# Effects of Initial Symmetry on the Global Symmetry of One-Dimensional Legal Cellular Automata

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### The Discrete Walsh Analysis

**Figure 1.**(

**a**) Examples of the two-dimensional discrete Walsh functions for $m=n=8$. Black represents +1, and white represents –1; and (

**b**) four types of symmetry in terms of the discrete Walsh function.

## 3. Calculations

^{5}, we can obtain the 32 possible legal totalistic rules. Wolfram [7,8] discovered that, although the patterns obtained with different rules differed in detail, they appeared to fall into four qualitative classes:

**Figure 2.**Examples of final patterns in each class. (

**a**) is an example of Class I, (

**b**) of Class II, (

**c**) Class III, and (

**d**) is an example of Class IV.

^{6}× 1) and the final pattern. The number of cells is 2

^{6}× 2

^{6}because, to measure the number of symmetries, this number of cells is the minimum needed [4]. In the present study, we can use only two kinds of initial condition, horizontal symmetry and double symmetry, because we consider n = 2 in the discrete Walsh function W

_{mn}(as in Figure 3).

**Figure 3.**An example of initial symmetry. (

**a**) Double Symmetry is one example of $D{S}_{0}$, and (

**b**) Horizontal Symmetry is one example of $H{S}_{0}$. In this study, the initial condition was limited to n = 2 because 2

^{1}= 2 = n. Thus, (

**a**) shows the case of m = even, and (

**b**) shows the case of m = odd.

## 4. Results

#### 4.1. Class

#### 4.2. Four Types of Symmetry

**Figure 4.**Pie charts of the three types of initial condition ($H{S}_{0}$, $D{S}_{0}$ and $R{a}_{0}$). For $H{S}_{0}$ and $R{a}_{0}$, all classes have all four types of symmetry: vertical symmetry, horizontal symmetry, centro symmetry, and double symmetry. In contrast, $D{S}_{0}$ with all classes had two types of symmetry: vertical symmetry and double symmetry. Regarding vertical symmetry and double symmetry, the relational expression was $H{S}_{0}$ < $R{a}_{0}$ < $D{S}_{0}$, and regarding horizontal symmetry and centro symmetry, it was $D{S}_{0}$ < $R{a}_{0}$ < $H{S}_{0}$.

**Figure 5.**Examples of final patterns with regular initial conditions. (

**a**,

**b**) belong to $D{S}_{0}$, and (

**c**,

**d**) belong to $H{S}_{0}$.

#### 4.3. The Ratio of Final Formation

Class | Horizontal Symmetry | Double Symmetry | Random |
---|---|---|---|

Periodic | 12 | 12 | 100× |

Chaotic | 43 | 44 | 100× |

Complex | 6 | 6 | 100× |

## 5. Discussion

#### 5.1. Class

#### 5.2. Four Types of Symmetry

#### 5.3. The Ratio of Final Formation

## 6. Conclusions

## Acknowledgements

## Conflicts of Interest

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**MDPI and ACS Style**

Tanaka, I.
Effects of Initial Symmetry on the Global Symmetry of One-Dimensional Legal Cellular Automata. *Symmetry* **2015**, *7*, 1768-1779.
https://doi.org/10.3390/sym7041768

**AMA Style**

Tanaka I.
Effects of Initial Symmetry on the Global Symmetry of One-Dimensional Legal Cellular Automata. *Symmetry*. 2015; 7(4):1768-1779.
https://doi.org/10.3390/sym7041768

**Chicago/Turabian Style**

Tanaka, Ikuko.
2015. "Effects of Initial Symmetry on the Global Symmetry of One-Dimensional Legal Cellular Automata" *Symmetry* 7, no. 4: 1768-1779.
https://doi.org/10.3390/sym7041768