),hence h(B) = 0.

O1.2 : If B = B1∪ B2,the patterns in each set can’t be attached together

and h(B1) = h(B2) = 0 ,then h(B) = 0.

7

2-1 2 minimal cycle generators: ( )1∪ ( )1

Serial number Basic set

Method Spatial entropy (1.1) ∪ (1.2) B = {O, E} h(B) = O1 (1.1) ∪ (2.1) B = {O, b, t} h(B) = O1 (1.1) ∪ (3) B = {O, I, J} h(B) = O1 (1.1) ∪ (4.1) B = {O, e1, e2, e3, e4} ρ(S2;11) = g h(B) = P1 (1.1) ∪ (4.2) B = {O, e1, e2, e3, e4} h(B) = O1 (1.1) ∪ (5.1) B = {O, e1, e2, e3, e4, b} h(B) = O1 (1.1) ∪ (6.1) B = {O, e1, e4, J} h(B) = O1 (1.1) ∪ (6.3) B = {O, e2, e3, J} h(B) = O1 (1.1) ∪ (7.1) B = {O, e1, e4, e1, e4} h(B) = O1 (2.1) ∪ (2.2) B = {b, t, l, r} h(B) = O1 (2.1) ∪ (3) B = {b, t, I, J} h(B) = O1 (2)1∪ (4)1 B = {b, t, e1, e2, e3, e4} ρ(S2;41S2;14) = 2 h(B) = P1 (2.1) ∪ (5.1) B = {b, t, e1, e2, e3, e4} Γm×n(B) ≤ 6 · 3(n) h(B) = O1,1 (2.1) ∪ (5.3) B = {b, t, e1, e3, e2, e4, r} Γm×n(B) ≤ 7 · 2(m) h(B) = O1,1 (2.1) ∪ (6.1) B = {b, t, e1, e4, J} h(B) = O1 (2.1) ∪ (7.1) B = {b, t, e1, e4, e1, e4} h(B) = O1 (3) ∪ (4)1 B = {I, J, e1, e2, e3, e4} ρ(S2;14S2;41) = 2 h(B) = P1 (3) ∪ (5)1 B = {I, J, e1, e2, e3, e4, b} ρ(S2;41S2;14) = g h(B) = P1 (3) ∪ (6.1) B = {I, J, e1, e4} h(B) = O1 (3) ∪ (7.1) B = {I, J, e1, e4, e1, e4} h(B) = O1 (4.1) ∪ (4.2) B = {e1, e2, e3, e4, e1, e2, e3, e4} h(B) = O1 (4.1) ∪ (5.1) B = {e1, e2, e3, e4, e3, e4, b} h(B) = O1

(4.1) ∪ (6.1) B = {e1, e2, e3, e4, J} h(B) = O1 (4.1) ∪ (6.3) B = {e1, e2, e3, e4, e2, e3, J} h(B) = O1 (4.1) ∪ (7.1) B = {e1, e2, e3, e4, e1, e4} ⊂ (4.1) ∪ (4.2) h(B) = O2 (5.1) ∪ (5.2) B = {e1, e2, e3, e4, b, e3, e4, e1, e2, t} ⊃ (2.1) ∪ (4.1) h(B) = P2 (5.1) ∪ (5.3) B = {e1, e2, e3, e4, b, e3, e2, r} h(B) = O1 (5.1) ∪ (6.1) B = {e1, e2, e3, e4, b, e4, J} h(B) = O1 (5.1) ∪ (7.1) B = {e1, e2, e3, e4, b, e4, e1} h(B) = O1 (6.1) ∪ (6.2) B = {e1, e4, J, e2, e3, I} = (3) ∪ (4.1) h(B) = P2 (6.1) ∪ (6.3) B = {e1, e4, J, e2, e3} h(B) = O1 (6.1) ∪ (6.4) B = {e1, e4, J, e1, e4, I} = (3) ∪ (7.1) h(B) = O2 (6.3) ∪ (6.4) B = {e2, e3, J, e1, e4, I} = (3) ∪ (4.2) h(B) = P2 (6.1) ∪ (7.1) B = {e1, e4, J, e1, e4} ⊂ (3) ∪ (7.1) h(B) = O2 (6.1) ∪ (7.2) B = {e1, e4, J, e2, e3, e2, e3} h(B) = O1 (7.1) ∪ (7.2) B = {e1, e4, e1, e4, e2, e3, e2, e3} = (4.1) ∪ (4.2) h(B) = O2

Summary 2-1 2 minimal cycle generators Serial number Basic set h(B) = P1 or

Method Undetermined (1.1) ∪ (4.1) B = {O, e1, e2, e3, e4} ρ(S2;11) = g h(B) = P1 (2)1∪ (4)1 B = {b, t, e1, e2, e3, e4} ρ(S2;41S2;14) = 2 h(B) = P1 (3) ∪ (4)1 B = {I, J, e1, e2, e3, e4} ρ(S2;14S2;41) = 2 h(B) = P1 (3) ∪ (5)1 B = {I, J, e1, e2, e3, e4, b} ρ(S2;41S2;14) = g h(B) = P1

