Book Description

Each chapter devoted to single type of problem with accompanying commentary and set of practice problems. Amateur puzzlists, students of mathematics and geometry will enjoy this rare opportunity to match wits with civilization’s great mathematicians and witness the invention of modern mathematics.
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Reviews (3)

Very Good for the interested reader
Although not reasonable for most high school students, this book does bring to light the background of modern geometry through its treatment of impossible and hard to solve problems.

Excellent background reading for a teacher!

Classic problems of geometry made simple
The principles of geometry are both elegant and timeless. One cannot help but understand why the Greeks considered it a pinnacle of intellectual achievement. This book sets down several of the classic problems, explaining them in such a clear way that it is easy to forget that it sometimes took centuries of work before the problem was resolved. Of course, those who first toiled on the problems did not have some of the additional machinery that we so take for granted today.
The problems covered are trisecting an angle, squaring the circle, constructing regular polygons and constructing a cube whose volume is twice that of a given cube. The background needed to understand the problems and solutions is nothing more than a solid grounding in basic algebra and trigonometry. Calculus is mentioned, but not used. Problem sets are included in all chapters and solutions to all are in the back of the book. They are well-posed, solidly reinforcing the points made in the text.
This is a book that covers several thousand years of progress in geometry in a little over one hundred pages. It is done well and it can be used as a supplement in any course in geometry, from high school to college.

Well done..
Bold has a gem of a book here. It's only a little bit over a hundred pages, but it's packed full of the great geometry problems that occupied the minds of the world's greatest thinkers for the past 2000 years.

The title describes the book perfectly. These really are "Famous Problems from Geometry" and he does indeed explain how to solve them.

The book has four major sections/chapters. He discusses in detail the three problems from antiquity (one section each): squaring a circle, doubling a cube, and trisecting an angle. Furthermore, he spends significant time with constructions of regular polygons (the fourth section) - which ones can be constructed and why. He also discusses which ones cannot be constructed and why.

The reader will be expected to understand concepts from Modern Algebra, particularly the concept of a Field. While Bold does spend time explaining what a Field is, his definition is quick and is assumed to be more of a refresher for someone who has already learned about them. Bold also has a section on Complex Numbers where he derives one of the formulas used later in the book. Again - this section is assumed to be a refresher on Complex Numbers. High School Geometry or Algebra students would have significant trouble understanding his explanations and proofs.

Bold provides problems for the reader to work along the way. These are problems that logically lead to the proof of the problem being studied. The problems are good. As a third year college student majoring in mathematics, I found the explanations/solutions to be sometimes hard to follow. He assumes a great deal about the reader's level of proficiency in math and in geometry. As a result, he liberally skips steps in proofs that are assumed to be "obvious."

If you're expecting simple proofs to these problems, you're not going to find them. If they were simple, they wouldn't have taken 2000 years to solve. But they are explained clearly here in terms that anyone with a college degree should be able to understand.

Overall, a superb book. A must have for anyone interested in the famous problems from the history of Geometry. ... Read more

Book Description

This book emphasizes the beauty of geometry using a modern approach. Models & computer exercises help readers to cultivate geometric intuition.Topics include Euclidean Geometry, Hand Constructions, Geometer's Sketch Pad, Hyperbolic Geometry, Tilings & Lattices, Spherical Geometry, Projective Geometry, Finite Geometry, and Modern Geometry Research.Ideal for geometry at an intermediate level. ... Read more

Book Description

Covers topics in plane and solid (space) geometry.Also included are pictorial diagrams with thorough explanations on solving problems in congruence, parallelism, inequalities, similarities, triangles, circles, polygons, constructions, and coordinate/analytic geometry. ... Read more

Book Description

This two volume work on ... Read more

Reviews (2)

Just the facts, straightaway
This book presents geometry in a straightforward way. Emphasis is on the facts, without getting sidetracked in proofs, (although the purist might object to the fact that proofs are not given). The drawings are relevant and straightforward. The book is well organized and proceeds logically from beginning to end. There are conversational problems with answers in the text, and lots of multiple-choice test questions with answers in the appendix. The test questions are especially good, because they resemble the standardized tests schoolchildren are forced to take these days. This book, along with with a standard school textbook, should make high-school students highly proficient in this subject, and get them ready for more advanced courses such as calculus and trigonometry. Note: I also have the chemistry, physics, and trigonometry books in the Demystified Series and have found them to be of comparable quality.

