Read This!

The MAA Online book review column

Briefly Noted

May 2003

Winston Churchill is said to have remarked that it is good for uneducated persons to read a book of quotations, meaning presumably that such a book can be an education in itself. One might add that the better the book of quotations, the more its value as an education even for those who are only slightly uneducated — and which of us can claim to be completely educated? From that perspective, this is a book which will be of value to everyone, because its breadth and scope would allow even the most widely read persons to gain something from it. The many quotations come from many different sources (and in most cases precise references are given).

Here are some quotes I enjoyed:

True science is restrictively the study of useless things. For the useful things will get studied without the aid of scientists. To employ rare minds on such work is like running a steam engine by burning diamonds. (C.S. Peirce)

Logic is the art of going wrong with confidence. (Morris Kline)

That student is best taught who is told the least. (R.L. Moore)

Greg N. Frederickson, a Professor of Computer Science at Purdue University, has written two books on dissections, both published by Cambridge University Press: one in 1997, Dissections: Plane & Fancy, and, in 2002, Hinged Dissections: Swinging & Twisting. Both of them are intended, as the author writes in the preface of his second book, for those who "enjoy extraordinarily beautiful objects and relish the challenge of something new."

The book does not require more knowledge than high school geometry and is not a succession of theorems, followed by proofs, examples, applications and examples. Instead, the author includes detailed explanations of very ingenious new techniques, as well as puzzles (with solutions!) The book is a lot of fun to read, and the lively text includes hinged dissections for polygons (triangles, squares, stars, crosses, etc), as well as curved and three-dimensional figures. Since hinged dissections refer to the cutting of a geometric figure into hinged pieces that can be rearranged to form another figure, it is obvious that the book needed to include many illustrations. Of these, the author includes a wealth, and all of them are very good! Of course, it is even more fun to see some animations, and some of these can be found at http://www.cs.purdue.edu/homes/gnf/book2/hingdanim.html.

The above is just part of a much larger collection of new developments posted by the author at his web-page about the book. It is in this web-page that we find out about a paper by a McGill computer scientist and a few Harvard chemists describing animated dissections with chemical processes. The paper has been published in the Journal of the American Chemical Society, vol. 124, pp. 14508-14509.

This means that hinged dissections are not only fun and entertaining, but also useful outside mathematics — one more reason for everybody interested in finding entertaining and challenging mathematical puzzles, from high school students to mathematicians, to read Frederickson's book. [Mihaela Poplicher]

No more solid evidence exists concerning the incredible complexity of the integers and the remarkable ingenuity of mathematicians than the continued finding of new results in the series of Fibonacci numbers. From the simple definition of F(0) = 0, F(1) = 1 and F(n) = F(n-1) + F(n-2) for n > 1, books have been written on the formulas that they satisfy and the many places in nature where they appear. The authors of New Visual Perspectives on Fibonacci Numbers define several modifications of the basic sequence and then pursue the consequences. As the title implies, they often rely on diagrams to demonstrate their results.

In the first sections, integer sequences called 2-Fibonacci sequences are defined. Several different forms are used, but the basic definition is to use two sequences and interlace the terms. For example, starting with the initial terms a(0) = c, b(0) = d, a(1) = e and b(1) = f, one definition of the sequences is

a(n+2) = b(n+1) + b(n) n ³0
b(n+2) = a(n+1) + a(n) n ³ 0

There are many different consequences of these definitions, and they prove many theorems, although obviously there are an enormous number of possibilities. While the sequences are interesting, they are not all that new, as I have seen similar sequences over the past several years in material originating from Florentin Smarandache.

In section 2, weighted binary trees where some form of Fibonacci process is used to weight and place the nodes are described. Connections are made between the trees and Gray codes as well as other types of recurrence trees. Once again interesting, but not extensively so.

The real excitement in the book starts on the first page of part B and continues to the end of the book. Three sets of three-dimensional vectors are defined at the beginning of part B:

The nth Fibonacci vector FV(n) = (F(n-1), F(n), F(n+1))

The nth Lucas vector LV(n) = (L(n-1), L(n), L(n+1))

The nth generalized Fibonacci vector GV(n) = (G(n-1), G(n), G(n+1))

Geometric consequences of these definitions lead to the Fibonacci Honeyomb Plane, where a plane is partitioned into hexagons defined by integer nearest neighbors, trigonometry in the Fibonacci plane and many other vector sequences that can be generated. This was mathematics that was new to me, and the authors presented it very well. Their results were easy to follow, and like all good mathematical expositions, future directions were clear, even though they were not necessarily explicitly stated.

A goldpoint is defined as one that forms a golden section in a segment. The sets of goldpoints defined by a set of line segments were then used to construct goldpoint geometries. Forming new figures from the goldpoints of a geometric figure opens up a whole new area in the construction of new figures, including fractals. It is also possible to tile the plane using figures with goldpoints, and the authors demonstrate many such tessalations. As I read this section, I thought of many additional figures that could be drawn by performing slight modifications of their basic definitions. Although it is not likely, I hope that someday I can find the time to program the generation of the figures.

In this book, the authors explored many applications of the Fibonacci-type sequences that are new and point the way to many additional lines of study. Their ideas were original and well-described and I recommend this book to anyone interested in iterative processes on integers. [Charles Ashbacher]

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