9

3-1 3 minimal cycle generators: ( )1∪ ( )1∪ ( )1

Serial number Basic set

Method Spatial entropy (1.1) ∪ (2.1) ∪ (3) B = {O, b, t, I, J} h(B) = O1 (1)1∪ (2)1∪ (5)1 h(B) = P (1.1) ∪ (2.1) ∪ (5.1) B = {e1, e2, e3, e4, b, O, t} ρ(S2;11S2;14S2;41) = 2 h(B) = P1 (1.1) ∪ (2.2) ∪ (5.1) B = {e1, e2, e3, e4, b, O, l, r} ρ(S2;11S2;11S2;14S2;41) = 2 h(B) = P1 (1.1) ∪ (2.1) ∪ (6.1) B = {e1, e4, J, O, b, t} ρ(S4;14S4;41S4;11) = 3 h(B) = P1 (1.2) ∪ (2.1) ∪ (6.1) B = {e1, e4, J, E, b, t} h(B) = O1 (1)1∪ (2)1∪ (7)1 B = {e1, e4, e1, e4, O, b, t} ρ(S5;14S5;44S5;41S5;11) = 3.73 h(B) = P1 (1.1) ∪ (3) ∪ (6.1) B = {e1, e4, J, O, I} h(B) = O1 (1.2) ∪ (3) ∪ (6.1) B = {e1, e4, J, E, I} h(B) = O1 (1.1) ∪ (3) ∪ (7.1) B = {e1, e4, e1, e4, O, I, J} h(B) = O1 (1.1) ∪ (4.1) ∪ (5.1) B ⊃ (1.1) ∪ (4.1) h(B) = P2 (1.1) ∪ (4.2) ∪ (5.1) B = {e1, e2, e3, e4, b, O, e1, e2} h(B) = O1 (1.1) ∪ (4.1) ∪ (6.1) B ⊃ (1.1) ∪ (4.1) h(B) = P2 (1.1) ∪ (4.2) ∪ (6.1) B = {e1, e4, J, O, e1, e2, e3, e4} h(B) = O1 (1.2) ∪ (4.1) ∪ (6.1) B = {e1, e4, J, E, e2, e3} h(B) = O1 (1.2) ∪ (4.2) ∪ (6.1) B ⊃ [(1.1) ∪ (4.1)]1 h(B) = P2 (1.1) ∪ (4.1) ∪ (7.1) B ⊃ (1.1) ∪ (4.1) h(B) = P2 (1.1) ∪ (4.2) ∪ (7.1) B = {e1, e4, e1, e4, O, e2, e3} ⊂ (1.1) ∪ (4.2) ∪ (6.1) h(B) = O2 (1.1) ∪ (5.1) ∪ (6.1) B = {e1, e4, J, O, e2, e3, e4, b} ρ(S6;14S6;41S6;11S6;11) = 4.08 h(B) = P1 (1.2) ∪ (5.1) ∪ (6.1) B = {e1, e4, J, E, e2, e3, e4, b} h(B) = O1 (1)1∪ (5)1∪ (7)1 B = {e1, e2, e3, e4, b, O, e4, e1} ρ(S6;14S6;44S6;41S6;11) = 3.24 h(B) = P1 (1.1) ∪ (6.1) ∪ (7.1) B = {e1, e4, J, O, e1, e4} h(B) = O1 (1.2) ∪ (6.1) ∪ (7.1) B = {e1, e4, J, E, e1, e4} h(B) = O1 (1.1) ∪ (6.1) ∪ (7.2) B = {e1, e4, J, O, e2, e3, e2, e3} ⊃ (1.1) ∪ (4.1) h(B) = P2 (1.2) ∪ (6.1) ∪ (7.2) B = {e1, e4, J, E, e2, e3, e2, e3} h(B) = O1

(2)1∪ (3) ∪ (6)1 B = {e₁, e₄, J, b, t, I} ρ(S3;14S3;41S3;14S3;41S3;11) = 3 h(B) = P1 (2)1∪ (3) ∪ (7)1 B = {e1, e4, e1, e4, b, t, I, J} ⊃ (2.1) ∪ (3) ∪ (6.1) h(B) = P2 (2)1∪ (5)1∪ (6)1 h(B) = P (2.1) ∪ (5.1) ∪ (6.1) B = {e1, e2, e3, e4, b, t, e4, J} ρ(S7;14S7;41S7;14S7;41S7;11) = 3.41 h(B) = P1 (2.2) ∪ (5.1) ∪ (6.1) B = {e1, e2, e3, e4, l, r, e4, J} ρ(S6;14S6;41S6;11S6;11S6;11) = 2 h(B) = P1 (2.1) ∪ (5.1) ∪ (7.1) B = {e1, e2, e3, e4, b, t, e4, e1} Undetermined (2.2) ∪ (5.1) ∪ (7.1) B = {e1, e2, e3, e4, b, l, r, e4, e1} ρ(S4;11) = 1.46 h(B) = P1 (2)1∪ (6)1∪ (7)1 h(B) = P (2.1) ∪ (6.1) ∪ (7.1) B = {e1, e4, J, b, t, e1, e4} ρ(S4;14S4;41S4;11) = 2.61 h(B) = P1 (2.1) ∪ (6.1) ∪ (7.2) B = {e1, e4, J, b, t, e2, e3, e2, e3} ⊃ (2.1) ∪ (4.1) h(B) = P2 (3) ∪ (6.1) ∪ (7.1) B = {e1, e4, J, I, e1, e4} = (3) ∪ (7.1) h(B) = O2 (3) ∪ (6.1) ∪ (7.2) B = {e1, e4, J, I, e2, e3, e2, e3} ⊃ (3) ∪ (4.1) h(B) = P2 (4.1) ∪ (5.1) ∪ (6.1) B = {e1, e4, J, e2, e3, e3, e4, b} Undetermined (4.2) ∪ (5.1) ∪ (6.1) B = {e1, e4, J, e1, e2, e3, e4, e2, b} ρ(S6;14S6;44S6;41S6;11) = 1.58 h(B) = P1 (4.1) ∪ (5.1) ∪ (7.1) B = {e1, e2, e3, e4, b, e3, e4, e1} h(B) = O1 (4.1) ∪ (6.1) ∪ (7.1) B = {e1, e4, J, e2, e3, e1, e4} Undetermined (4.1) ∪ (6.1) ∪ (7.2) B = {e1, e4, J, e2, e3, e2, e3} = (6.1) ∪ (7.2) h(B) = O2 (4.2) ∪ (6.1) ∪ (7.2) B = {e1, e4, J, e1, e2, e3, e4, e2, e3} ρ(S6;14S6;41S6;11S6;11S6;11) = 3.4 h(B) = P1 (5.1) ∪ (6.1) ∪ (7.1) B = {e1, e2, e3, e4, b, e4, J, e1} Undetermined (5.1) ∪ (6.1) ∪ (7.2) B = {e1, e2, e3, e4, b, e4, J, e3, e2} ρ(S6;14S6;44S6;41S6;11) = 1.58 h(B) = P1

