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(Preface for *Rubik’s Cube and Its Application in Science*)

The DaMaoHuDong（大毛忽洞）is the name of my hometown village, a place where there are two salient characters: poor and blue. The poor sense has been pertained to the people and the view of the sky has been remaining the blue.

The scenery of my hometown village in summer can compete with anyplace in the world. The boundless stretch of field is artlessly colored by six colors of emblements: the light green of wheat, the blackish green of naked oats, the silvery white of buckwheat, the golden yellow of rapeseed, the off-white of potato and the red of green manured land. When you look around the field, you will feel as if you stand in a natural color phalanx. If you watch at the same location next year, the scenery of the field still looks like color squares, but the color of the location changes, because the land should be cultivated by turns.

The alternate cultivation of the plowland causes the permutation of the field’s colors, and the periods of the permutation is a mathematics problem.

It is very important that I found such question of mathematics in my hometown village during my childhood’s hard farming work, which introduces me into mathematics’ kingdom.

I had been interested in mathematics since childhood, and this avocation is never intermitted. No matter what in the season of reaping wheat by hand and in the holiday of Chinese New Year, I always keep my mind on some topic of mathematics. My hobby about mathematics helped me pass National College Entrance Examinations in 1977 after the ten- year stoppage of the Great Cultural Revolution.

My hometown village is very poor, but its rural scenery seems like that in poetry and picture. Even the name of my hometown village also has poetic meaning, similar with that of the Monkey King’s hometown.

Early in 1970s, the students in grade three and grade five shared a single classroom due to the lack of resources. When our grade five had mathematics class, the grade three read aloud Chairman Mao’s poetry, “……, Yang’s flower and Liu’s catkin gently raise and directly fly to the highest of heavens”.

It was by unconscious mix of mathematics and poetry that the mathematics that I studied at my hometown village contained poetic rhyme and picture’s beauty.

Right from 1973 to 1976, I was a secondary school student and many great events occurred in China during these three years, but I was only interested in the publication of *Marx’s Mathematics Manuscript* in Beijing in 1975. My hometown village and my school were far away from the city where bookstores sold *Marx’s Mathematics Manuscript* and my economic status did not allow me to travel to buy *Marx’s Mathematics Manuscript*. Afterwards I learned that there existed a Wangfujing bookstore where they accepted mail orders in Beijing, then I immediately remitted five Yuan. About one month later, a brand-new *Marx’s Mathematics* *Manuscript* was mailed to me and the balance was also returned in the form of stamps together with the book. Subsequently I again enquired the books concerning mathematics by the letter and only *Operational Research* and *Optimal Seeking Method* by Hua Luogeng occurred in the mailed listing besides *Marx’s Mathematics* *Manuscript*, which revealed the lack of mathematical books in Wangfujing book store.

It was well known that *Marx’s Mathematics Manuscript* was not a textbook or an academic monograph. At that time I only recognized that Marx is a great scholar and I could learn something from his book. Afterwards when I went to Jilin University I also brought this book with me and keep it in good condition till today.

After graduation from high school, I returned to my the DaMaoHuDong, which just agreed with the words of advice given at parting by me to my classmates that “ The water of Yellow River comes from the sky and again returns to the aerosphere after the evaporation”. Really in 1976 what other ideas can you have?

Even in those years we could not have ideas I still had dense interest in mathematics and warily collected the mathematics games popular in the folk such as Tang Wang Luan Dian Bing and Han Xin Li Ma Fen You and established the corresponding mathematical models. In 1976 September for my talent in mathematics I was appointed the mathematics teacher at a village middle school. I got the notice just one day before starting work, in other words, the news was beyond my imagination. In order to become a village teacher in 1976 required equivalent family background while I had no family background. I liked Marx’s logion that “dialectics adores nothing”, while I was indistinctly aware that “mathematics also worships nothing.

In 1977 the news of resuming of the National College Entrance Examinations was just like an intense 10-grade earthquake that the after-effect even sent to my hometown village far away from the hypocenter. After simple review I went to the examination room. Our examination room was so strict that two posse men with the guns stood guarding on each gate in spite of the snow outsides and no cheat phenomena occur. Our way of taking part in the National College Entrance Examinations was so special that we brought our own bedrolls and food with ourselves and lived in the villagers’ home enjoying the boiled water and hypocaust provided by the villagers. Altogether, in that winter, I ate my own solid food, drank the villager’s boiled water, slept on the villager’s hypocaust and finally completed my examination.

On March 5 1978, I received the matriculation advice note from Jilin University by registered post and there was only 7 days left from the beginning of the term. My will was mathematics while the matriculation major was metal physics, which might be due to that my hometown, was close to a steel company located at Baotou. The most profound impression of university life to me was Kaoliang and the memory of Kaoliang was a physical memory to me just like the ability to ride bicycle. Kaoliang of northeast in those days was both red and excessive so that I ate it for four years for twice a day. Four years later, faced with graduation assignment, I selected two places according to symmetry theory, namely,

The luggage labels in those years were just written in the above format, which had some resemblance with the patterns of the mini cube. 20 years have passed since then.

