Customer Reviews

The Zen of Magic Squares, Circles, and Stars: An Exhibition of Surprising Structures across Dimensions

5 star:		(8)
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3 star:		(2)
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This is a book about mathematical artifacts, but it has practically no mathematical content of its own. A casual reader who wants to gaze at these beautiful objects and come away impressed but with little understanding will find this a marvellous book. However, a mathematically inclined reader is not satisfied with someone declaring that an object has such-and-such a property, he wants to know WHY.

Chapter 1 of this book gives dozens of fascinating constructions, but for most of them not a shred of proof is offered that the arrays produced are the magic squares Pickover claims. It leaves me wondering whether or not Pickover could produce such proofs himself, even for the more simple constructions in the book.

Pickover describes some interesting computer experiments at the end of the chapter but seems completely stymied as to why they work. The demonstration is a lovely, but simple, piece of matrix theory that I would expect my first or second year Linear Algebra students to be able to perform.
He shows two "brute-force" proofs for the order 3 case, one by Hendricks and "another" by Johnson (at least here is an attempt at including a proof), but annoyingly seems unaware that the second is just a minor variation on the first. I wonder if Pickover actually tried to follow these proofs himself or if he just copied them for his book.

Mathematics is not a collection of statements that the hearer must accept on "authority", it is a systematic development of theory in which every statement can be, at least in principle, demonstrated by a logical argument. The mathematics is in understanding "why", not in the acceptance of fact. Without demonstration of the claims, all that is left is the shell with no life. Beautiful, like other shells we find along the shore, but not the genuine article itself.

I am reminded somewhat of Stephen Hawking's popularizations of physics in which the reader is deeply impressed with the beauty of the subject, but comes away knowing practically no actual physics to speak of, for the author carefully seals the machinery of physics from his reader and presents only the glamorous face. In the case of Hawking, however, the author's authority is unquestionable; I'm sure he could, if pressed, demonstrate every claim in his books from first principles. I suspect that Pickover could not.

Aside from a few excusable errors of fact, the book shares a serious omission with almost every book on magic squares that I have seen, in that it does not present what is surely the most elementary construction known for magic squares of any odd order, as the sum of a circulant and a back-circulant matrix. Even Pickover would be able to prove that this construction works, since the reason it works is extremely obvious. Given the connection of this construction to the very important subject of orthogonal Latin Squares, you would think a serious writer would devote some space to it.

Aside from all of the above, the material in the book is comprehensive and fascinating, drawing on a number of sources, displaying many artifacts that have titillated dabblers for millennia. As a museum piece I'd have to give the book an "A", but as a piece of mathematics, only a "D".

While I am writing this in late February, it is still a safe bet to conjecture that this is the best recreational mathematics book that will be published this year. Magic squares are a fascinating area of mathematics, and Pickover covers a great deal of ground in bringing the field up to date. A magic square is a square grid of numbers where the row and column sums are the same. They appear throughout history and the most famous person to create them was the immensely talented Benjamin Franklin.
Magic squares can be created using many different formulas, including the moves of a knight on a board, using operations other than addition, and the embedding of magic squares inside magic squares. If you have not followed the development of the field, you will be amazed at how many different ways they can be constructed.
Magic squares have also been extended to include magic cubes of three and four dimensions. The star of the book is John Hendrick, an incredible person who seems blessed with some form of magic as he creates ever more complicated magic structures. Hendrick uses only a programmable calculator in his searches for larger and more complex magic figures, which makes his work all the more remarkable. Additional magic structures are the star and circle, where the points of intersection are marked with numbers and the sums of the points along lines are equal.
Pickover writes with his usual style and straightforward simplicity in this book. The material is presented well and can be understood by anyone with a basic middle school mathematics background. This is a cool book!

OK, there were a couple of typos -- keeps you on your toes. Lots and lots of examples of different variations on the magic square theme -- and puzzles for the reader to solve. Some of those puzzles are quite easy and some are quite difficult and have yet to be solved by anyone. You can't be a mathphobe to read this book, but you don't need to be a math whiz either. Anybody who likes the challenge of a good crossword or crossnumber puzzle should like this.

A magic square is an array of numbers in which the sums of numbers in rows, columns, and diagonals are equal. A magic square uses consecutive numbers from 1 to N. Here's an example,


4 9 2
3 5 7
8 1 6


This book is different from all others I've seen on the subject, and I don't know any other books that present the large range of patterns that you'll find here. The book also focuses on discoveries in the last few years. As Pickover says, the book is essentially an exhibit of magnificent forms discovered through the centuries. All sorts of historical and quirky-human aspects are also described. Centuries ago, people believed that magic squares to had special, magical powers....