11

Summary 3-1 3 minimal cycle generators: ( )1∪ ( )1∪ ( )1

Serial number Basic set h(B) = P1 or

Method Undetermined (1)1∪ (2)1∪ (5)1 h(B) = P (1.1) ∪ (2.1) ∪ (5.1) B = {e1, e2, e3, e4, b, O, t} ρ(S2;11S2;14S2;41) = 2 h(B) = P1 (1.1) ∪ (2.2) ∪ (5.1) B = {e1, e2, e3, e4, b, O, l, r} ρ(S2;11S2;11S2;14S2;41) = 2 h(B) = P1 (1.1) ∪ (2.1) ∪ (6.1) B = {e1, e4, J, O, b, t} ρ(S4;14S4;41S4;11) = 3 h(B) = P1 (1)1∪ (2)1∪ (7)1 B = {e1, e4, e1, e4, O, b, t} ρ(S5;14S5;44S5;41S5;11) = 3.73 h(B) = P1 (1.1) ∪ (5.1) ∪ (6.1) B = {e1, e4, J, O, e2, e3, e4, b} ρ(S6;14S6;41S6;11S6;11) = 4.08 h(B) = P1 (1)1∪ (5)1∪ (7)1 B = {e1, e2, e3, e4, b, O, e4, e1} ρ(S6;14S6;44S6;41S6;11) = 3.24 h(B) = P1 (2)1∪ (3) ∪ (6)1 B = {e1, e4, J, b, t, I} ρ(S4;14S4;41S4;11) = 3 h(B) = P1 (2)1∪ (5)1∪ (6)1 h(B) = P (2.1) ∪ (5.1) ∪ (6.1) B = {e1, e2, e3, e4, b, t, e4, J} ρ(S7;14S7;41S7;14S7;41S7;11) = 3.41 h(B) = P1 (2.2) ∪ (5.1) ∪ (6.1) B = {e1, e2, e3, e4, l, r, e4, J} ρ(S6;14S6;41S6;11S6;11S6;11) = 2 h(B) = P1 (2.1) ∪ (5.1) ∪ (7.1) B = {e1, e2, e3, e4, b, t, e4, e1} Undetermined (2.2) ∪ (5.1) ∪ (7.1) B = {e1, e2, e3, e4, b, l, r, e4, e1} ρ(S4;11) = 1.46 h(B) = P1 (2.1) ∪ (6.1) ∪ (7.1) B = {e1, e4, J, b, t, e1, e4} ρ(S4;14S4;41S4;11) = 2.61 h(B) = P1 (4.1) ∪ (5.1) ∪ (6.1) B = {e1, e4, J, e2, e3, e3, e4, b} Undetermined (4.2) ∪ (5.1) ∪ (6.1) B = {e1, e4, J, e1, e2, e3, e4, e2, b} ρ(S6;14S6;44S6;41S6;11) = 1.58 h(B) = P1 (4.1) ∪ (6.1) ∪ (7.1) B = {e1, e4, J, e2, e3, e1, e4} Undetermined (4.2) ∪ (6.1) ∪ (7.2) B = {e1, e4, J, e1, e2, e3, e4, e2, e3} ρ(S6;14S6;41S6;11S6;11S6;11) = 3.4 h(B) = P1 (5.1) ∪ (6.1) ∪ (7.1) B = {e1, e2, e3, e4, b, e4, J, e1} Undetermined (5.1) ∪ (6.1) ∪ (7.2) B = {e1, e2, e3, e4, b, e4, J, e3, e2} ρ(S6;14S6;44S6;41S6;11) = 1.58 h(B) = P1

3-2 3 minimal cycle generators: ( )1∪ ( )2

Serial number Basic set

Method Spatial entropy (1.1) ∪ (2.1) ∪ (2.2) B = {O, b, t, l, r} h(B) = O1 (1.1) ∪ (1.2) ∪ (2.1) B = {O, E, b, t} h(B) = O1 (1.1) ∪ (1.2) ∪ (3) B = {O, E, I, J} h(B) = O1 (1)1∪ (5)2 h(B) = P (1.1) ∪ (5.1) ∪ (5.2) B = {O, e1, e2, e3, e4, b, e3, e4, e1, e2, t} ⊃ (1.1) ∪ (4.1) h(B) = P2 (1.1) ∪ (5.1) ∪ (5.3) B = {O, e1, e2, e3, e4, b, e3, e2, r} ρ(S6;14S6;44S6;41S6;11) = 3.54 h(B) = P1 (1)2∪ (5)1 B = {O, E, e1, e2, e3, e4, b} ρ(S4;14S4;41S4;11S4;11) = 2 h(B) = P1 (1.1) ∪ (6.1) ∪ (6.2) B = {O, e1, e4, J, e2, e3, I} ⊃ (1.1) ∪ (4.1) h(B) = P2 (1.1) ∪ (6.1) ∪ (6.3) B = {O, e1, e4, J, e2, e3} h(B) = O1 (1.1) ∪ (6.1) ∪ (6.4) B = {O, e1, e4, J, e1, e4, I} = (1.1) ∪ (3) ∪ (7.1) h(B) = O2 (1.1) ∪ (6.3) ∪ (6.4) B = {O, e2, e3, J, e1, e4, I} ⊃ (3) ∪ (4.2) h(B) = P2 (1.1) ∪ (1.2) ∪ (6.1) B = {O, E, e1, e4, J} h(B) = O1 (1)1∪ (7)2 B = {O, e1, e4, e1, e4, e2, e3, e2, e3} ⊃ (1.1) ∪ (4.1) h(B) = P2 (1.1) ∪ (1.2) ∪ (7.1) B = {O, E, e1, e4, e1, e4} h(B) = O1 (2.1) ∪ (2.2) ∪ (3) B = {b, t, l, r, I, J} h(B) = O1 (2)1∪ (5)2 h(B) = P (2.1) ∪ (5.1) ∪ (5.2) B = {b, t, e1, e2, e3, e4, e3, e4, e1, e2} ⊃ (2.1) ∪ (4.1) h(B) = P2 (2.1) ∪ (5.1) ∪ (5.3) B = {b, t, e1, e2, e3, e4, e3, e2, r} ⊃ [(2.2) ∪ (5.1) ∪ (7.1)]1 h(B) = P2 (2.1) ∪ (5.3) ∪ (5.4) B = {b, t, e1, e3, e2, e4, r, e2, e4, e1, e3, l} ⊃ (2.1) ∪ (4.1) h(B) = P2 (2.1) ∪ (2.2) ∪ (5.1) B = {b, t, l, r, e1, e2, e3, e4} Γm×n(B) ≤ 8 · 3(n) h(B) = O1,1 (2)1∪ (6)2 h(B) = P (2.1) ∪ (6.1) ∪ (6.2) B = {b, t, e1, e4, J, e2, e3, I} ⊃ (2.1) ∪ (4.1) h(B) = P2 (2.1) ∪ (6.1) ∪ (6.3) B = {b, t, e1, e4, J, e2, e3} ρ(S5;14S5;41) = 1.73 h(B) = P1 (2.1) ∪ (6.1) ∪ (6.4) B = {b, t, e1, e4, J, e1, e4, I} = (2.1) ∪ (3) ∪ (7.1) h(B) = P2 (2.1) ∪ (2.2) ∪ (6.1) B = {b, t, l, r, e1, e4, J} h(B) = O1

