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书名:《幻方和魔方》 英文书名:Magic squares and cubes
丛书系列: 数学专业英语系列 图书编号:∑173
作者:[美]安德鲁斯著 出版社:哈尔滨工业大学出版社
ISBN:978-7-5603-3576-6 开本:787mm×960mm 1/16
版次:2012年5月第1版 2012年5月第1次印刷 印张:34  字数:375千字千字
定价:68.00元元 页数:535

 

【内容提要】

       本书列举诸多幻方和魔方的例子,研究幻方和魔方所具备的特性及构筑方法,生动地展示幻方和魔方的神奇之处。主要包括幻方的数学研究,六阶幻方,幻方类型,构造方法,幻圆等十五章内容。适合在校学生的学习研究,以及幻方和魔方爱好者作为兴趣读物。

 


  

PUBLISHERS'PREFACE

The essays which comprise this volume appeared first in The Monist at different times during the years 1905 to 1916, and under different circumstances, Some of the diagrams were photographed from the authors' drawings, others were set in type, and different authors have presented the results of their labors in different styles. In compiling all these in book form the original presentation has been largely preserved, and in this way uniformity has been sacrificed to some extent. Clarity of presentation was deemed the main thing, and so it happens that elegance of typographical appearance has been considered of secondary importance. Since mathematical readers will care mainly for the thoughts presented, we hope they will overlook the typographical shortcomings. The first edition contained only the first eight chapters, and these have now been carefully revised. The book has been doubled in volume through the interest aroused by the first edition in mathematical minds who have contributed their labors to the solution of problems along the same line.

In conclusion we wish to call attention to the title vignette which is an ancient Tibetan magic square borne on the back of the cosmic tortoise.

 


  

INTRODUCTION

The peculiar interest of magic squares and all lusus numerorum in general lies in the fact that they possess the charm of mystery. They appear to betray some hidden intelligence which by a preconceived plan produces the impression of intentional design, a phenomenon which finds its close analogue in nature.

Although magic squares have no immediate practical use, they have always exercised a great influence upon thinking people. It seems to me that they contain a lesson of great value in being a palpable instance of the symmetry of mathematics, throwing thereby a clear light upon the order that pervades the universe wherever we turn, in the infinitesimally small interrelations of atoms as well as in the immeasurable domain of the starry heavens, and order which, although of a different kind and still more intricate, is also traceable in the development of organized life, and even in the complex domain of human action.

Pythagoras says that number is the origin of all things, and certainly the law of number is the key that unlocks the secrets of the universe. But the law of number possesses an immanent order,which is at first sight mystifying,

but on a more intimate acquaintance we easily understand it to be intrinsically necessary; and this law of number explains the wondrous consistency of the laws of nature. Magic squares are conspicuous instances of the intrinsic harmony of number, and so they will serve as an interpreter of the cosmic order that dominates all existence. Though they are a mere intellectual play they not only illustrate the nature of mathematics, but also, incidentally, the nature of existence dominated by mathematical regularity.

In arithmetic we create a universe of figures by the process of counting; in geometry we create another universe by drawing lines in the abstract field of imagination, laying down definite directions; in algebra we produce magnitudes of a still more abstract nature, expressed by letters. In all these cases the first step producing the general conditions in which we move, lays down the rule to which all further steps are subject, and so every one of these universes is dominated by a consistency, producing a wonderful symmetry.

There is no science that teaches the harmonies of nature more clearly than mathematics, and the magic squares are like a mirror which reflects the symmetry of the divine norm immanent in all things, in the immeasurable immensity of the cosmos and in the construction of the atom not less than in the mysterious depths of the human mind.

 

PAUL CARUS

  



  


