Book Description

The Budapest semesters in mathematics were initiated with the aim of offering undergraduate courses that convey the tradition of Hungarian mathematics to English-speaking students. This book is an elaborate version of the course on 'Conjecture and Proof'. It gives miniature introductions to various areas of mathematics by presenting some interesting and important, but easily accessible results and methods. The text contains complete proofs of deep results such as the transcendence of e, the Banach-Tarski paradox and the existence of Borel sets of arbitrary (finite) class. One of the purposes is to demonstrate how far one can get from the first principles in just a couple of steps. Prerequisites are kept to a minimum, and any introductory calculus course provides the necessary background for understanding the book. Exercises are included for the benefit of students. However, this book should prove fascinating for any mathematically literate reader. ... Read more

Reviews (1)

The basics of proof techniques covered in sufficient depth
As the title indicates, the legendary Paul Erdos was involved in the creation of this book. In 1983, Erdos and other Hungarian mathematicians started the Budapest Semester in Mathematics (BSM), a program for American and Canadian undergraduate students. One of the courses in this program involves creative problem solving, which was the motivation for the material in this book. As is the case with books on problem solving, no particular area of mathematics is examined. The emphasis is on proof techniques, which are largely independent of the mathematical topic.
Of course, the quality of any book of this type is largely dependent on the choice of problems that are described, and in this case the chosen topics are excellent. The book is split into two main sections, which are further split into the following subsections:

I) Proofs of Impossibility, Proofs of Nonexistence.
1) Proofs of Irrationality.
2) The Elements of the Theory of Geometric Constructions.
3) Constructible Regular Polygons.
4) Some Basic Facts About Linear Spaces and Fields.
5) Algebraic and Transcendental Numbers.
6) Cauchy's Functional Equation.
7) Geometric Decompositions.

II) Constructions, Proofs of Existence.
8) The Pigeonhole Principle.
9) Liouville Numbers.
10) Countable and Uncountable Sets.
11) Isometries of R^n.
12) The Problem of Invariant Measures.
13) The Banach-Tarski Paradox.
14) Open and Closed Sets in R. The Cantor Set.
15) The Peano Curve.
16) Borel Sets.
17) The Diagonal Method.

While each of these topics is introduced, that does not mean that the coverage is superficial. The book is advertised as having more than elementary coverage, and I concur with that assessment. Detailed proofs of the main ideas are included with exercises at the end of each section. Hints for the solution of many of the problems are included in an appendix.
This is an excellent short introduction to many of the proof techniques that are the staple of working mathematicians. I strongly recommend it as a primary or secondary text for any course where the goal is to teach basic proof techniques to advanced undergraduates. ... Read more

Book Description

Boiled-down essentials of the top-selling Schaum's Outline series for the student with limited time

What could be better than the bestselling Schaum's Outline series? For students looking for a quick nuts-and-bolts overview, it would have to be Schaum's Easy Outline series. Every book in this series is a pared-down, simplified, and tightly focused version of its predecessor. With an emphasis on clarity and brevity, each new title features a streamlined and updated format and the absolute essence of the subject, presented in a concise and readily understandable form.

Graphic elements such as sidebars, reader-alert icons, and boxed highlights stress selected points from the text, illuminate keys to learning, and give students quick pointers to the essentials.

• Designed to appeal to underprepared students and readers turned off by dense text
• Cartoons, sidebars, icons, and other graphic pointers get the material across fast
• Concise text focuses on the essence of the subject
• Delivers expert help from teachers who are authorities in their fields
• Perfect for last-minute test preparation
• So small and light that they fit in a backpack!

Book Description

Book Description

Reviews (2)

Only for PhD's.1945 Vintage, Mathmatical proofs
I bought the book with the hope that I would be able to build some geometric models with the information.Instead you get a ton of theory.This book has a 1940's feel to it, complete with muddy photos and very few explanitory drawings.It seems that so many books of that time period assume you already know the material.Instead of making the information as simple as possible, they jam the pages with endless symbols. All I want is enough theory to build the models, not write a thesis.