This book is awesome! It seems as if Cliff Pickover has journeyed around the world to find unusual people and their fascinating magic squares, circles, stars, and other mathematical wonders. Topics include: Benjamin Franklin's "most magical" magic square, John Hendricks' four-dimensional magic tesseracts and other gems from prisoners, scientists, little-known artists, and computer programmers. Just last year, Pickover came across a wonderful collection of magic figures designed by the late, great Fubine. Fubine, whose real name was Cipriano Ferraris, died in 1958. Fubine's designs ranged from simple squares through a wide variety of linear geometric shapes and three-dimensional figures. Rows, columns, spokes, and diameters consisted of lines of numbers, no single one of which was repeated and whose totals were always the same. Pickover say that in 1929, Fubine lost all his money in the great stock market crash. He found himself in near suicidal state and distracted himself by creating ever-larger magic squares.

What a smorgasbord for children, laypeople, and even seasoned mathematicians! In this book, you'll find information on magic square creation, classification, and history, and graphical representations that can be quite beautiful. The book contains math and art. Although, the literature on magic squares is vast, this book contains some magnificent structures discovered in the last few years. I don't think there is any other book that presents such a huge range of patterns.

If you love numbers, magic squares, geometry and mental calisthenics, read this book. It is the most complete source of information available on this topic and the author is exceedingly thorough and precise in his treatment of it. I was thrilled to discover new gems that I never knew existed before. As a result, a few extra neurons connected in my brain! That alone was worth the price of admission.

The book starts on a classical way, describing old french methods of magic construction : de la Loubère, Bachet de Méziriac, and de la Hire. But also other methods like Lee Fults, Strachey, diagonal, knight's move, lozenge,...

In my point of view, the very very interesting parts are the 230 pages of chapters 2 and 3, the heart of the book.
Cliff explains and classifies the different and numerous magic objects according to their properties.
And you will find the state of the art about magic squares, magic cubes but also about magic objects using more than our current 3-dimensional space !
The excellent and recent work of John Hendricks, one of the world's main specialist on magic square, is oftenly presented and described.

The end of the book describes very strange magic objects that will really astonish you : circles, spheres, stars, hexagons, flowers ( !), spider, ...

An excellent book, THE NEW REFERENCE on the subject !
With a lot of figures, notes and references.

My only regret is self-centred : my new record for multimagic squares were probably too recent to be in the book.
Cliff's book, page 136 : « A magic square is p-multimagic if the square formed by replacing each element by its kth power (for k=1, 2, ..., p) is also magic. As we discussed, a 2-multimagic square is called a bimagic square, and a 3-multimagic square is called a trimagic square. I do not know if a quadramagic or pentamagic square exists and welcome feedback from readers ».
So, feedback : both squares have been discovered in 2001. Our pentamagic square, 1024x1024 sized, is magic as far as the 5th power.

Cliff, I hope that the new record for multimagic squares, by André Viricel and Christian Boyer, will be in the next edition !
And you will probably have a next edition, I am sure that your book will be really successful.

The enthusiasm of the author for the subject comes through very clearly. He covers the history of magic squares and along with his obvious mathematical genius, he covers with great reverence the mystical and religious aspects of the subject. There is even a chapter that gives a number of methods for one to learn to make "instant" magic squares any time or place. The book is chock full of about every magic square, star, cube (or any other geometric construction) one could imagine. There are even tessaracts. Just looking at these magical constructs and making an effort to understand something about them is worth the price of admission. The book is so comprehensive on the subject that one could spend a very long time studying it-or one could benefit greatly by a good serious reading of it.

William Swyter

Magic squares have fascinated us for many centuries. Even in ancient Babylonian times, people considered these squares to have magical powers. Albrecht Dürer, the painter and printmaker, used them in his artworks.

Most of the ideas in this book can be explored with just a pencil and paper! You can even discover new patterns in old magic squares that no one has ever found before. Even the famous eighteenth-century American Benjamin Franklin loved magic squares although he once considered them a waste of time.

Pickover presents interesting people and their magic squares. From Benjamin Franklin's magic squares to four-dimensional magic tesseracts, the patterns fascinate us with their elegance. The book is a treasure and has gotten some rave reviews in the press. I enjoyed the magic spheres best of all, but I think each reader will find something new and interesting as they browse. A lot of magic squares deal with the chess board. Some focus on DNA sequences! A few were made by prisioners in jail. The author has certainly searched far and wide to assemble this massive collection.

This book contains print and mathematical errors. A cute book but because of the math misprints [I refuse to believe the author cannot add] a shoddy publication very uncharacteristic of Princeton