13 (2)1∪ (7)2 B = {b, t, e₁, e₄, e₁, e₄, e₂, e₃, e₂, e₃} ⊃ (2.1) ∪ (4.1) h(B) = P2 (2.1) ∪ (2.2) ∪ (7.1) B = {b, t, l, r, e1, e4, e1, e4} h(B) = O1 (3) ∪ (6.1) ∪ (6.2) B = {I, J, e1, e4, e2, e3} = (3) ∪ (4.1) h(B) = P2 (3) ∪ (6.1) ∪ (6.3) B = {I, J, e1, e4, e2, e3} = {I, J, e1, e4} ∪ {I, J, e2, e3} h(B) = O1,2 (3) ∪ (6.1) ∪ (6.4) B = {I, J, e1, e4, e1, e4} h(B) = O1 (3) ∪ (7)2 B = {I, J, e1, e4, e1, e4, e2, e3, e2, e3} ⊃ (3) ∪ (4.1) h(B) = P2 (4)1∪ (5)2 h(B) = P (4.1) ∪ (5.1) ∪ (5.2) B = {e1, e2, e3, e4, e3, e4, b, e1, e2, t} ⊃ (2.1) ∪ (4.1) h(B) = P2 (4.1) ∪ (5.1) ∪ (5.3) B = {e1, e2, e3, e4, e3, e4, b, e2, r} ρ(S6;14S6;41S6;11S6;11S6;11) = 3 h(B) = P1 (4)2∪ (5)1 B = {e1, e2, e3, e4, e1, e2, e3, e4, b} ρ(S2;11S2;14S2;41) = g h(B) = P1 (4.1) ∪ (6.1) ∪ (6.2) B = {e1, e2, e3, e4, J, I} ⊃ (3) ∪ (4.1) h(B) = P2 (4.1) ∪ (6.1) ∪ (6.3) B = {e1, e2, e3, e4, J, e2, e3} h(B) = O1 (4.1) ∪ (6.1) ∪ (6.4) B = {e1, e2, e3, e4, J, e1, e4, I} ⊃ (3) ∪ (4.1) h(B) = P2 (4.1) ∪ (6.3) ∪ (6.4) B = {e1, e2, e3, e4, e2, e3, J, e1, e4, I} ⊃ (3) ∪ (4.1) h(B) = P2 (4)2∪ (6)1 B = {e1, e2, e3, e4, e1, e2, e3, e4, J} = (4.2) ∪ (6.1) ∪ (7.2) h(B) = P2 (4.1) ∪ (7.1) ∪ (7.2) B = {e1, e2, e3, e4, e1, e4, e2, e3} = (4.1) ∪ (4.2) h(B) = O2 (4.1) ∪ (4.2) ∪ (7.1) B = {e1, e2, e3, e4, e1, e2, e3, e4} = (4.1) ∪ (4.2) h(B) = O2 (5.1) ∪ (6.1) ∪ (6.2) B = {e1, e2, e3, e4, b, e4, J, e3, I} ⊃ (3) ∪ (4.1) h(B) = P2 (5.1) ∪ (6.1) ∪ (6.3) B = {e1, e2, e3, e4, b, e4, J, e2} h(B) = O1 (5.1) ∪ (6.1) ∪ (6.4) B = {e1, e2, e3, e4, b, e4, J, e1, I} ρ(S2;14S2;41) = g h(B) = P1 (5)2∪ (6)1 h(B) = P (5.1) ∪ (5.2) ∪ (6.1) B = {e1, e4, J, e2, e3, e4, b, e3, e1, e2, t} ⊃ (2.1) ∪ (4.1) h(B) = P2 (5.1) ∪ (5.3) ∪ (6.1) B = {e1, e4, J, e2, e3, e4, b, e3, e2, r} ρ(S5;14S5;41S5;11S5;11) = 4.41 h(B) = P1 (5.2) ∪ (5.3) ∪ (6.1) B = {e1, e4, J, e3, e1, e2, t, e4, r} ρ(S7;14S7;41S7;11S7;11) = 2.30 h(B) = P1 (5)1∪ (7)2 B = {e1, e2, e3, e4, b, e4, e1, e2, e3} = (4.1) ∪ (4.2) ∪ (5.1) h(B) = P2 (5.1) ∪ (5.2) ∪ (7.1) B = {e1, e4, e1, e4, e2, e3, b, e3, e2, t} ⊃ (2.1) ∪ (4.1) h(B) = P2 (5.1) ∪ (5.3) ∪ (7.1) B = {e1, e4, e1, e4, e2, e3, b, e3, e2, r} ⊃ (4.1) ∪ (5.1) ∪ (5.3) h(B) = P2 (5.1) ∪ (5.4) ∪ (7.1) B = {e1, e4, e1, e4, e2, e3, b, l} = (5.1) ∪ (5.4) h(B) = O2

(6)1∪ (7)2 B = {e₁, e₄, J, e₁, e₄, e₂, e₃, e₂, e₃} = (4.2) ∪ (6.1) ∪ (7.2) h(B) = P2 (6.1) ∪ (6.2) ∪ (7.1) B = {e1, e4, e1, e4, J, e2, e3, I} ⊃ (3) ∪ (4.1) h(B) = P2 (6.1) ∪ (6.3) ∪ (7.1) B = {e1, e4, e1, e4, J, e2, e3} = (4.2) ∪ (6.1) h(B) = O2 (6.1) ∪ (6.4) ∪ (7.1) B = {e1, e4, e1, e4, I, J} = (3) ∪ (7.1) h(B) = O2 (6.2) ∪ (6.3) ∪ (7.1) B = {e1, e4, e1, e4, e2, e3, I, e2, e3, J} ⊃ (3) ∪ (4.1) h(B) = P2

Summary 3-2 3 minimal cycle generators: ( )1∪ ( )2

Serial number Basic set h(B) = P1or

Method Undetermined (1.1) ∪ (5.1) ∪ (5.3) B = {O, e1, e2, e3, e4, b, e3, e2, r} ρ(S6;14S6;44S6;41S6;11) = 3.54 h(B) = P1 (1)2∪ (5)1 B = {O, E, e1, e2, e3, e4, b} ρ(S4;14S4;41S4;11S4;11) = 2 h(B) = P1 (2.1) ∪ (6.1) ∪ (6.3) B = {b, t, e1, e4, J, e2, e3} ρ(S5;14S5;41) = 1.73 h(B) = P1 (4.1) ∪ (5.1) ∪ (5.3) B = {e1, e2, e3, e4, e3, e4, b, e2, r} ρ(S6;14S6;41S6;11S6;11S6;11) = 3 h(B) = P1 (4)2∪ (5)1 B = {e1, e2, e3, e4, e1, e2, e3, e4, b} ρ(S2;11S2;14S2;41) = g h(B) = P1 (5.1) ∪ (6.1) ∪ (6.4) B = {e1, e2, e3, e4, b, e4, J, e1, I} ρ(S2;14S2;41) = g h(B) = P1 (5.1) ∪ (5.3) ∪ (6.1) B = {e1, e4, J, e2, e3, e4, b, e3, e2, r} ρ(S5;14S5;41S5;11S5;11) = 4.41 h(B) = P1 (5.2) ∪ (5.3) ∪ (6.1) B = {e1, e4, J, e3, e1, e2, t, e4, r} ρ(S7;14S7;41S7;11S7;11) = 2.30 h(B) = P1