Don't bother with this one.

Great.
Thi is an excellent text that I recommend to anyone seriously interested in geometry.

The book enumerates all n-dimensional polytopes and aids in visualizing multidimensional space.

The book could be improved withcomputer graphics of the polytopes, but otherwise it is perfect. ... Read more

Book Description

In this book, completed shortly before his death, Robin Evans recasts the idea of the relationship between geometry and architecture, drawing on mathematics, engineering, art history, and aesthetics to uncover processes in the imagining and realizing of architectural form. He shows that geometry does not always play a stolid and dormant role but, in fact, may be an active agent in the links between thinking and imagination, imagination and drawing, and drawing and building. ... Read more

Reviews (2)

A must read!
This book is a must read for any architect interested in the geometries and shapes of buildings (which I hope is every architect) If you have second thoughts about buying it, buy it... it is informative, entertaining the diagrams and pictures are beautiful and it will take 2 (amazing) months to go through it.

Typical Evans. Clear, breath takingly obvious.
Robin Evans has a knack of getting right to the point of many a subject with expertise. Extremely versatile and knowledgable, he uses this base to write profoundly. Evans takes criticism to another level by getting to 'the obvious' quickly, then building on pre conceived theory with frightening clarity to form an original alternative view. This is a marvelously laid out book with fantastic illustrations and plates from Renaissance history to Eisenman. He is not caught up in the hype of self-preserving discourse or traditional methods of interrogation. The book has a wealth of information that acts like a reference book. So easy to read and so refreshing in opinion. RIP Robin, this is a classic work. ... Read more

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Reviews (1)

RULER-AND-COMPASS IMPOSSIBILITY PROOFS DEMYSTIFIED
This thoroughly readable and enjoyable book for a general audience was written by an accomplished and respected mathematician, former Professor and Chair at two of the world's best Departments of Mathematics, the University of Michigan and the University of Buffalo.Unlike many other writers of popular mathematics books, Kazarinoff does not betray his beloved field by over-simplification, pontification, or dogmatism.On the contrary, Kazarinoff has deep faith in the intelligence and critical abilities of his readers and he makes ever effort to help them to become genuine participants in a small, but richly fascinating and beautiful corner of mathematics.His aim is to help the readers to gain personal knowledge of several mathematically, philosophically, historically and culturally important mathematical facts.Although these facts can be stated in short simple sentences, they were discovered to be facts only after centuries of intense mathematical research by some of the world's greatest minds.The simplest example is the fact that it is impossible to trisect the angle, i.e. there exists no general construction method or algorithm using only straight-edge and compass for trisecting an arbitrarily given angle. "RULER" in the title refers to the straight-edge; "ROUND" refers to the compass.
The Kazarinoff book, again contrary to the vast majority of popular mathematics books, carefully explains the nature of the mathematical facts to be proved: the relevant fundamentals of geometry, what a construction is, exactly what can and can not be done with the straight-edge, exactly what can and can not be done with the compass.It has an intriguing and pedagogically effective discussion of the differences between what I call the collapsing compass and the non-collapsing compass.With the collapsing compass one draws a circle given the center and a point on the circumference, but one can not carry the length of the radius to other points to make copies of the circle, the compass "collapses". But with the non-collapsing compass, once one circle has been drawn it can be copied over and over wherever a center is given.An appendix presents the reader with enough practice using the straight-edge and collapsing compass that there will be no confusion or uncertainty about which facts are being proved.The practice is not just exercise: it is used to give a cogent and accessible proof of the fact, justifiably called astounding by Kazarinoff, that any figure constructible from given points by means of the straight-edge and non-collapsing compass can be done with the straight-edge supplemented only by a collapsing compass.Moreover, and here, as far as I know, RULER AND THE ROUND is absolutely unique: it provides a brief but informative discussion of exactly what a mathematical proof is.
In each case, Kazarinoff wants the readers to know exactly which fact is being argued for and exactly what a proof of it would be like -- so that the readers can make their own judgments of whether Kazarinoff has actually proved it.On page 5, at the end of the section called "PROOF" he says to the reader concerning the arguments to be presented: "I hope they convince you too". In what other popular mathematics book have you seen such respect for the reader, such openness, such modesty? In what other popular mathematics book have you seen concern for the reader's opinion? This book is an implicit insult to the elitist high-priests of popularization with their breezy enthusiasm, their hocus-pocus "proofs", their mumbo-jumbo, their scientistic dogmatism.
Ironically, it is Kazarinoff's openness the leads him to temper his realism with what to my mind seems to be an unacceptable level of cultural relativism and to temper his egalitarianism with a sometimes hard-edged elitism. Nevertheless, his frankness and independence are truly refreshing and his sincere effort to share with non-experts his profound mastery of the material can only evoke gratitude. Of course, there is room for disagreement about the details and about how well he fulfilled his goals. Judgments on these issues are to some extent subjective and will depend on the background of the person making the judgment.
I first read a library copy of this book in 1970 when it first appeared.A few months later, when I decided to reread it, the library copy was on-loan with a long waiting list.I tried to buy a copy but by then it was out of print. Recently, I went to Amazon.com to try to get a used copy and was thrilled to learn that it is back in print with a 2003 date.
Who should read this book? Mathematics majors should look at this short 130-page book as early as possible because it might reveal to them what subject they have chosen:, or at the very least it will reveal to them what a serious, accomplished mathematician thinks the subject is. Mathematics teachers, especially those who complain that their students do not know what a proof is, might pick up a pointer or two from reading it.Logicians might learn something from it, especially from the section on pages 5 and 6 about what proofs are.Historians and philosophers of mathematics will find many original and thought-provoking perspectives in this book. Kazarinoff does not belong to any of the identifiable "schools" of philosophy of mathematics-he gives no signs in this book of having paid any of them the slightest attention. He is not selling anything and he is not spinning anything.I can not think of a better book for people curious about mathematics. - Frango Nabrasa, Manatee FL ... Read more