【目  录】

CHAPTER . MAGIC SQUARES  1

THE ESSENTIAL CHARACTERISTICS OF MAGIC SQUARES  1

ASSOCIATED OR REGULAR MAGIC SQUARES OF ODD NUMBERS  2

ASSOCIATED OR REGULAR MAGIC SQUARES OF EVEN NUMBERS  24

THE CONSTRUCTION OF EVEN MAGIC SQUARES BY DE LA HIRE'S METHOD  43

COMPOSITE MAGIC SQUARES  55

CONCENTRIC MAGIC SQUARES  58

GENERAL NOTES ON THE CONSTRUCTION OF MAGIC SQUARES  67

CHAPTER . MAGIC CUBES  80

THE ESSENTIAL CHARACTERISTICS OF MAGIC CUBES  80

ASSOCIATED OR REGULAR MAGIC CUBES OF ODD NUMBERS  81

ASSOCIATED OR REGULAR MAGIC CUBES OF EVEN NUMBERS  96

GENERAL NOTES ON MAGIC CUBES  104

CHAPTER . THE FRANKLIN SQUARES  110

AN ANALYSIS OF THE FRANKLIN SQUARES  119

CHAPTER . REFLECTIONS ON MAGIC SQUARES  139

THE ORDER OF FIGURES  139

MAGIC SQUARES IN SYMBOLS  148

THE MAGIC SQUARE IN CHINA  150

THE JAINA SQUARE  155

CHAPTER . A MATHEMATICAL STUDY OF MAGIC SQUARES  159

A NEW ANALYSIS  159

NOTES ON NUMBER SERIES USED IN THE CONSTRUCTION OF MAGIC SQUARES  170

CHAPTER . MAGICS AND PYTHAGOREAN NUMBERS  181

MR.BROWNES SQUARE AND LUSUS NUMERORUM  197

CHAPTER . SOME CURIOUS MAGIC SQUARES AND COMBINATIONS  204

CHAPTER . NOTES ON VARIOUS CONSTRUCTIVE PLANS BY WHICH MAGIC SQUARES MAY BE CLASSIFIED  222

THE MATHEMATICAL VALUE OF MAGIC SQUARES  233

CHAPTER . MAGIC CUBES OF THE SIXTH ORDER  235

A FRANKLIN CUBE OF SIX  236

A MAGIC CUBE OF SIX  245

MAGIC CUBE OF SIX  250

CHAPTER . VARIOUS KINDS OF MAGIC SQUARES  258

OVERLAPPING MAGIC SQUARES  258

ODDLY EVEN MAGIC SQUARES  271

NOTES ON ODDLYEVEN MAGIC SQUARES  283

NOTES ON PANDIAGONAL AND ASSOCIATED MAGIC SQUARES  288

SERRATED MAGIC SQUARES  305

LOZENGE MAGIC SQUARES  308

CHAPTER . SUNDRY CONSTRUCTIVE METHODS  313

A NEW METHOD FOR MAKING MAGIC SUQARES OF ODD ORDERS  313

THE CONSTRUCTION OF MAGIC SQUARES AND RECTANGLES BY THE METHOD OF COMPLEMENTARY DIFFERENCES  325

NOTES ON THE CONSTRUCTION OF MAGIC SQUARES OF ORDERS IN WHICH n IS OF THE GENERAL FORM 4 p+2  337

NOTES ON THE CONSTRUCTION OF MAGIC SQUARES OF ORDERS IN WHICH n IS OR THE GENERAL FORM 8p+2  351

GEOMETRIC MAGIC SQUARES AND CUBES  358

CHAPTER . THE THEORY OF REVERSIONS  373

CHAPTER MAGIC CIRCLES, SPHERES AND STARS  406

MAGIC CIRCLES  406

MAGIC SPHERES  417

MAGIC STARS  426

CHAPTER XIV. MAGIC OCTAHEDROIDS  440

MAGIC IN THE FOURTH DIMENSION  440

FOURFOLD MAGICS  455

CHAPTER XV.ORNATE MAGIC SQUARES  474

GENERAL RULE FOR CONSTRUCTING ORNATE MAGIC SQUARES OF ORDERS0 (mod 4)  474

ORNATE MAGIC SQUARES OF COMPOSITE ODD ORDERS  483

THE CONSTRUCTION OF ORNATE MAGIC SQUARES OF ORDERS 8, 12 AND 16 BY TABLES  492

THE CONSTRUCTION OF ORNATE MAGIC SQUARES OF ORDER 16 BY MAGIC RECTANGLES  508

PANDIAGONALCONCENTRIC MAGIC SQUARES OF ORDERS 4m  515

编辑手记  523

 


