Mathematical Vistas

1. Front Matter

   Pages i-xiv

   PDF

2. Paradoxes in Mathematics

   • Peter Hilton, Derek Holton, Jean Pedersen

   Pages 1-21

3. Not the Last of Fermat

   • Peter Hilton, Derek Holton, Jean Pedersen

   Pages 23-47

4. Fibonacci and Lucas Numbers: Their Connections and Divisibility Properties

   • Peter Hilton, Derek Holton, Jean Pedersen

   Pages 49-69

5. Paper-Folding, Polyhedra-Building, and Number Theory

   • Peter Hilton, Derek Holton, Jean Pedersen

   Pages 71-125

6. Are Four Colors Really Enough?

   • Peter Hilton, Derek Holton, Jean Pedersen

   Pages 127-158

7. From Binomial to Trinomial Coefficients and Beyond

   • Peter Hilton, Derek Holton, Jean Pedersen

   Pages 159-198

8. Catalan Numbers

   • Peter Hilton, Derek Holton, Jean Pedersen

   Pages 199-233

9. Symmetry

   • Peter Hilton, Derek Holton, Jean Pedersen

   Pages 235-269

10. Parties

    • Peter Hilton, Derek Holton, Jean Pedersen

    Pages 271-297

11. Back Matter

    Pages 299-337

    PDF

Focusing YourAttention We have called this book Mathematical Vistas because we have already published a companion book MathematicalRefiections in the same series;1 indeed, the two books are dedicated to the same principal purpose - to stimulate the interest ofbrightpeople in mathematics.Itis not our intention in writing this book to make the earlier book aprerequisite, but it is, of course, natural that this book should contain several references to its predecessor. This is especially - but not uniquely- true of Chapters 3, 4, and 6, which may be regarded as advanced versions of the corresponding chapters in Mathematical Reflections. Like its predecessor, the present work consists of nine chapters, each devoted to a lively mathematical topic, and each capable, in principle, of being read independently of the other chapters.' Thus this is not a text which- as is the intention of most standard treatments of mathematical topics - builds systematically on certain common themes as one proceeds 1Mathematical Reflections - In a Room with Many Mirrors, Springer Undergraduate Texts in Math­ ematics, 1996; Second Printing 1998. We will refer to this simply as MR. 2There was an exception in MR; Chapter 9 was concerned with our thoughts on the doing and teaching of mathematics at the undergraduate level.

• Mathematica
• Windows
• binomial
• design
• development
• form
• mathematics
• number theory
• reflection
• symmetry

From the reviews:

MAA ONLINE

"…much of the material is about the authors’ own research. This is a positive thing; its good for students at all levels to get information straight from the horse’s mouth, not only for accuracy but also for enthusiasm and authenticity. The authors put their writing where their talents are, and students get to see just how alive mathematics is…there is much to comment the book. It contains plenty of interesting mathematics, often going in unusual directions. I like the diagrams; the authors have chosen mathematics that involves especially pretty ones. And readers with some mathematical experience are sure to find things here and there that will delight them. In elementary school, before first-year algebra had made me conscious of being a mathematician, I called my favorite pastime ‘number tricks.’ The section ‘A Number Trick and its Explanations’ brought back fond memories."

 

P.Hilton, D.Holton, and J. Pedersen

Mathematical Vistas

From a Room with Many Windows

"The authors put their writing where their talents are, and students get to see just how alive mathematics is . . . there is much to comment the book. It contains plenty of interesting mathematics, often going in unusual directions. I like the diagrams; the authors have chosen mathematics that involves especially pretty ones. And readers with some mathematical experience are sure to find things here and there that will delight them."—THE MATHEMATICAL ASSOCIATION OF AMERICA

"Readers will find in … Vistas more than another couple of collections of popular tidbits. … a lively selection of mathematical topics delivered in a casual writing style, in which technical arguments are richly interspersed with comments on the guiding principles of mathematical investigation … . Finally, a word of praise is due for the many illustrations appearing throughout … these are carefullyrendered and appropriate, never gratuitous. … Enjoy!" (John Grant McLoughlin, Crux Mathematicorum Mathematical Mayhem, 2003)

"All chapters have in common that they speak about their subject in a light manner. But in each chapter there are so-called breaks consisting of problems designed to enable the readers to test their understanding of the material. Answers to some of the problems appear at the end of the book. The reading of this book can be recommended for high school and college students who are interested in mathematics, and it may stimulate them to read more about it." (Helmut Koch, Zentralblatt MATH, Vol. 1011, 2003)

"The purpose of this book … is to provide a relaxed and informal treatment of several mathematical topics that will convey to the reader … a sense of the pleasurable excitement that accompanies genuine mathematical discovery. … topics range from Andrew Wile’s famous proof of Fermat’s Last Theorem … to the discussion of paradoxes in which all relevant arguments are given in full. Each chapter is provided with ‘Breaks’, in which readers can test their understanding of the material just presented by answering questions." (Zentralblatt für Didaktik der Mathematik, August, 2003)