Book Description

This book, aimed at future teachers of mathematics, contains selected topics from the history of geometry, with "modern" proofs of some of the results, as well as a fully modern treatment of selected basic issues in geometry. All too often the geometry which goes into the syllabus for teacher-students presents the material as pedantic and formalistic, suppressing its dynamic character and its role as part of the foundation for our common cultural heritage. The motivation for the book is to open up these aspects of the field. Another motivation is to provide an invitation to mathematics in general. It is an unfortunate fact that today, at a time when mathematics and knowledge of mathematics are more important than ever, phrases like math avoidance and math anxiety are very much in the public vocabulary. ... Read more

Reviews (1)

A good classroom account of geometry past and present
Noticeable weaknesses are the typos and lapses in grammar on virtually every page. Otherwise the history and current state of geometry are nicely presented, and my objections are again to failings I find to pervade recent geometry and mathematics in general.

I will focus here on the author's discussion of foundations of mathematics (pp.170-176) preceding his more detailed presentations. He speaks characteristically of a high level of precision in present-day logic and set theory, yet I see numerous misconceptions.

He describes logic in terms of "boolean functions", by which a statement P is either true or false, and "No other alternatives exist". This has been contested since Aristotle. A familiar counterexample is in asking "Did the husband stop or not stop beating his wife?" about a husband who never beat her. The concerned Law of Excluded Middle is accordingly restricted.

The author in that vein also assigns to a statement P customary "truth-values", T and F, applied in "truth tables" for composite statements like "P implies Q". Briefly, that statement is held, e.g., true if P is false and Q is true. Beside this conflict with the common understanding of implication, also phrased as "if P then Q", the example conflictingly supposes P true in the implication and false in the truth-value.

One other failing I might consider in that part of the book is the misunderstanding (by Russell also) and consequent widespread influence of "Russell's paradox". It posits

THE SET OF ALL SETS NOT MEMBERS OF THEMSELVES.

For instance, the set of all chairs (not itself a chair) qualifies, but not the set of all non-chairs. It is then observed that if the posited set is a member of itself then, since not in the set, it is contradictorily not a member of itself, and if it is not then, since it qualifies, it is.