【编辑手记】

世界级数学游戏大师马丁・加德纳于20105月逝世。但世界人民对数学游戏的热情并没有随着大师的离世而消散。201232841在美国亚特兰大举行了第十届马丁・加德纳聚会。这项旨在向数学游戏泰斗马丁・加德纳致敬的聚会创立于1993年,地点选在马丁・加德纳晚年幽居的亚特兰大市,目前每两年举行一次,可以算得上是全世界数学游戏爱好者的盛会。据参加者介绍,为期5天的聚会异彩纷呈,几天中的耳濡目染,让参加者真切感受到了数学游戏在欧、美、日等国家和地区历史之悠久、发展水平之高。应该看到,在数学游戏领域,除了魔方、数独等具有竞技性的玩具或游戏之外,中国大陆至今尚未与国际接轨。不仅水平十分有限,而且非常缺乏与国外的交流,所以经常是闭门造车,低水平重复。另外一个影响因素是由于数学游戏中的某一类如幻方源于中国,所以国人往往或多或少地将推广数学游戏与弘扬传统文化之类的载道思维联系起来,而这往往会冲淡对数学游戏本身价值的追求。所以要想使趣味数学有趣起来,我们要做三件事:

一是,引进国外优秀趣味数学经典,奉行拿来主义。

二是,放弃文以载道的传统思维模式。让数学的归于数学,让历史的归于历史。

三是,淡化功利心态,将趣味当成目的而不是训练所谓思维的手段。逐渐使人们认识到除了革命人生观、财富人生观以外,趣味人生观也是一种正当选择。

这是一本旧书,兼有收藏和研究价值。

本雅明在谈论藏书的时候说:

Magic Squares and Cubes

“对一个真正的收藏家,获取一本旧书之时乃是此书的再生之日。”

幻方起源于我国,但年代久远,现在流传的仅是传说而已。最早可追溯到大禹治水,说是从洛水中浮出一只玄色的龟,背负一幅神秘之图即洛书。《拾遗记》中称:玄龟,河精之使者也。《周易》中记有:河出图,洛出书,圣人则之。《大戴礼・明堂》中有:明堂者,古有之也,凡九室……二九四七五三六一八。

1956111,时任中宣部部长陆定一在南京欢迎苏联驻华大使尤金博士学术演讲的报告会上提出新概念:“美国没文化”,此时正是中国人民志愿军入朝参战的初期,陆定一以如此口吻批判、否定美国文化是可以理解的。但是从幻方的研究和史料挖掘及整理上看,美国人还是很有文化的。可以说到目前为止中国还没有一部著作将幻方理论讲得如此完备,而且资料搜集得如此全面。这倒让我们感到没文化的心虚。

历史学的最基本的学科规范、学术要求是“无徵不立”。所谓“徵”,主要是历史文献,没有文献,便没有依据。

1956年,我国考古学家在西安原元代的安西王府旧址,发现了一块铸造在铁板上的幻方,即安西王府幻方。据史学家考证元世祖忽必烈入主中原为中统元年(1260年),亡于顺帝至正十八年。世祖至元二十八年有阿拉伯学者扎马鲁丁为安西王推算历法,所以推测此铁板幻方可能为阿拉伯人所作。但后来幻方的发展中阿拉伯世界日趋没落,美、欧、日于1920世界开始大放异彩。有人说,中国写意画养生,被尊为文人画鼻祖的吴道子活了近80岁;而西方油画伤神,拉斐尔、华托、莫迪格利阿尼、梵高、卡拉瓦乔等几位西方油画大师都只活了37岁。幻方虽在中国不绝如缕,但大多用力不勤。在杨辉的《续古摘奇算法》(1275年),程大位的《算法统宗》(卷十七1592年),方中道的《数度衍》(1661年),梅彀的《增删算法统宗》(十一卷1760),张潮的《心斋杂俎》,清代保其寿的《增补算法浑圆图》中均有研究。但后由于闭关锁国,所以与世界主流渐行渐远。

亚当斯密曾说:

“今日之广运万里地球中第一大国而受制于小夷……有待于夷者,独船坚炮利一事耳。”

后来在与世界重新交融之后越来越多的文献逐渐被国人所知。其中比较重要的三篇文章是C.A.BrowneHarry A. SaylesJohn worthington所写。均收录在本书中。可见本书在幻方研究中之重要。

其实在我国数学界对幻方一直有人在研究。如李俨、舒文中、陶照民、欧阳录等均有著述发表。

中国计算数学界有“三徐”。徐桂芳,徐献瑜,徐利治。其中的徐献瑜是西安交通大学数学系教授,专治计算数学和组合数学,对幻方也有深入研究,著有《纯幻方的构造》。

总之这是一个既有趣又有用的研究专题。值得爱好者一试身手!

 

刘培杰

2012412

于哈工大


   
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