"Peter Hilton, Derek Holton and Jean Pedersen have produced a gem … . The authors have written a set of independent mathematical essays … for which there is rarely room in the undergraduate curriculum. … To be sure, there is a lot … that forces reader participation and that is, frankly, fun. … a joy to read, and I’m certain … a delight to teach from! … The writing is seamless, thought-provoking and entertaining. You and your students deserve to read … !" (Marvin Schaefer, MAA Online, July, 2002)

"The reader should not only approach this delightful book with pen and paper but also will find it worthwhile to consider attempting the geometric constructions." (F. J. Papp, Mathematical Reviews, Issue 2003e)

"It’s good for students … to get information straight from the horse’s mouth, not only for accuracy but also for enthusiasm and authenticity. The authors put their writing where their talents are, and students get to see just how alive mathematics is. … there is much to commend the book. It contains plenty of interesting mathematics, often going in unusual directions. … And readers with some mathematical experience are sure to find things here and there that will delight them." (Marion Cohen, MAA, May, 2003)

"The first purpose of this book is to stimulate the interest … in mathematics. … One of the features of the book is the presence in each chapter of a number of breaks consisting of problems that permit to the reader to test his understanding of the given material. This book is a good motivation for mathematics and has to be recommended to all students beginning studies in mathematics." (Yves Félix, Belgian Mathematical Society – Simon Stevin Bulletin, 2003)

"This trio of authors will be familiar to many … . Their new book is a sequel to the original … . Vistas contains three chapters which build on subject matter in the earlier book. … These two volumes are an impressive achievement. There are many books on the market … but few of them are prepared to go into as much detail as you will encounter here. … I thoroughly recommend both books … ." (Gerry Leversha, The Mathematical Gazette, Vol. 87 (509), 2003)

"The present book … consists of nine chapters devoted to different mathematical problems. … One of the special features of the text is that a number of ‘breaks’ are included in each chapter; these consist of problems so that the readers can test their understanding of the presented material. Selected answers … are given at the end of the book. … The book can be recommended for students and for all people who consider mathematics as a part of human culture." (European MathematicalSociety Newsletter, December, 2002)

"The goal of ‘Mathematical Vistas’ is to stimulate the interest … in mathematics. The book consists of nine related mathematical essays which will intrigue and inform the curious reader. … Some chapters can be understood completely with little background, others can be thought of as appetizers for further study. A number of breaks are included in each chapter. These are problems designed to test the reader’s understanding. This book is a sequel to the author’s popular book ‘Mathematical Reflections’ and can be read independently." (L’Enseignement Mathematique, Vol. 48 (1-2), 2002)

"The book aims to inspire readers with the pleasure that comes from genuine mathematical discovery. Much of the discussion is of the mathematics itself, rather than its applications. … I found the book very enjoyable to read, and I imagine that many other physicists will as well. … One of the strengths of the book is its frequent text breaks, which consist of questions that have been designed to test the readers’ understanding of the preceding information." (Sally Jordan, Physics World, September 2002)

"The book is written in an economical yet ‘chatty’ and relaxed style. The subject matter is developed from the concrete to the abstract with many clear examples and frequent diagrams and illustrations. The book will hold the interest of the bright interested person. … There are undergraduate mathematics students with sufficient self-motivation to read and profit from the book … . There is also the opportunity for the sympathetic instructor to include the book or chapters into required reading for formal courses … ." (W.P. Galvin, European Mathematical Society Newsletter, December, 2002)

"This book presents a broad spectrum of exciting developments in mathematics that will intrigue the curious reader with an interest in mathematics. The nine related essays cover topics, such as paradoxes inmathematics, Fermats last theorem, Fibonacci numbers, paper-folding, polyhedra building, and much more. This book is written in the same spirit as the authors highly popular Mathematical Reflections." (Amazon.de, May, 2002)

"Mathematical Vistas covers a single mathematical problem in each of its nine sections, including gems such as the Four-Colour Problem and Fermat’s Last Theorem. The authors … describe the ideas involved while encouraging the reader to investigate it through regular ‘breaks’. References provide pointers to more detailed information. Such an approach seems well suited to undergraduate teaching, giving students an overview of topics, then enabling them to investigate further the ones that interest them most. … mathematics students will enjoy this book." (Andrew Bowler, New Scientist, March, 2002)

• Mathematical Sciences Department, SUNY at Binghamton, Binghamton, USA

  Peter Hilton

• Department of Mathematics and Statistics, University of Otago, Dunedin, New Zealand

  Derek Holton

• Department of Mathematics and Computer Science, Santa Clara University, Santa Clara, USA

  Jean Pedersen

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