A like paradox has been noted in considering

A MAN WHO AS BARBER IN A VILLAGE SHAVES ALL MEN IN IT NOT SHAVING THEMSELVES.

It has been tried to dismiss this, in efforts to view the paradox as a problem in set theory. But the problem is merely an unrelated apparent contradiction.

If the barber shaves ALL those not shaving themselves, he cannot be among them, since he would contradictorily shave himself. And if he shaves ONLY those not shaving themselves, he must be among them, since he would otherwise again contradictorily shave himself. Hence he cannot both shave all and only those in question, but either the first or the second.

Likewise, the concerned set cannot include both all and only those in question, but either one or the other.
... Read more

Book Description

This revised, expanded edition covers the theory, design, geometry and manufacture of all types of gears and gear drives.An invaluable reference for designers, theoreticians, students, and manufacturers, the second edition includes advances in gear theory, gear manufacturing, and computer simulation. Among the new topics are: new geometry for gears and pumps; new design approaches for planetary gear trains and bevel gear drives; an enhanced approach for stress analysis; new methods of grinding and gear shaving; and new theory on the simulation and its application. First Edition published by Pearson Education Hb (1994): 0-132-11095-4 ... Read more

Reviews (5)

Reader from Paradise, Texas
I personally have tried the formulas for square roots and higher roots and have been successful at obtaining accurate answers. Although these formulas are difficult and mistakes can easily be made, they are accurate if one is careful and pays attention to detail. Although the dot used in the formula is a separator and not a decimal. This may be the cause of inaccurate answers.

Faulty
Disappointing! At first glance this book does have lots potential in terms of simplifying concepts and formulas, but it is loaded with mistakes - not only in the answer sections, but also regarding some of the methods. I applied their "Method for Finding Square Roots" to numbers other than the given examples, and it did not work. This is not a good buy - even if one merely needs a refresher in math.

Mathematics made simple
This is a helpful math book,it covers a lot of subjects from basic math to some pretty difficult topics.My only problem with it is that there are a bunch of wrong answers in the "answer" section,but it just made me hunt and work that much harder to find the correct answers.I hope they do a revision of this book because I would like to give it as a gift,but not at this time with all the mistakes in it.

Great review - lousy math
Nice review of basic math principles, but there are an incredible number of mistakes in the answer section in the back of the book. Really confusing.

It would be good, if it weren't for the wrong answers
Overall, the book seems to be a good, quick overview of math. Great if you've been out of school for a while. But it's tarnished by the unbelievable number of wrong answers in the back of the book. Math is an absolute, either right or wrong. How can these errors be allowed in a MATH book? ... Read more

Book Description

In Euclidean geometry, constructions are made with ruler and compass. Projective geometry is simpler: its constructions require only a ruler. In projective geometry one never measures anything, instead, one relates one set of points to another by a projectivity. The first two chapters of this book introduce the important concepts of the subject and provide the logical foundations. The third and fourth chapters introduce the famous theorems of Desargues and Pappus. Chapters 5 and 6 make use of projectivities on a line and plane, repectively. The next three chapters develop a self-contained account of von Staudt's approach to the theory of conics. The modern approach used in that development is exploited in chapter 10, which deals with the simplest finite geometry that is rich enough to illustrate all the theorems nontrivially. The concluding chapters shows the connections among projective, Euclidean, and analytic geometry. ... Read more

Book Description

This book is an exposition of semi-Riemannian geometry (also called pseudo-Riemannian geometry)--the study of a smooth manifold furnished with a metric tensor of arbitrary signature. The principal special cases are Riemannian geometry, where the metric is positive definite, and Lorentz geometry. For many years these two geometries have developed almost independently: Riemannian geometry reformulated in coordinate-free fashion and directed toward global problems, Lorentz geometry in classical tensor notation devoted to general relativity. More recently, this divergence has been reversed as physicists, turning increasingly toward invariant methods, have produced results of compelling mathematical interest. ... Read more

Reviews (4)

Very good contents but..
The only drawback, and it is a serious one, is the binding. For a such expensive book, one could expect a DECENT binding, but the outcome is a SHAME.