NOT RECOMMENDED IF YOU NEED HELP IN GEOMETRY
because of the topic coverage. I picked this up because, though a math teacher, I really don't like Geometry. Never have. This book covers some of what a high school student would need in just a few chapters. So, Why aren't I recommending it to students needing help? Because the author completely skips proofs, which is what most students are having trouble with. It also has no chapter dealing with circles and theorems related to them. What it does cover it covers in a highly interesting and original way (why does a stool have three legs instead of four?). It is also filled with topics not covered in a high school geometry course, but which are very interesting on their own. Given the authors other books' titles, this is perhaps "geometry for electricians and hobbyists". If you are someone with bad memories of geometry, but you would like to try revisiting it, then this is highly recommended. It would also be a good outside source for students doing well in Geometry but wanting to read about some higher level topics (including 3 - 4 - and higher dimensional geometry.) The book has loads of multiple choice test questions, so you can see how well you are understanding what you are reading, but it has no detailed solutions in the back--just the correct answers. (techinical point: readers should know that the author teaches polar coordinates "backwards" from the way we teach it in Trigonometry. The form is (r, theta), NOT (theta, r).) ... Read more

Book Description

Recent advances in computing and algorithms make it easier to do many classical problems in algebra. Suitable for graduate students, this book brings advanced algebra to life with many examples. The first three chapters provide an introduction to commutative algebra and connections to geometry. The remainder of the book focuses on three active areas of contemporary algebra: homological algebra; algebraic combinatorics and algebraic topology; and algebraic geometry. ... Read more

Book Description

Many secondary mathematics teachers and teacher educators feel ill-equipped to teach a course in geometry. Most feel fortunate to stay one chapter ahead of the very students they are teaching! Most college-level geometry texts don't address their specific needs. Advanced Euclidian Geometry was written to fill this void in many ways: • Provides both in-service and pre-service teachers with a solid review of the essentials of the high school geometry course - AND MORE! • Offers further elaboration of those concepts (advanced Euclidian geometry, the 19th and 20th century expansions of Euclid's work) to give teachers more confidence in guiding student explorations and answering questions. • Contains hundreds of illustrations created in The Geometer's Sketchpad(r) Dynamic Geometry(tm) software. • Is packaged with a dual (Windows(r)/Macintosh(r)) platform CD-ROM containing over 100 interactive sketches using The Geometer's Sketchpad (assumes that users has access to the Sketchpad program) ... Read more

Reviews (1)

A must-have book for every geometry teacher
As a high-school geometry teacher, I have often wished for a book like this. Sadly, the typical one-year geometry course comes to an end just as students are within reach of some truly beautiful and intriguing theorems. Dr. Posamentier's book begins where most high-school geometry textbooks end, and presents many wonderful results that lie just beyond their purview: the nine-
point circle; the golden rectangle; the theorems of Ceva, Menelaus, Ptolemy, Pascal, Desargues, and Brianchon; excircles and incircles; cyclic quadrilaterals; and much more. This book provides a rich geometric feast.

There are several books that cover much of the same ground that Dr. Posamentier surveys: GEOMETRY REVISITED, by Coxeter and Greitzer, and EPISODES IN NINETEENTH AND TWENTIETH CENTURY EUCLIDEAN GEOMETRY, by Honsberger, are two of the best. What makes Dr. Posamentier's book stand out is its usefulness as a textbook. Theorems are fully proved, and arranged in a logically coherent sequence. (The book is Euclidean in format as well as in subject matter.)

The book is thoughtfully designed. The pages are large, the type is easy to read, the diagrams are clear, and the book lies flat when opened.

EVERY HIGH-SCHOOL GEOMETRY TEACHER SHOULD HAVE THIS BOOK. It's a rich source of supplementary material for regular sections, and an ideal textbook for the second semester of an honors-level class, or for a student who wants to pursue the study of geometry on an independent-study basis.

