Five recent book reviews
This interesting book describes piano-hinged dissections, a new type of hinged dissection that generates many challenges. The book starts with mathematical descriptions, definitions and models of piano-hinged dissections (a piano-hinge connects two flat pieces that are side-by-side on the same level and forces one piece to flip on top of the other). In the book the author provides over 150 dissections and outlines methods for discovering them. He has also prepared an excellent CD with video, which provides step-by-step demonstrations for creating new dissections. The author also presents the lost manuscript on geometric dissections written by Ernest Irving Freese, an architect in Los Angeles, who at the end of his life became passionate about dissections. The book gives an overview of Freese's work in its five-part series spread throughout the text. This brilliant book can be recommended to students of geometry and teachers of mathematics, as well as students and all people who are interested in geometric dissections. Every creative reader will find new material for his own discoveries. The reader can easily experiment with the piano-hinge dissections because their mechanism can be simulated by folding a piece of paper without special mathematical knowledge, materials, computer programs, etc.
This book is an advanced, comprehensive text covering branches of the problems of models of Peano arithmetic that are not treated in Kayes’s book, “Models of Peano arithmetic”, a ‘bible’ of the study of models of arithmetic. It is assumed that the reader is familiar with Kaye’s book. Roughly speaking, the two basic themes of the book are the substructure lattices of elementary submodels of a given model of Peano arithmetic and recursive and arithmetic saturation. The first theme is mostly developed in the first four chapters (extensions in chapter 2, minimal and other types in chapter 3 and interstructure lattices in chapter 4). The second main topic can be found in chapters 5–11. Automorphism groups of recursively saturated models are studied in chapters 8 and 9, indiscernible generators are treated in chapter 5, the ω1-like models in chapter 10 and a classification of reducts in chapter 11. Chapter 6 deals with generics and forcing. ‘Twenty questions’ is the name of the last chapter. Exercises of various difficulties are included as an integral part of each chapter. The book is a welcome, comprehensive and useful publication.
This book, written in elegant French by a mathematician and historian of mathematics J.-P. Pier, is a special and very interesting contribution to the history of mathematics, the history of science and science itself. The author asks some provocative, creative and motivational questions and he tries to answer them from many points of view (such as mathematics, philosophy, history and sociology). He describes what mathematics means for mathematicians and others and he shows the position of mathematics in our scientific system. He also deals with the influence of mathematics on the historical development of philosophical ideas, daily life, technology and culture, etc. He discusses why we like or dislike mathematics, why we divide mathematics into two parts: pure and applied mathematics or classical and modern mathematics, and how we apply and use mathematics in technological progress, our daily life and work. Each of his ideas is described and illustrated with perfectly chosen subjects or examples from number theory, algebra, geometry and analysis. To underline his ideas, the author gives many interesting quotations and sentences from the works of principal mathematicians, physicists and philosophers. The book is written like a remarkable and easily readable essay and can be recommended to all people who are interested in mathematics, the history of mathematics, philosophy and the history of science.
There are various problems one can have with shoelaces (broken shoelaces, undone shoelaces, missing shoelaces, too long or too short shoelaces and so on). Mathematically minded people however might find interest in yet another set of problems, very different from those mentioned. These include questions like: ‘What is the shortest/longest/strongest/weakest way to lace the shoes?’ and ‘How many ways are there to lace the shoes?’. This beautiful and amusing book attacks all these problems from a mathematical point of view. It all started with an innocent article published by the author in 2002 in the journal Nature, which attracted an enormous amount of publicity and great interest in the mathematics of shoelaces from many people in all walks of life. The mathematics of shoelaces is a lovely combination of combinatorics and elementary analysis. It has nice and surprising connections to things like the travelling salesman problem or calculating the area of simply closed planar polygons. The book also has sections on the history of shoelacing, shoelace superstitions, style and fashion, and at the end it tackles the difficult philosophical question: what is the best way to lace the shoes? A very enjoyable book indeed.
In the six chapters of the book, selected problems at a Mathematical Olympiad level are solved. The first chapter, “Strategies in problem solving”, explains in detail the main strategic steps in solving a concrete problem. The following five chapters contain solutions to many problems from number theory, algebra, analysis, and Euclidean and analytic geometry. The sixth chapter, “Sundry examples”, is dedicated to interesting (and quite funny) problems that have something in common with combinatorics and game theory. At the end of some solved problems, there are additional exercises whose solutions can be obtained in a similar manner. It is an important feature of the book that problems are not only solved but that various solving strategies and methods are also demonstrated, which make problems easier and help to find the best route towards a solution. Without any exaggeration, I can say that the book gave me enormous pleasure. It not only portrays the author’s enthusiasm for the beauty of mathematics but also his ability to explain problems and their solutions to young people who are at the beginning of their mathematical career. And this is one of the reasons why the book will be useful for pupils and students who are interested in mathematics. It can also be recommended to mathematics teachers working with gifted students and will undoubtedly make their lectures more attractive.