So 5 star for the contents an 0 for the binding

Addendum
This book is now available at Amazon.co.uk!

Its contents are: Manifold Theory. Tensors. Semi-Riemannian Manifolds. Semi-Riemannian Submanifolds. Riemannian and Lorentz Geometry. Special Relativity. Constructions. Symmetry and Constant Curvature. Isometries. Calculus of Variations. Homogeneous and Symmetric Spaces. General Relativity; Cosmology. Schwarzschild Geometry. Causality in Lorentz Manifolds.

Let's go buy it!

Excellent for beginner and experienced mathematicians
This is one of the best books on Differential Geometry I've ever read. It includes a clear exposition of all the basic results and then goes on to the most deep aspects of the subject, making it useful for undergraduate and graduate students, as well as experienced working mathematicians. It's a pitty that it's no longer available.

I laughed, I cried, it became a part of me.
Ever wonder how to mathematically model what goes on inthe vicinity of the sun? Here's the book for you. ... Read more

Reviews (2)

Written for the physicist in mind
This book, written by some of the master expositors of modern mathematics, is an introduction to modern differential geometry with emphasis on concrete examples and concepts, and it is also targeted to a physics audience. Each topic is motivated with examples that help the reader appreciate the essentials of the subject, but rigor is not sacrificed in the book.

In the first chapter the reader gets a taste of differentiable manifolds and Lie groups, the later gving rise to a discussion of Lie algebras by considering, as usual, the tangent space at the identity of the Lie group. Projective space is shown to be a manifold and the transition functions explicitly written down. The authors give a neat example of a Lie group that is not a matrix group. A rather quick introduction to complex manifolds and Riemann surfaces is given, perhaps too quick for the reader requiring more details. Homogeneous and symmetric spaces are also discussed, and the authors plunge right into the theory of vector bundles on manifolds. Thus there is a lot packed into this chapter, and the authors should have considered spreading out the discussion more, as it leaves the reader wanting for more detail.

The authors consider more fundamental questions in smooth manifolds in chapter 3, with partitions of unity used to prove the existence of Riemannian metrics and connections on manifolds. They also prove Stokes formula, and prove the existence of a smooth embedding of any compact manifold into Euclidean space of dimension 2n + 1. Properties of smooth maps, such as the ability to approximate a continuous mapping by a smooth mapping, are also discussed. A proof of Sard's theorem is given, thus enabling the study of singularities of a mapping. The reader does get a taste of Morse theory here also, along with transversality, and thus a look at some elementary notions of differential topology. An interesting discussion is given on how to obtain Morse functions on smooth manifolds by using focal points.

Notions of homotopy are introduced in chapter 3, along with more concepts from differential topology, such as the degree of a map. A very interesting discussion is given on the relation between the Whitney number of a plane closed curve and the degree of the Gauss map. This leads to a proof of the important Gauss-Bonnet theorem. Degree theory is also applied to vector fields and then to an application for differential equations, namely the Poincare-Bendixson theorem. The index theory of vector fields is also shown to lead to the Hopf result on the Euler characteristic of a closed orientable surface and to the Brouwer fixed-point theorem.

Chapter 4 considers the orientability of manifolds, with the authors showing how orientation can be transported along a path, thus giving a non-traditional characterization as to when a connected manifold is orientable, namely if this transport around any closed path preserves the orientation class. More homotopy theory, via the fundamental group, is also discussed, with a few examples being computed and the connection of the fundamental group with orientability. It is shown that the fundamental group of a non-orientable manifold is homomorphic onto the cyclic group of order 2. Fiber bundles with discrete fiber, also known as covering spaces, are also discussed, along with their connections to the theory of Riemann surfaces via branched coverings. The authors show the utility of covering maps in the calculation of the fundamental group, and use this connection to introduce homology groups. A very detailed discussion of the action of the discrete group on the Lobachevskian plane is given.