I know that I'll be turning to this book again and again next year, and for as long as I teach geometry. In fact, I plan to buy another copy so that I'll have one at home and one in my classroom.

Bravo, Dr. Posamentier, and thank you! ... Read more

Book Description

Over the last forty years, researchers have made great strides in elucidating the laws of image formation, processing, and understanding by animals, humans, and machines. This book describes the state of knowledge in one subarea of vision, the geometric laws that relate different views of a scene. Geometry, one of the oldest branches of mathematics, is the natural language for describing three-dimensional shapes and spatial relations. Projective geometry, the geometry that best models image formation, provides a unified framework for thinking about many geometric problems relevant to vision. The book formalizes and analyzes the relations between multiple views of a scene from the perspective of various types of geometries. A key feature is that it considers Euclidean and affine geometries as special cases of projective geometry.

Images play a prominent role in computer communications. Producers and users of images, in particular three-dimensional images, require a framework for stating and solving problems. The book offers a number of conceptual tools and theoretical results useful for the design of machine vision algorithms. It also illustrates these tools and results with many examples of real applications. ... Read more

Book Description

This book exposes readers to many practical applications of geometry, especially those involving measurement. A three- part organization divides topics into Problem Solving, Geometric Shapes, and Measurement; Formal Synthetic Euclidean Geometry; and Alternate Approaches to Plane Geometry. ... Read more

Reviews (1)

An outstanding introduction to geometric thought
This is one of the few introductory level texts I have seen that gives some of the real flavor of mathematics, without being too challenging for beginning students. The initial section on problem solving is modelled on the famous book by Polya, "How to solve it," and has many simple but thought-stimulating problems. The following sections develop plane and solid geometry with many illustrated problems and interesting historical notes. The final chapters carefully introduce geometric proofs. There are also review sections on simple algebraic manipulations and basic logic, as well as a short section on the implications of alternate parallel postulates. Overall, the text has a well thought out development of basic skills and concepts, and enough interesting tidbits from more "advanced" topics to challenge the imagination of any student. ... Read more

Book Description

Titles in Barron's Painless Series are textbook supplements designed especially for classroom use by middle school and high school students. The approach of each title is an appeal to students who think that the subject is boring, or too difficult, or both. The authors, all experienced educators, take a light approach, showing kids what is most interesting about each subject, and how seemingly difficult problems can be transformed into fun quizzes, brain-ticklers, and challenging puzzles with rational solutions. Geometry becomes painless—and even fun—once students learn the subject's basic components and see how solving any geometric problem is fitting parts together to solve an intriguing puzzle. They learn the meaning of postulates and theorems, discover angles of all kinds, find the relationships that exist between parallel and perpendicular lines, and discover the characteristics of shapes such as triangles, quadrilaterals, and circles. The author introduces real-world geometry experiments to make concepts less abstract, offers study strategies, and demonstrates how mini-proofs are the first step toward understanding formal geometry proofs. ... Read more

Reviews (2)

Not Very Good
On page 16, it is stated that the area of a circle is pi times the diameter. Is there anybody out there who DOESN'T know that the area of a circle is pi times the square of the radius? That error wouldn't such a big deal, except that there are plenty more to come. I don't recommend this book to anyone.

Definitely NOT for homeschool!
Since I'm homeschooling my high school sophomore this year, I've been spending time looking at math books. "Painless Geometry" seemed like a good bet. Profusely illustrated (albeit with silly monkey pictures) and written in plain English, it looked like just what we'd want.

That's until I started actually using the book. First of all, who ever heard of a 300-page reference book with only three pages of index? How are you supposed to find things that way? It's missing things like the base of a triangle (the index has neither "base" nor "triangle:base") and how to label an angle. The information's in the book, but you certainly can't find it using the index. Not only that, but the pages aren't labeled like a normal book, with the name and number of the chapter at the top or bottom of each page. You can't find your place in a book that way!

