20 of 21 people found the following review helpful:
3.0 out of 5 stars
No Math, but pretty anyway, January 9, 2004
This review is from: The Zen of Magic Squares, Circles, and Stars: An Exhibition of Surprising Structures across Dimensions. (Hardcover)
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".
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14 of 15 people found the following review helpful:
5.0 out of 5 stars
Great book on pure mathematical fun, February 21, 2002
This review is from: The Zen of Magic Squares, Circles, and Stars: An Exhibition of Surprising Structures across Dimensions. (Hardcover)
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!
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8 of 9 people found the following review helpful:
5.0 out of 5 stars
What fun!, December 28, 2002
This review is from: The Zen of Magic Squares, Circles, and Stars: An Exhibition of Surprising Structures across Dimensions. (Hardcover)
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.
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