Absolute and relative homotopy groups are introduced in chapter 5, and many examples are given of their calculation. The idea of a covering homotopy leads to a discussion of fiber spaces. The most interesting discussion in this chapter is the one on Whitehead multiplication, as this is usually not covered in introductory books such as this one, and since it has become important in physics applications. The authors do take a stab at the problem of computing homotopy groups of spheres, and the discussion is a bit unorthodox since it depends on using framed normal bundles.

The theory of smooth fiber bundles is considered in the next chapter. The physicist reader should pay close attention to this chapter is it gives many insights into the homotopy theory of fiber bundles that cannot be found in the usual books on the subject. The discussion of the classification theory of fiber bundles is very dense but worth the time reading. Interestingly, the authors include a discussion of the Picard-Lefschetz formula, as an example of a class of "fiber bundles with singularities". Those interested in the geometry of gauge field theories will appreciate the discussion on the differential geometry of fiber bundles.

Dynamical systems are introduced in chapter 7, first as defined over manifolds, and then in the context of symplectic manifolds via Hamaltonian mechanics. Liouville's theorem is proven, and a few examples are given from relativistic point mechanics. The theory of foliations is also discussed, although the discussion is too brief to be of much use. The authors also consider variational problems, and given its importance in physics, they continue the treatment in the last chapter of the book, giving several examples in general relativity, and in gauge theory via a consideration of the vacuum solutions of the Yang-Mills equation. The physicist reader will appreciate this discussion of the classical theory of gauge fields, as it is good preparation for further reading on instantons and the eventual quantization of gauge fields.

A masterful sequel!
Novikov et al's first volume was the defining book on differential geometry (S-V 93). The second volume picks up on the detailed theory of manifolds and topology and other advanced theories of differential geometry, including homotopy groups, Lie algebras and digressing into physical theories as well (eg.Yang-Mills) giving one of the juciest books on the subject - an utter delight! ... Read more

Book Description

Aimed at the layman as well as the student or professional in mathematics, this book explains the structure of the nine regular solids—tetrahedron, cube, octahedron, dodecahedron, icosahedron among them—as well as many semiregular solids, and demonstrates how they can be used to explain mathematics visually rather than by symbol systems. Instructions for constructing cardboard models of each. Over 300 photographs and drawings. Index.
... Read more

Reviews (3)

Little Gem
Alan Holden must be a very dedicated man. This short book on polyhedra is filled with pictures of intricate paper models, all made by the author. In the final section, showing how to construct similar models, one can see a photo of his workshop with hundreds of models arrayed neatly on shelves behind him. This book was clearly a work of love.

Most of the book is occupied with a treatment of regular and semiregular polyhedra, prisms and antiprisms. These are examined in some depth--for example, all nine regular polyhedra are constructed. The last fifty pages introduce other topics, such as packing, lattices, and knots; the treatment here is very brief, somewhat disappointing and leaving a desire for more depth. The same can be said of the final section, on construction--Holden gives general guidelines but leaves the reader to compute the dimensions of all the faces of his models himself.

The prose is clear and concise, rare for a mathematics book. But the real substance lies in the photographs of polyhedra models. These are contructed in such a way that it is always easy to see the details of the solid: faces of different shapes are made of different shades of paper, complicated models are shown in intermediate stages of construction, polyhedra to be compared (such as duals) are shown as individuals and interpenetrating. The great icosidodecahedron photo on page 112 (or its companion that might go by the same name on page 98) is almost worth the price of the book by itself.

This is not a rigorous treatment of the subject, but it is a beautiful one.

An excellent introduction to Archimedean Star Polyhedra
This well-made and inexpensive book is brim-full of pictures of Alan Holden's models of polyhedra. It is a book for the hobbyist and the enthusiastic closet Pythagorean, more than it is for the professional mathematician. It is especially useful as an introduction to Archimedean star polyhedra, which are surely as beautiful as anything in geometry, and which were not fully catalogued until the 1950s.