There's little depth to the book. There are experiments with pencil and paper, but no real-world examples of where you'd use geometry. Area is calculated in "square units" with no discussion of real units of measure. Pi is introduced with a single paragraph. No explanation is given of its rich history, how it's calculated, or applicability throughout mathematics.

The oversimplifications in this book may make life difficult later. The book states that all angles are measured in degrees, and the degrees symbol is generally omitted. Whatever happened to radians? In one of the problems, she asks for the area of a circle with diameter of ten. The correct answer is 100 times pi. The book states the answer as 314. That's an approximation, not an answer!

Then we started finding the mistakes. Typos like "Computer the area of a circle" (page 184) I can live with. It's hard core mistakes like these I can't tolerate:

The reader is asked to identify what type of triangle has angles of 120, 35, and 35 degrees (page 101). The answer says it's isosceles and obtuse. In reality, it's not a triangle at all, as the angles don't add up to 180 degrees!

How's this for a statement of the Side-Angle-Side postulate (page 126)? "If two sides and the included angle of one triangle are congruent to two triangles and the included angle of a second triangle, then the triangles are congruent." Huh?

There's a "super brain tickler" on page 163 which indicates, according to the answers in the book, that for squares, rhombuses, rectangles, and parallelograms, all four sides are parallel! No. Four parallel line segments wouldn't ever meet. Those four shapes have two sets of parallel sides, not one set of four parallel sides!

.... That tends to leave us with drek like "Painless Geometry."

All in all, I found this book to be poorly proofread, ridded with errors, badly indexed, oversimplified, and disconnected from the real world. It may be good as an adjunct for a student having trouble with a real geometry book, but only if there's someone around to explain what "Painless Geometry" omits or misstates. ... Read more

Book Description

No number has captured the attention and imaginations of people throughout the ages as much as the ratio of a circle's circumference to its diameter.With incisive historical insight and a refreshing sense of humor, David Blatner explores the many facets of pi and humankind's fascination with it-from the ancient Egyptians and Archimedes to Leonardo da Vinci and the modern-day Chudnovsky brothers, who have calculated pi to billions of digits with a homemade supercomputer. New in paperback. ... Read more

Reviews (28)

History of a Mystery
David Blatners book on the history of Pi is inspiring and concise. Going through the history, East and West, where the reader the will discover all the errors in formulating this most mysterious infinite number. Blatner also discusses how to remember the number, if need be, as well as some tidbits of information where pi has shown up, either in movies or books or wherever. The book essentially avoids mathematical equations, although they are displayed to those that understand them, the book is written so anyone can appreciate the the work involved in getting to pi. A fun read and reccomended as an introduction to finding the ratio of the circumference of a circle to its diameter of a circle.

Not great but also not bad...
"The Joy of Pi" is a short synopsis on the history of pi. Even if you are a math, engineering, life science, or economics major, there are bound to be details in this book that will enlighten you. I was somewhat disappointed in the lack of mathematical jargon in the book though. Also, the book is somewhat difficult to read due to the writing style of the author and the page setup of the book. Text is located in boxes or triangles or circles on nearly every page. It almost appears the author wanted to fill the space and decided to use an irregularly styled page format. Yet, if you are wanting something to read on your next flight or at the swimming pool, this could be a book for you. If the subject matter interests you enough, you might find that there are no references to pursue this information further. This is a drawback to the book. Equations containing pi are included in the book but it is up to you to seek out the proof for these equations.

A little too easy
Though the subject is very attractive, but i think it's too easy for a normal-educated person who loves maths or natural science. If you want tp buy a book of this kind, "the history of pi" will do.

Mostly a Good Review of Pi and its history
Unique and eye catching format. The author, however, does seem to dismiss the fact that tht Bible discovered Pi and did so quite accurately before most else. He seems to do so for no good reason. He neglects that the Bible commonly rounds numbers (this isn't a blueprint!). And the Hebrew mechanics itself do reveal a percise value for pi (3.14150943...) This is only a difference of 0.0000832 with actual pi, making the Bible's description of pi the most accurate in antiquity! The author embarasses himself and reveals is lack of research when he calls this "dubious."