3-3 3 minimal cycle generators: ( )3

Serial number Basic set

Method Spatial entropy (5.1) ∪ (5.2) ∪ (5.3) B = {e1, e2, e3, e4, b, e3, e4, e1, e2, t, r}

⊃ (2.1) ∪ (4.1) h(B) = P2

(6.1) ∪ (6.2) ∪ (6.3) B = {e1, e4, J, e2, e3, I, e2, e3, e1, e4}

15

4-1 4 minimal cycle generators: ( )1∪ ( )1∪ ( )1∪ ( )1

Serial number Basic set

Method Spatial entropy (1.1) ∪ (3) ∪ (6.1) ∪ (7.1) B = {e1, e4, J, O, I, e1, e4} = (1.1) ∪ (3) ∪ (7.1) h(B) = O2 (1.1) ∪ (3) ∪ (6.1) ∪ (7.2) B = {e1, e4, J, O, I, e2, e3, e2, e3} ⊃ (1.1) ∪ (4.1) h(B) = P2 (1.2) ∪ (3) ∪ (6.1) ∪ (7.2) B = {e1, e4, J, E, I, e2, e3, e2, e3} ⊃ (3) ∪ (4.1) h(B) = P2 (1.1) ∪ (4.1) ∪ (5.1) ∪ (6.1) B ⊃ (1.1) ∪ (4.1) h(B) = P2 (1.1) ∪ (4.1) ∪ (5.1) ∪ (6.3) B ⊃ (1.1) ∪ (4.1) h(B) = P2 (1.1) ∪ (4.2) ∪ (5.1) ∪ (6.1) B = {e1, e2, e3, e4, b, O, e1, e2, e4, J} ρ(S6;14S6;44S6;41S6;11) = 17.41 h(B) = P1 (1.1) ∪ (4.2) ∪ (5.1) ∪ (6.3) B = {e1, e2, e3, e4, b, O, e1, e2, J} Undetermined (1.1) ∪ (4.1) ∪ (5.1) ∪ (7.1) B = {e1, e2, e3, e4, b, O, e3, e4, e1} ⊃ (1.1) ∪ (4.1) h(B) = P2 (1.1) ∪ (4.2) ∪ (5.1) ∪ (7.1) B = {e1, e2, e3, e4, b, O, e1, e2, e4} ρ(S6;14S6;44S6;41S6;11) = 3.24 h(B) = P1 (1.1) ∪ (4.1) ∪ (6.1) ∪ (7.1) B ⊃ (1.1) ∪ (4.1) h(B) = P2 (1.1) ∪ (4.1) ∪ (6.1) ∪ (7.2) B ⊃ (1.1) ∪ (4.1) h(B) = P2 (1.1) ∪ (4.2) ∪ (6.1) ∪ (7.1) B = {e1, e4, J, O, e1, e2, e3, e4} = (1.1) ∪ (4.2) ∪ (6.1) h(B) = O2 (1.1) ∪ (4.2) ∪ (6.1) ∪ (7.2) B = {e1, e4, J, O, e1, e2, e3, e4, e2, e3} ⊃ (1.1) ∪ (4.1) h(B) = P2 (1.2) ∪ (4.1) ∪ (6.1) ∪ (7.1) B = {e1, e4, J, E, e2, e3, e1, e4} Undetermined (1.1) ∪ (5.1) ∪ (6.1) ∪ (7.1) B = {e1, e2, e3, e4, b, O, e4, J, e1} ρ(S6;14S6;41S6;11S6;11) = 12.98 h(B) = P1 (1.1) ∪ (5.1) ∪ (6.1) ∪ (7.2) B = {e1, e2, e3, e4, b, O, e4, J, e3, e2} ⊃ (1.1) ∪ (4.1) h(B) = P2 (1.1) ∪ (5.1) ∪ (6.3) ∪ (7.1) B = {e1, e2, e3, e4, b, O, e2, J, e4, e1} ⊃ (1.1) ∪ (5.1) ∪ (6.1) ∪ (7.1) h(B) = P2 (1.1) ∪ (5.1) ∪ (6.3) ∪ (7.2) B = {e1, e2, e3, e4, b, O, e2, J, e3} ρ(S6;14S6;44S6;41S6;11) = 3.24 h(B) = P1 (4.1) ∪ (5.1) ∪ (6.1) ∪ (7.1) B = {e1, e2, e3, e4, b, e3, e4, J, e1} Undetermined (4.1) ∪ (5.1) ∪ (6.1) ∪ (7.2) B = {e1, e2, e3, e4, b, e3, e4, J, e2} ρ(S6;14S6;44S6;41S6;11) = 1.58 h(B) = P1 (4.1) ∪ (5.1) ∪ (6.3) ∪ (7.1) B = {e1, e2, e3, e4, b, e3, e4, e2, J, e1} ⊃ (4.1) ∪ (5.1) ∪ (6.1) ∪ (7.2) h(B) = P2

Summary 4-1 4 minimal cycle generators: ( )1∪ ( )1∪ ( )1∪ ( )1

Serial number Basic set h(B) = P1 or

Method Undetermined (1.1) ∪ (4.2) ∪ (5.1) ∪ (6.1) B = {e1, e2, e3, e4, b, O, e1, e2, e4, J} ρ(S6;14S6;44S6;41S6;11) = 17.41 h(B) = P1 (1.1) ∪ (4.2) ∪ (5.1) ∪ (6.3) B = {e1, e2, e3, e4, b, O, e1, e2, J} Undetermined (1.1) ∪ (4.2) ∪ (5.1) ∪ (7.1) B = {e1, e2, e3, e4, b, O, e1, e2, e4} ρ(S6;14S6;44S6;41S6;11) = 3.24 h(B) = P1 (1.2) ∪ (4.1) ∪ (6.1) ∪ (7.1) B = {e1, e4, J, E, e2, e3, e1, e4} Undetermined (1.1) ∪ (5.1) ∪ (6.1) ∪ (7.1) B = {e1, e2, e3, e4, b, O, e4, J, e1} ρ(S6;14S6;41S6;11S6;11) = 12.98 h(B) = P1 (1.1) ∪ (5.1) ∪ (6.3) ∪ (7.2) B = {e1, e2, e3, e4, b, O, e2, J, e3} ρ(S6;14S6;44S6;41S6;11) = 3.24 h(B) = P1 (4.1) ∪ (5.1) ∪ (6.1) ∪ (7.1) B = {e1, e2, e3, e4, b, e3, e4, J, e1} Undetermined (4.1) ∪ (5.1) ∪ (6.1) ∪ (7.2) B = {e1, e2, e3, e4, b, e3, e4, J, e2} ρ(S6;14S6;44S6;41S6;11) = 1.58 h(B) = P1