If you find this material as compellingly fascinating as I do, you may want to follow up this book with these two:

"Polyhedron Models," by Magnus Wenninger, has a more thorough and systematic treatment of the Archimedean star polyhedra than Holden's book. These include some incredibly complicated models of "snub" star polyhedra -- spectacular stuff that is not included here. (On the other hand, Wenninger's book costs a good deal more.)

"Regular Polytopes," by H.S.M. Coxeter, is an elegantly written introduction to polyhedra in 3 and 4 dimensions. Coxeter himself wrote the first systematic treatment of the Archimedean star-polyhedra, and helped to discover the last few in the process. This book's illustrations are nowhere near as nice as the other ones', but this is balanced by its more rigorous mathematical treatment of the theme. Somebody needs to come up with a better way (using computer graphics?) to illustrate higher-dimensional polyhedra. In the meantime, this inexpensive book is the best I know on the subject.

A beautiful, simple and elegant book on polyhedra
A friend of mine lent me this book in 1975. (I still haven't given it back). Although this 200 page book is very simply and clearly written, I have never been able to sit down and read it from start to finish. Each part I read makes me stop and contemplate. After 24 years I still find new things in it! The book starts out describing the five Platonic solids. Next it explores the dualities: between the octahedron and cube, between the dodecahedron and icosohedron, and between the tetrahedron with itself. Holden talks about solids discovered by Kepler and Poinsot, space filling solids other than the cube, Nolids, lattices and a whole lot more. He also describes how to make your own models with cardboard and Elmers' glue. Doug Kendall's photographs of Holden's models are very pleasing. This is my favorite book. ... Read more

Book Description

Inspiring popular video games like Tetris while contributing to the study of combinatorial geometry and tiling theory, polyominoes have continued to spark interest ever since their inventor, Solomon Golomb, introduced them to puzzle enthusiasts several decades ago. In this fully revised and expanded edition of his landmark book, the author takes a new generation of readers on a mathematical journey into the world of the deceptively simple polyomino. Golomb incorporates important, recent developments, and poses problems, inviting the reader to play with and develop an understanding of the extraordinary properties of polyominoes. ... Read more

Reviews (2)

Ultimate book on polyominoes by the inventor.
Polyominoes is an entire sub-domain of geometrical puzzles and
this book is the epitomie of the subject. Packed full of
results and puzzles old and new. An extensive bibliography
is provided. Another one for the bookshelf of all mathematical puzzlers.

Even better the second time through
One of the most eagerly anticipated second editions in the history of mathematics, the wait was worth it. Literally defining a whole new area of recreational mathematics, the problems and proof techniques introduced in the first edition have kept an entire generation of mathematical thinkers busy. Although easily understood, some of the problems have defied solution for decades after the publication of the first edition in 1965. It now appears that all of the problems listed as unsolved in the first edition have been resolved, the last succumbing in 1993. As befits the enormous interest in these problems, three people announced solutions simultaneously, this reviewer being one of them.
While there is not a lot of material that was not part of the first book, it still stirs the mathematical heart. For these problems and proofs are timeless things of beauty. Even a child can understand how to put polyominoes together - my two-year-old daughter is an existence proof of that. And the proofs are sometimes so clever in their simplicity that one is tempted to use the phrase attributed to Paul Erdos, "That one is from God's little book." Who among us fails to appreciate some of the proofs of placing pentominoes on a checkerboard that relies on nothing more than the number of spaces colored black versus red. Even the proofs of the problems that took decades to resolve can be understood by those with only a rudimentary knowledge of mathematics. Sometimes, if you can count to 60, you can understand the proof.
Truly a jewel in the crown of mathematical royalty, this book deserves to be a runaway bestseller.