3.141592.........
I enjoyed this book immensely. It is neither too long to be ponderous for its subject - a non-technical review of the fascination of the number pi - nor too brief to be trivial. Its graphic nature and content went a long way to maintaining my interest and stimulating my own ideas. Maybe I do have a way to square the circle!

Just kidding!! ... Read more

Book Description

Symmetry groups are of intense interest to mathematicians, physicists, chemists, and designers, as well as to the "philomorphs" who are attracted to the ideas of Design Science. The author of this book has beautifully illustrated active symmetry through the observation of the orbit of a plain cube as it creates an arrangement of cube replicas with overall symmetry under such a specific group. For the general reader, the beauty and utility of the resulting pictures is evident. The mathematician, scientific investigator, or student will further welcome a full description of all finite symmetry groups with relevant drawings, calculations and data.In the detailed study of the cube's orbits, a complete enumeration of cube compounds is introduced and illustrated by photographs, patterns, and constructable models. This group theoretical approach demonstrates a step forward in the long attempt at full classification of polyhedral compounds, whose history is treated in a special chapter in the book. The method includes construction of single polyhedra and allows a logical creation of their compounds never before published. A more complete understanding of symmetry is thus granted.The book will appeal to a wide range of researchers and students, even at the undergraduate level, in geometry, crystallography, structural chemistry, engineering, architecture -- and the developing discipline of Design Science. ... Read more

Reviews (1)

All Compounds of Cubes Classified
A really marvellous book, that is theoretical as well as practical, andwhich classifies all compounds of cubes. The theory behind this isexplained clearly in the first part of the book. In the second part thecompounds are reviewed, often with a picture of a beautifully built model.It also contains some templates for building some of the models, simple andas well as complex ones. ... Read more

Book Description

Liberal Arts mathematics books often cover much more material than can be addressed in a one-semester course. Karl Smith has created a solution to this problem with his new book: THE NATURE OF PROBLEM SOLVING IN GEOMETRY AND PROBABILITY. Loyal customers of Karl Smith's books laud his clear writing, coverage of historical topics, selection of topics, and emphasis on problem solving. Based on the successful NATURE OF MATHEMATICS text, this new book is designed to give you only the chapters and information you need, when you need it. Smith takes great care to provide insight into precisely what mathematics is--the nature of mathematics--what it can accomplish, and how it is pursued as a human enterprise. At the same time, Smith emphasizes Polya's problem-solving method throughout the text so students can take from the course an ability to estimate, calculate, and solve problems outside the classroom. Moreover, Smith's writing style gives students the confidence and ability to function mathematically in their everyday lives. This new text emphasizes problem solving and estimation, which, along with numerous in-text study aids, encourage students to understand the concepts as well as mastering techniques. ... Read more

Reviews (1)

Review for "Conjecture & Proof" by D.D. Schwartz
This book is primarily designed as a textbook for a "transition" college course (the kind of course at the sophomore level that helps students make the transition from calculus and calculation intensivecourses to more abstract mathematics), and it does an excellent job. Whilethe structure of the book is quite standard for a text of this kind,students find it accessible and user-friendly. To achieve the goal ofhelping students grow comfortable with abstract mathematical structures,Dr. Diane Driscoll Schwartz presents at first an introduction to logic andits use in the construction of mathematical proofs. The topics included inthe two chapters cover several standard proof techniques, such as the useof the contrapositive and of the principle of mathematical induction. Thispart is followed by the discussion of a number of mathematical topics suchas sets, functions, relations, groups, modular arithmetic, combinatoricsand cardinality. While the sets of exercises include a sufficient number ofproblems, it would be nice to see this number increased. Some of theproblems are designed for group-work and provide a good starting point forclassroom discussion and for "non standard" homework assignments.The book is accompanied by an Instructor's Guide, which includessuggestions for class preparation and hints for some of the problems. Forthose schools that provide a technology-intensive environment, the text canbe use with the computer language ISETL. The software can be downloadedfrom the Web Sites indicated in the book. ... Read more