17

4-2 4 minimal cycle generators: ( )1∪ ( )1∪ ( )2

Serial number Basic set

Method Spatial entropy (1.1) ∪ (2.1) ∪ (2.2) ∪ (3) B = {O, b, t, l, r, I, J} h(B) = O1 (1.1) ∪ (1.2) ∪ (2.1) ∪ (3) B = {O, E, b, t, I, J} h(B) = O1 (11) ∪ (2)1∪ (6)2 h(B) = P (1.1) ∪ (2.1) ∪ (6.1) ∪ (6.2) B = {O, b, t, e1, e4, J, e2, e3, I} ⊃ (1.1) ∪ (4.1) h(B) = P2 (1.1) ∪ (2.1) ∪ (6.1) ∪ (6.3) B = {O, b, t, e1, e4, J, e2, e3} ρ(S5;14S5;41) = 2.14 h(B) = P1 (1.1) ∪ (2.1) ∪ (6.1) ∪ (6.4) B = {O, b, t, e1, e4, J, e1, e4, I} ρ(S4;14S4;41S4;11) = 10 h(B) = P1 (1.1) ∪ (2.1) ∪ (6.3) ∪ (6.4) B = {O, b, t, e2, e3, J, e1, e4, I} ⊃ (3) ∪ (4.2) h(B) = P2 (1.1) ∪ (2.1) ∪ (2.2) ∪ (6.1) B = {e1, e4, J, O, b, t, l, r} ρ(S4;14S4;41S4;11) = 3.14 h(B) = P1 (1.2) ∪ (2.1) ∪ (2.2) ∪ (6.1) B = {e1, e4, J, E, b, t, l, r} h(B) = O1 (12) ∪ (2)1∪ (6)1 h(B) = P (1.1) ∪ (1.2) ∪ (2.1) ∪ (6.1) B = {e1, e4, J, b, t, O, E} ρ(S4;14S4;41S4;11) = 3 h(B) = P1 (1.1) ∪ (1.2) ∪ (2.2) ∪ (6.1) B = {e1, e4, J, l, r, O, E} ρ(S3;14S3;41S3;11S3;11) = 2 h(B) = P1 (1.1) ∪ (3) ∪ (6.1) ∪ (6.3) B = {O, I, J, e1, e4, e2, e3} = {O, I, J, e1, e4} ∪ {I, J, e2, e3} h(B) = O1;2 (1.1) ∪ (3) ∪ (6.1) ∪ (6.4) B = {O, I, J, e1, e4, e1, e4} = (1.1) ∪ (3) ∪ (7.1) h(B) = O2 (1.1) ∪ (3) ∪ (6.3) ∪ (6.4) B = {O, I, J, e2, e3, e1, e4} ⊃ (3) ∪ (4.2) h(B) = P2 (1.1) ∪ (3) ∪ (6.1) ∪ (6.2) B = {O, E, I, J, e1, e4} h(B) = O1 (1)1∪ (3) ∪ (7)2 B ⊃ [1]1∪ [7]2 h(B) = P2 (1.1) ∪ (1.2) ∪ (3) ∪ (7.1) B = {O, E, I, J, e1, e4, e1, e4} h(B) = O1

(1.1) ∪ (4.2) ∪ (6.3) ∪ (6.4) B = {O, e1, e2, e3, e4, e1, e4, J} h(B) = O1 (1.1) ∪ (4.2) ∪ (6.1) ∪ (6.4) B = {O, e1, e2, e3, e4, e1, e4, J, I} ⊃ (3) ∪ (4.2) h(B) = P2 (1.1) ∪ (4.2) ∪ (6.3) ∪ (6.4) B = {O, e1, e2, e3, e4, I, J} ⊃ (3) ∪ (4.2) h(B) = P2 (11) ∪ (5)1∪ (6)2 h(B) = P (1.1) ∪ (5.1) ∪ (6.1) ∪ (6.1) B = {e1, e2, e3, e4, b, O, e4, J, e3, I} ⊃ (1.1) ∪ (4.1) h(B) = P2 (1.1) ∪ (5.1) ∪ (6.1) ∪ (6.3) B = {e1, e2, e3, e4, b, O, e4, J, e2} ρ(S6;14S6;41S6;11S6;11) = 4.08 h(B) = P1 (1.1) ∪ (5.1) ∪ (6.1) ∪ (6.4) B = {e1, e2, e3, e4, b, O, e4, J, e1, I} ρ(S2;14S2;41) = g h(B) = P1 (1.1) ∪ (5.1) ∪ (6.3) ∪ (6.4) B = {e1, e2, e3, e4, b, O, e2, J, e3, I} ⊃ (3) ∪ (4.2) h(B) = P2 (1.1) ∪ (6.1) ∪ (6.2) ∪ (7.1) B = {O, e1, e4, e1, e4, J, e2, e3, I} ⊃ (1.1) ∪ (4.1) h(B) = P2 (1.1) ∪ (6.1) ∪ (6.3) ∪ (7.1) B = {O, e1, e4, e1, e4, J, e2, e3} = (1.1) ∪ (4.2) ∪ (6.1) h(B) = O2 (1.1) ∪ (6.1) ∪ (6.4) ∪ (7.1) B = {O, e1, e4, e1, e4, J, I} = (1.1) ∪ (3) ∪ (7.1) h(B) = O2 (1.1) ∪ (6.2) ∪ (6.3) ∪ (7.1) B = {O, e1, e4, e1, e4, e2, e3, I, e2, e3, J} ⊃ (1.1) ∪ (4.1) h(B) = P2 (1.1) ∪ (6.2) ∪ (6.4) ∪ (7.1) B = {O, e1, e4, e1, e4, e2, e3, I} ⊃ (1.1) ∪ (4.1) h(B) = P2 (1.1) ∪ (6.2) ∪ (6.4) ∪ (7.1) B = {O, e1, e4, e1, e4, e2, e3, J, I} ⊃ (3) ∪ (4.2) h(B) = P2 (1.1) ∪ (1.2) ∪ (6.1) ∪ (7.1) B = {e1, e4, J, O, E, e1, e4} h(B) = O1 (1.1) ∪ (6.1) ∪ (6.3) ∪ (7.1) B = {e1, e4, J, O, E, e2, e3, e2, e3} ⊃ (1.1) ∪ (4.1) h(B) = P2 (2)2∪ (5)1∪ (7)1 h(B) = P (2.1) ∪ (2.2) ∪ (5.1) ∪ (7.1) B = {e1, e2, e3, e4, b, t, l, r, e4, e1} ρ(S4;11) = g h(B) = P1 (2.1) ∪ (2.2) ∪ (5.1) ∪ (7.2) B = {e1, e2, e3, e4, b, t, l, r, e3, e2} ρ(S4;11) = g h(B) = P1 (3) ∪ (6.1) ∪ (6.2) ∪ (7.1) B = {I, J, e1, e4, e1, e4, e2, e3} ⊃ (3) ∪ [4]1 h(B) = P2 (3) ∪ (6.1) ∪ (6.4) ∪ (7.1) B = {I, J, e1, e4, e1, e4} = (3) ∪ (7.1) h(B) = O2 (3) ∪ (6.2) ∪ (6.3) ∪ (7.1) B = {I, J, e1, e4, e1, e4, e2, e3, e2, e3} ⊃ (3) ∪ (4.1) h(B) = P2 (4)1∪ (5)1∪ (6)2 h(B) = P (4.1) ∪ (5.1) ∪ (6.1) ∪ (6.2) B = {e1, e2, e3, e4, b, e3, e4, J, I} ⊃ (3) ∪ (4.1) h(B) = P2 (4.1) ∪ (5.1) ∪ (6.1) ∪ (6.3) B = {e1, e2, e3, e4, b, e3, e4, J, e2} ρ(S6;14S6;44S6;41S6;11) = 1.58 h(B) = P1 (4.1) ∪ (5.1) ∪ (6.1) ∪ (6.4) B = {e1, e2, e3, e4, b, e3, e4, J, e1, I} ⊃ (3) ∪ (4.1) h(B) = P2 (4.1) ∪ (5.1) ∪ (6.3) ∪ (6.4) B = {e1, e2, e3, e4, b, e3, e4, e2, J, e1, I} ⊃ (3) ∪ (4.1) h(B) = P2