Published in Journal of Recreational Mathematics, reprinted with permission ... Read more

Book Description

This book offers a new treatment of the topic, one which is designed to make differential geometry an approachable subject for advanced undergraduates. Professor McCleary considers the historical development of non-Euclidean geometry, placing differential geometry in the context of geometry students will be familiar with from high school. The text serves as both an introduction to the classical differential geometry of curves and surfaces and as a history of a particular surface, the non-Euclidean or hyperbolic plane. The main theorems of non-Euclidean geometry are presented along with their historical development.The author then introduces the methods of differential geometry and develops them toward the goal of constructing models of the hyperbolic plane.While interesting diversions are offered, such as Huygen's pendulum clock and mathematical cartography, the book thoroughly treats the models of non-Euclidean geometry and the modern ideas of abstract surfaces and manifolds. ... Read more

Reviews (2)

a great book!
This is a great book.The author develops the differential geometry of curves and surfaces.The endpoint is the vindication of Euclid's parallel postulate.I thoroughly enjoyed reading this book.Very readable.

Excellent text connecting classical to differential geometry
This book is ideal for those with a long time interest in mathematics or the student just becoming interested in advanced topics.It successfully takes the concepts of classic geometry (Euclidean), clearly explains how the parallel postulate interacts with the other postulates and then introduces differential geometry as a natural outgrowth of hyperbolic geometry.McLeary's book succeeds by demonstrating the connection of modern differential geometry to the concepts in which we were educated.This is not a book for the casual reader, but includes many problems and solutions to the more interesting of them ... Read more

Book Description

Working Skills in Geometric Dimensioning and Tolerance Mike Fitzpatrick ISBN: 0-8273-4900-9 ... Read more

Book Description

Keith Van Rijsbergen demonstrates how different models of information retrieval (IR) can be combined in the same framework used to formulate the general principles of quantum mechanics. All the standard results can be applied to address problems in IR, such as pseudo-relevance feedback, relevance feedback and ostensive retrieval. The relation with quantum computing is examined. Appendices with background material on physics and mathematics are also included. ... Read more

Reviews (2)

Systematic approach
I am hoping to start a new company that will provide physicians with accelerated access to medical research papers as part of their daily clinical practice.I've been looking for an inclusive theory to information retrieval end-to-end that I can use to guide us in technology evaluation and selection.

This book has been very helpful in providing this end-to-end perspective.It can be challenging mathematically at times, but is worth the effort.Most other books I've gotten so far seem to be more of a collection of techniques, which isn't bad if an overall theoretical basis is in place.But lacking that basis makes it difficult if not impossible to objectively evaluate candidate approaches to meet the goals of the end product.

I recommend this book to those looking for an inclusive theory to information retrieval but not looking for specific technologies, software, algorithms, etc.In addition, it may be most relevant to readers with a good math background in matrix theory/Hilbert spaces, or to those who are willing to wade through those portions without frustration.While the author provides good introductions to the core concepts where needed, in practice it can be difficult to put all the background necessary into the book without re-writting an extensive amount of literature that spans heavy mathematics plus quantum mechanics (each of which is a heavy-weight on it's own much less in combination).

The book also contains a very good list of references which are useful for filling in any mathematical or quantum mechanics theoretical gaps.

Bottom-line pro: end-to-end understanding of a sort of unifying theory for information retrieval

Bottom-line con: heavy mathematics with some physics requiring patience in reading or a good background a priori

Please don't let this last comment scare you though.I have a mediocre background in math and physics but I still pushed through this in a weekend to get the key points.I'll follow up on important details as I read the reference literature cited by the author.

Interesting in many ways
Here are three cases where you can find this book very interesting:

1) You are interesting in the mathematical foundations of Information Retrieval, and a new and powerful approach to them catch your attention.

2) You are not very familiar with Quantum Theory, but you are not willing to go across lots of physics (movement of particles, experiments, etc.) and just want to learn about its most relevant logical features. The book explains them in a very clear way.

3) You are familiar with Quantum Theory, and suspect that its scope goes beyond physics, to the realm of information itself. Then you find this book, where it is proposed that the quantum language is a natural choice to formulate the foundations of Information Retrieval, a science of information handling.

Whatever your case is, do not think that the level of abstraction of the subjects treated will make this a difficult book to read.The argumentations are clear, simple, and the way it is written isvery entertaining . ... Read more