It was in March 1982, namely, not long after my graduation from university that I first contacted Rubik’s cube. My colleague forwardly lend his Rubik’s cube to me and I restored two layers in my own way in two days and gained the high evaluation of my colleague for my talent. Afterwards I read an articles concerning Rubik’s cube in *Scientific America* in the library and wrote down the basic information. At that time I felt calm towards Rubik’s cube, so put no special devotion to it.

In 1987 winter, I carried out my master dissertation in the laboratory of the Materials Science Department of Jilin University. My experimental work, the creep-alike experiment, centered on the superplastic tension of aluminum alloy with the addition of rare earth metals and it generally took several hours to complete one specimen. While keeping my eyes on the experiment, I thought the Eurler angle. At that time ODF (orientation distribution function in crystal) was a hot issue and I decided to adopt the ODF to explain my experimental results of superplasticity. I designed 50 kinds of Eurler transformation, which connect two three-dimensional spaces by the Eurler rotation.

It should be mentioned that, Group Theory, master degree course of solid-state physics major of Jilin University had great influences on me. At that time, more than 100 graduate students took the course and the teacher is Professor Ding Peizhu who taught excellently by self-compiled textbook. During the study session held by Professor Ding, my confusion of the rotation matrix resulted in the different result from the answer in the textbook despite the flawless deduction and it took Professor Ding nearly one hour to check the problem and find the faults, which gave me profound impression. The final examination gave me more profound impression. The whole test questions was divided into two parts: 1/3 open book and 2/3 close book. There is only one question in the open book examination. The problem was not difficult but involves the multiplication table of O group. More than 100 graduates took part in the examination and all found no ready multiplication table of O group, so we had to construct it by ourselves. Actually the construction of O group multiplication table was not difficult, but improper symbol system would lead to large quantities of work because any two-two multiplications of 24 elements make 276 combinations. In fact the difficulty did not lie in the operation number of the matrix multiplication but the representation of 24 spatial orientations. Then I made use of the crystallography symbols (namely, the Miller indexes) to represent the corresponding rotation axis of 24 elements and gave the complete multiplication table of O group. Among 276 results only three were involved in the test question, so most students didn’t like such kind of low-efficient work, while I felt that constructing the multiplication table of O group is of more significance than the examination itself. During that study session I had mastered the nature of the rotation matrix and again looked over all the matrices of the point group.

In 1990 I returned to the problem of ODF and realized the theory subtly connected the realistic two-dimensional space with the abstract three-dimensional Eurler space where the transformation matrix utilized from the angular monument theory of quantum mechanics. The result of thinking was published in the 4th issue of *Physical Testing and Chemical Analysis- Physical Testing *in 1990 in the form of the paper named *The Physical Idea of ODF*. Before that I completed the BASICA program design of ODF and made comprehensive self-training from the model to the algorithm, from the algorithm to the programming language and from the programming language to the debugging. All these work laid solid foundation for my cube theory and the cube computer programs.

In 1991 winter, I began to think the mathematical model of Rubik’s cube. I collected all the concerning information such as the class notes of Group Theory, the multiplication table of O group and the Eurler rotation of the coordinate system, thus proposing a project for myself. The results showed that, the method of describing the cube’s rotation by the rotation matrix of group theory holds true for some cubies, but not for other cubies. I had strong conviction in the rotation matrix, so I believed the problem lies in how to dispose the coordinates of the cubes unsuitable for the rotation matrix, therefore, I introduced the techniques of the mirror image treatment and right-handed treatment. During the time less than two weeks, I established the mathematical model of Rubik’s cube and designed BASICA program for the cube’s rotation and debugged the programs on the computer.

In the publication of *Rubik’s Cube and its Application* in 1992, my cube computer program had been available at hand, but I had not written the mathematical model and the computer program into the book because I believed that the opportunity was far from mature. I also did not care other people designed the same model in a short time and the waiting is 10 years.

In 2000, the National Natural Science Foundation of China established the special branch for spreading the science. In 2001, I submitted the proposal of *The Scientific Metaphor of Rubik’s Cube and its Computer Presentation*. In the end of the year, my proposal about Rubik’s cube was approved by NSFC.

I had been interested in Rubik’s cube for 20 years, but the *Rubik’s Cube and its Application in Science* only took me one year. During the writing of this book, I scanned almost all the website and web pages concerning Rubik’s cube, which gives me more confidence in my own model and program. I felt my Miller Symbol System had pronounced advantages over others’ Cube Operation Systems. Especially for the high-order cube, my system seems very concise.

By Rubik’s cube I understood how to do scientific research and experienced the meaning of carry out the originally creative work. Doing research resembles climbing the mountain. Climbing the mountain following others is laborsaving and safer, but the landscape in the front is always sheltered by them. If you have your own theory and gradually perfect it each day, your work is originally creative. If you utilize or remark others’ theory, first, you shall make clear others’ theory and then use others’ theory to solve your own problem but it is difficult to understand the working background of others when they established the theory.

The problem lies in that how to have your own theory or model, in other words, how to establish your own theory or model. I believe the unique path is accumulation and years of accumulation. If you can insist on thinking one problem for ten years or twenty years, the obtained results will be surely larruping. It shall be emphasized that, the problem to be selected should have considerable difficulty, otherwise cannot sustain the thinking of ten years or twenty years.

The above words regarding the historical background of this book are dedicated to the reader in the name of my hometown village.

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