19 (4.1) ∪ (6.1) ∪ (6.2) ∪ (7.1) B = {e1, e2, e3, e4, e1, e4, J, I} = (3) ∪ (4.1) h(B) = P2 (4.1) ∪ (6.1) ∪ (6.3) ∪ (7.1) B = {e1, e2, e3, e4, e1, e4, J, e2, e3} = (4.2) ∪ (6.1) ∪ (7.2) h(B) = P2 (4.1) ∪ (6.3) ∪ (6.4) ∪ (7.1) B = {e1, e2, e3, e4, e1, e4, J, e2, e3} = (3) ∪ (4.1) h(B) = P2 (4.1) ∪ (6.3) ∪ (6.4) ∪ (7.1) B = {e1, e2, e3, e4, e1, e4, I} = (4.1) ∪ (6.4) h(B) = O2 (5)1∪ (6)2∪ (7)1 h(B) = P (5.1) ∪ (6.1) ∪ (6.2) ∪ (7.1) B = {e1, e2, e3, e4, b, e4, e1, J, e3, I} ⊃ (3) ∪ (4.1) h(B) = P2 (5.1) ∪ (6.1) ∪ (6.3) ∪ (7.1) B = {e1, e2, e3, e4, b, e4, e1, J, e2} = (5.1) ∪ (6.3) ∪ (7.1) h(B) = P2 (5.1) ∪ (6.3) ∪ (6.4) ∪ (7.1) B = {e1, e2, e3, e4, b, e4, e1, J, I} ⊃ (3) ∪ (5.1) h(B) = P2 (5.1) ∪ (6.2) ∪ (6.3) ∪ (7.1) B = {e1, e2, e3, e4, b, e4, e1, e3, I, e2, J} = ((3) ∪ (4.1) h(B) = P2 (5)2∪ (6)1∪ (7)1 B ⊃ [5]2∪ [6]1 h(B) = P2

Summary 4-2 4 minimal cycle generators: ( )1∪ ( )1∪ ( )2

Serial number Basic set h(B) = P1or

Method Undetermined (1.1) ∪ (2.1) ∪ (6.1) ∪ (6.3) B = {O, b, t, e1, e4, J, e2, e3} ρ(S5;14S5;41) = 2.14 h(B) = P1 (1.1) ∪ (2.1) ∪ (6.1) ∪ (6.4) B = {O, b, t, e1, e4, J, e1, e4, I} ρ(S4;14S4;41S4;11) = 10 h(B) = P1 (1.1) ∪ (2.1) ∪ (2.2) ∪ (6.1) B = {e1, e4, J, O, b, t, l, r} ρ(S4;14S4;41S4;11) = 3.14 h(B) = P1 (1.1) ∪ (1.2) ∪ (2.1) ∪ (6.1) B = {e1, e4, J, b, t, O, E} ρ(S4;14S4;41S4;11) = 3 h(B) = P1 (1.1) ∪ (1.2) ∪ (2.2) ∪ (6.1) B = {e1, e4, J, l, r, O, E} ρ(S3;14S3;41S3;11S3;11) = 2 h(B) = P1 (1.1) ∪ (5.1) ∪ (6.1) ∪ (6.3) B = {e1, e2, e3, e4, b, O, e4, J, e2} ρ(S6;14S6;41S6;11S6;11) = 4.08 h(B) = P1 (1.1) ∪ (5.1) ∪ (6.1) ∪ (6.4) B = {e1, e2, e3, e4, b, O, e4, J, e1, I} ρ(S2;14S2;41) = g h(B) = P1 (2.1) ∪ (2.2) ∪ (5.1) ∪ (7.1) B = {e1, e2, e3, e4, b, t, l, r, e4, e1} ρ(S4;11) = g h(B) = P1 (2.1) ∪ (2.2) ∪ (5.1) ∪ (7.2) B = {e1, e2, e3, e4, b, t, l, r, e3, e2} ρ(S4;11) = g h(B) = P1 (4.1) ∪ (5.1) ∪ (6.1) ∪ (6.3) B = {e1, e2, e3, e4, b, e3, e4, J, e2} ρ(S6;14S6;44S6;41S6;11) = 1.58 h(B) = P1

4-3 4 minimal cycle generators: ( )2∪ ( )2

Serial number Basic set

Method Spatial entropy (1)2∪ (2)2 B = {O, E, b, t, l, r} h(B) = O1 (1)2∪ (5)2 B ⊃ [1]1∪ [5]2 h(B) = P2 (1.1) ∪ (1.2) ∪ (6.1) ∪ (6.2) B = {O, E, e1, e4, J, e2, e3, I} ⊃ (1.1) ∪ (4.1) h(B) = P2 (1.1) ∪ (1.2) ∪ (6.1) ∪ (6.3) B = {O, E, e1, e4, J, e2, e3} = {O, e1, e4, J} ∪ {E, J, e2, e3} h(B) = O1,2 (1.1) ∪ (1.2) ∪ (6.1) ∪ (6.4) B = {O, E, e1, e4, J, e1, e4, I} = (1.1) ∪ (1.2) ∪ (3) ∪ (7.1) h(B) = O2 (4)2∪ (7)2 B = (4)2 h(B) = O2

