Geometric Etudes in Combinatorial Mathematics

ISBN-13: 9780387754697 / Angielski / Miękka / 2010 / 264 str.

Alexander Soifer

Opis

A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas. A painter makes patterns with shapes and colours, a poet with words. A painter may embody an 'idea, ' but the idea is usually commonplace and unimportant. In poetry, ideas count for a great deal more; but as Housman insisted, the importance of ideas in poetry is habitually exaggerated... A mathematician, on the other hand, has no material to work with but ideas, and so his patterns are likely to last longer, since ideas wear less with time than words. The mathematician's patterns, like the painter's or the poet's, must be beautiful; the ideas, like the colors or the words, must 't together in a harmonious way. Beauty is the ?rst test: there is no permanent place in the world for ugly mathematics. --G.H.Hardy, A Mathematician's Apology, 1940 Har, pp. 24-25] I grew up on books by Isaac M. Yaglom and Vladimir Bolty- ski. I read their books as a middle and high school student in Moscow. During my college years, I got to know Isaak Moiseevich Yaglom personally and treasured his passion for and expertise in geometry and ?ne art. In the midst of my xxv xxvi Preface college years, a group of Moscow mathematicians, including Isaak Yaglom, signed a letter protesting the psychiatric - prisonment of the famous dissident Alexander Esenin-Volpin.

Recenzje

From the book reviews:

"This book itself has also a good chance to occupy a permanent place in the mathematical literature. Among its virtues is the lively and fluent style, in which it introduces and explains the problems. ... In summing up, we warmly recommend this book to any interested reader: take and read, and dip into the exercises and the problems ... ." (Gábor Gévay, Acta Scientiarum Mathematicarum (Szeged), Vol. 77 (3-4), 2011)

Characteristically, each of the topics included in the book require very little in the way of preparation and evolve fast into open questions and research level conjectures...This is a delightful book that will be welcomed enthusiastically by students and organizers of mathematical circles and mathematics fans.---Alexander Bogomolny

Boltyanski and Soifer have titled their monograph aptly, inviting talented students to develop their technique and understanding by grappling with a challenging array of elegant combinatorial problems having a distinct geometric tone. The etudes presented here are not simply thoese of Czerny, but are better compared to the etudes of Chopin, not only technically demanding and addressed to a variety of specific skills, but at the same time possessing an exceptional beauty that characterizes the best of art...Keep this book at hand as you plan your next problem solving seminar. ---The American Mathematical Monthly

Spis treści

-Forewords to the Second Edition.-Forewords to the First Edition.-Preface to the Second Edition.-Preface to the First Edition.-Part I. Original Etudes.-1. Tiling a Checker Rectangle.- 1.1 Introduction.-1.2 Tiling Rectangles by Trominoes.- 1.3 Tetrominoes and 'Color' Reasoning.-1.4Tiling by Linear Polyominoes.- 1.5 Polyominoes and Rotational.-1.6 Symmetries.-1.7 Tiling on Other Surfaces.- 2. Proofs of Existence.- 2.1 The Pigeonhole Principle in Geometry.- 2.2 An Infinite Flock of Pigeons.-3. A Word About Graphs.-3.1 Combinatorics of Acquaintance, or Introduction to Graph Theory. -3.2 More About Graphs.-3.3 Planarity.-3.4 The Intersection Index and the Jordan Curve Theorem.-4. Ideas of Combinatorial Geometry.-4.1 What are Convex Figures?.-4.2 Decomposition of Figures Into Parts of Smaller Diameters.-4.3 Figures of Constant Width.-4.4 Solution of the Borsuk Problem for Figures in the Plane.-4.5 Illumination of Convex Figures.-4.6 Theorems of Helly and Szökefalvi-Nagy.-Part II. New Landscape or the View 18 Years Later.-5. Mitya Karabash and a Tiling Conjecture.-6. Norton Starr’s 3-Dimensional Tromino Tiling.-7. Large Progress in Small Ramsey Numbers.-8. The Borsuk Problem Conquered.-9. Etude on the Chromatic Number of the Plane.-10. Farewell to the Reader.-References.-Notation.-Index.

Blurb

The etudes presented here are not simply those of Czerny, but are better compared to the etudes of Chopin, not only technically demanding and addressed to a variety of specific skills, but at the same time possessing an exceptional beauty that characterizes the best of art... Keep this book at hand as you plan your next problem solving seminar.

—Don Chakerian

THE AMERICAN MATHEMATICAL MONTHLY

 

Alexander Soifer’s Geometrical Etudes in Combinatorial Mathematics is concerned with beautiful mathematics, and it will likely occupy a special and permanent place in the mathematical literature, challenging and inspiring both novice and expert readers with surprising and exquisite problems and theorems… He conveys the joy of discovery as well as anyone, and he has chosen a topic that will stand the test of time.

—Cecil Rousseau

MEMPHIS STATE UNIVERSITY

Each time I looked at Geometrical Etudes in Combinatorial Mathematics I found something that was new and surprising to me, even after more than fifty years working in combinatorial geometry. The new edition has been expanded (and updated where needed), by several new delightful chapters. The careful and gradual introduction of topics and results is equally inviting for beginners and for jaded specialists. I hope that the appeal of the book will attract  many young mathematicians to the visually attractive problems that keep you guessing how the questions will be answered in the end.

—Branko Grünbaum

UNIVERSITY OF WASHINGTON, SEATTLE

All of Alexander Soifer’s books can be viewed as excellent and artful entrees to mathematics in the MAPS mode... Different people will have different preferences among them, but here is something that Geometric Etudes does better than the others: after bringing the reader into a topic by posing interesting problems, starting from a completely elementary level, it then goes deep. The depth achieved is most spectacular in Chapter 4, on Combinatorial Geometry, which could be used as part or all of a graduate course on the subject, but it is also pretty impressive in Chapter 3, on graph theory, and in Chapter 2, where the infinite pigeon hole principle (infinitely many pigeons, finitely many holes) is used to prove theorems in an important subset of the set of fundamental theorems of analysis.

—Peter D. Johnson, Jr.

AUBURN UNIVERSITY

 

This interesting and delightful book … is written both for mature mathematicians interested in somewhat unconventional geometric problems and especially for talented young students who are interested in working on unsolved problems which can be easily understood by beginners and whose solutions perhaps will not require a great deal of knowledge but may require a great deal of ingenuity ... I recommend this book very warmly.

—Paul Erdos