Book Description

In this superb topology text, the readers not only learn about knot theory, 3-dimensional manifolds, and the topology of embedded graphs, but also their role in understanding molecular structures.Most results described in the text are motivated by the questions of chemists or molecular biologists, though they often go beyond answering the original question asked.No specific mathematical or chemical prerequisites are required. The text is enhanced by nearly 200 illustrations and 100 exercises. With this fascinating book, undergraduate mathematics students escape the world of pure abstract theory and enter that of real molecules, while chemists and biologists find simple and clear but rigorous definitions of mathematical concepts they handle intuitively in their work. ... Read more

Reviews (1)

How can topology be applied to chemistry?
This question is answered in a remarkable new book by Erica Flapan, knottheorist and professor of mathematics at Pomona College.Stereochemistry,the study of the three-dimensional structure of molecules, is a recurringtheme in chemistry and related fields.Topology, the study of geometricalproperties which are invariant under continuous transformations, is asimilarly popular area for mathematicians.While it is not immediatelyobvious that the two fields have anything in common, both fields owe a debtto the other.Although she initiates her book with a historicalperspective and detailed expository on basic aspects of low dimensionaltopology (e.g., stereoisomers, chirality, nonrigid symmetries, knot andlink types, three-dimensional manifolds, and link polynomials) she doesproceed into more advanced subject matter in later chapters includingMöbius ladders, symmetries of embedded graphs, and hierarchies ofautomorphisms.Of particular interest to the molecular biologist, thefinal chapter is devoted entirely to the topology of DNA and includestopological considerations of supercoiling, toroidal winding, enzymeaction, and tangle theory. The arguments are clearly presented within theframework of interesting and relevant molecular structures, yet there isenough mathematical rigor to satisfy dyed-in-the-wool mathematicians aswell.Although there will likely be something of interest to the averageworking chemist, the supramolecular scientist, organic chemist, molecularbiologist, and biophysicist, in particular, stand to gain the most by thecontents of this book. ... Read more

Book Description

The basic goals of the book are: (i) to introduce the subject to those interested in discovering it, (ii) to coherently present a number of basic techniques and results, currently used in the subject, to those working in it, and (iii) to present some of the results that are attractive in their own right, and which lend themselves to a presentation not overburdened with technical machinery. ... Read more

Book Description

This unique approach to combinatorics is centered around challenging examples, fully-worked solutions, and hundreds of problems---many from Olympiads and other competitions, and many original to the authors. Each chapter highlights a particular aspect of the subject and casts combinatorial concepts in the guise of questions, illustrations, and exercises that are designed to encourage creativity, improve problem-solving techniques, and widen the reader's mathematical horizons.

Topics encompass permutations and combinations, binomial coefficients and their applications, recursion, bijections, inclusions and exclusions, and generating functions. The work is replete with a broad range of useful methods and results, such as Sperner's Theorem, Catalan paths, integer partitions and Young's diagrams, and Lucas' and Kummer's Theorems on divisibility. Strong emphasis is placed on connections between combinatorial and graph-theoretic reasoning and on links between algebra and geometry.

The authors' previous text, 102 Combinatorial Problems, makes a fine companion volume to the present work, which is ideal for Olympiad participants and coaches, advanced high school students, undergraduates, and college instructors. The book's unusual problems and examples will stimulate seasoned mathematicians as well. A Path to Combinatorics for Undergraduates is a lively introduction not only to combinatorics, but to mathematical ingenuity, rigor, and the joy of solving puzzles. ... Read more

Book Description

This book is the first systematic and comprehensive study ofthe theory of continuous selections of multivalued mappings. Thisinteresting branch of modern topology was introduced by E.A. Michaelin the 1950s and has since witnessed an intensive development withvarious applications outside topology, e.g. in geometry of Banachspaces, manifolds theory, convex sets, fixed points theory,differential inclusions, optimal control, approximation theory, andmathematical economics. The work can be used in different ways: the first part is anexposition of the basic theory, with details. The second part is acomprehensive survey of the main results. Lastly, the third partcollects various kinds of applications of the theory. Audience: This volume will be of interest to graduate studentsand research mathematicians whose work involves general topology,convex sets and related geometric topics, functional analysis, globalanalysis, analysis on manifolds, manifolds and cell complexes, andmathematical economics. ... Read more