4-4 4 minimal cycle generators: ( )1∪ ( )3

( )1∪ ( )3 B ⊃ ( )3 h(B) = P2

4-5 4 minimal cycle generators: ( )4

21

5-1 5 minimal cycle generators: ( )1∪ ( )1∪ ( )1∪ ( )1∪ ( )1

Serial number Basic set

Method Spatial entropy (1.1) ∪ (4.2) ∪ (5.1) ∪ (6.1) ∪ (7.1) B = {e1, e2, e3, e4, b, O, e1, e2, e4, J}

⊃ (1.1) ∪ (4.2) ∪ (5.1) ∪ (6.1) h(B) = P2

(1.1) ∪ (4.2) ∪ (5.1) ∪ (6.3) ∪ (7.2) B = {e1, e2, e3, e4, b, O, e1, e2, J, e3}

⊃ [(1.1) ∪ (4.2) ∪ (5.1) ∪ (7.1)]1 h(B) = P2

5-2 5 minimal cycle generators: ( )1∪ ( )1∪ ( )1∪ ( )2

(1.1) ∪ (1.2) ∪ (3) ∪ (6.1) ∪ (7.1) B = {O, E, I, J, e1, e4, e1, e4} = (1.1) ∪ (1.2) ∪ (3) ∪ (7.1) h(B) = O2 (1.1) ∪ (1.2) ∪ (3) ∪ (6.1) ∪ (7.2) B = {O, E, I, J, e1, e4, e2, e3, e2, e3} ⊃ (1.1) ∪ (4.1) h(B) = P2 (1.1) ∪ (3) ∪ (6.1) ∪ (6.2) ∪ (7.1) B = {O, I, J, e1, e4, e2, e3, e1, e4} ⊃ (1.1) ∪ (4.1) h(B) = P2 (1.1) ∪ (3) ∪ (6.1) ∪ (6.3) ∪ (7.1) B = {O, I, J, e1, e4, e2, e3, e1, e4} ⊃ (3) ∪ (4.2) h(B) = P2 (1.1) ∪ (3) ∪ (6.1) ∪ (6.4) ∪ (7.1) B = {O, I, J, e1, e4, e1, e4} ⊃ (1.1) ∪ (3) ∪ (7.1) h(B) = O2 (1.1) ∪ (3) ∪ (6.1) ∪ (6.4) ∪ (7.2) B = {O, I, J, e1, e4, e1, e4, e2, e3, e2, e3} ⊃ (1.1) ∪ (4.1) h(B) = P2 (1.1) ∪ (4.2) ∪ (6.1) ∪ (6.3) ∪ (7.1) B = {O, e1, e2, e3, e4, e1, e4, J} = (1.1) ∪ (4.2) ∪ (6.1) h(B) = O2 (1.1) ∪ (4.2) ∪ (6.1) ∪ (6.4) ∪ (7.1) B = {O, e1, e2, e3, e4, e1, e4, J, I} = (3) ∪ (4.2) h(B) = P2

5-3 5 minimal cycle generators: ( )1∪ ( )2∪ ( )2

(1.1) ∪ (1.2) ∪ (3) ∪ (6.1) ∪ (6.2) B = {O, E, I, J, e1, e4, e2, e3} ⊃ (1.1) ∪ (4.1) h(B) = P2 (1.1) ∪ (1.2) ∪ (3) ∪ (6.1) ∪ (6.3) B = {O, E, I, J, e1, e4, e2, e3} = {O, I, J, e1, e4} ∪ {E, I, J, e2, e3} h(B) = O1,2 (1.1) ∪ (1.2) ∪ (3) ∪ (6.1) ∪ (6.4) B = {O, E, I, J, e1, e4, e1, e4} = (1.1) ∪ (1.2) ∪ (3) ∪ (7.1) h(B) = O2 (1.1) ∪ (1.2) ∪ (6.1) ∪ (6.2) ∪ (7.1) B = {O, E, e1, e4, e1, e4, J, e2, e3, I} ⊃ (1.1) ∪ (4.1) h(B) = P2 (1.1) ∪ (1.2) ∪ (6.1) ∪ (6.3) ∪ (7.1) B = {O, E, e1, e4, e1, e4, J, e2, e3} B ⊃ [(1.1) ∪ (4.1)]1 h(B) = P2 (1.1) ∪ (1.2) ∪ (6.1) ∪ (6.4) ∪ (7.1) B = {O, E, e1, e4, e1, e4, J, I} = (1.1) ∪ (1.2) ∪ (3) ∪ (7.1) h(B) = O2

6 6 minimal cycle generators: ( )1∪ ( )1∪ ( )2∪ ( )2

(1.1) ∪ (1.2) ∪ (3) ∪ (6.1) ∪ (6.4) ∪ (7.1) B = {O, E, I, J, e1, e4, e1, e4}

Appendix _A

A.1. _{The symmetries of D4 and S2 of 17 minimal cycle-generators in Cc(2) are} listed in Table A.1.

Minimal cycle-generator ρ ρ2 ρ3 m mρ mρ2 mρ3 0 ↔ 1 (1) 1 × 1 {O} • • • • • • • (2) (2) 1 × 1 {E} • • • • • • • (1) (3) 1 × 2 {t, b} (4) • (4) • (4) • (4) • (4) 2 × 1 {l, r} (3) • (3) • (3) • (3) • (5) 2 × 2 {I, J} • • • • • • • • (6) 2 × 2 {e1, e2, e3, e4} • • • • • • • (7) (7) 2 × 2 {e1, e2, e3, e4} • • • • • • • (6) (8) 3 × 2 {r, e1, e3, e2, e4} (10) (9) (11) (9) (10) • (11) (9) (9) 3 × 2 {l, e2, e4, e1, e3} (11) (8) (10) (8) (11) • (10) (8) (10) 2 × 3 {t, e3, e4, e1, e2} (9) (11) (8) • (8) (11) (9) (11) (11) 2 × 3 {b, e1, e2, e3, e4} (8) (10) (9) • (9) (10) (8) (10) (12) 3 × 3 {e1, e4, J} (13) • (13) (13) • (13) • (15) (13) 3 × 3 {e2, e3, I} (12) • (12) (12) • (12) • (14) (14) 3 × 3 {e2, e3, J} (15) • (15) (15) • (15) • (13) (15) 3 × 3 {e1, e4, I} (14) • (14) (14) • (14) • (12) (16) 4 × 4 {e2, e3, e2, e3} (17) • (17) (17) • (17) • • (17) 4 × 4 {e1, e4, e1, e4} (16) • (16) (16) • (16) • • Table A.1 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.

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 colored 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 Math-ematicae, 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 two colors. To be published .