No comments about Gear Geometry and Applied Theory.

3 comments about Geodesic Domes.

1. Don't be fooled by the title of this book. It is not an architectural treatise, it's a children's book with cutouts that let you make a paper model. If you want a good book on the architecture of Geodesic Domes, buy a biography of Buckminster Fuller.



2. This is classified as a children's book, but it isn't so simple. It would be suitable for teenagers but the mathematics is relevant whatever your age and it is described clearly and concisely. However, if you're looking for a book on dome architecture this is not for you. There are two models that demonstrate the ideas about the subdivision of triangles to approximate the sphere based on the octahedron and the icosahedron. Then there are three 'architectural' dome models and lastly a model of the carbon molecule bukminsterfullerene. Each model is included for a reason and there's an explanatory text to accompany each one. The book does not cover real examples of domes much (only a sketch of the Expo 67 pavilion), although the introduction briefly explains the history and reason for them. It is important to understand that this is not a book about dome architecture: it covers the math behind the concept and explains it using cut-out models.



3. The book provides basic insights into geodesic design of higher frequency structures from the fundamental icosehdral blocks. Cut out patterns provide some hands on builds. Worth the money.

3 comments about Differential Geometry and its Applications (Classroom Resource Materials) (Classroom Resource Materials).

1. This book is not to be used as a rigorous introduction to differential geometry. There are some definitions and theorems that are casually described, and the motive behind particular definitions are vague. Those not interested in MAPLE might find constant instructions for MAPLE annoying. Not to be completely negative, there are some good excercizes in the text that I especially enjoyed.



2. I found this book to be a fine introduction to this subject. I was particularly pleased with the practical examples outlined in the book. Even though I am not extremely proficient with Maple, I found the exercises using this software provided important illustrations of applications.



3. This is a very well-written text on modern differential geometry for undergraduates. The content of the book is similar to O'Neill's "Elementary Differential Geometry" (e.g. covariant derivatives, shape operators), but it's easier to read. There are many undergrad texts around -- O'Neill, do Carmo, Pressley -- but this one is the most lucidly written one hands-down.
   
   Afer going through Oprea, one might like to tackle O'Neill's "Elementary Differential Geometry" and Vols 2-4 of Spivak's "Comprehensive Introduction to D.G."
   
   Like O'Neill, Oprea develops surface theory using the shape operator. But Oprea takes shortcuts and doesn't develop the theory in quite the same generality as O'Neill does. For example, Oprea doesn't introduce differential forms and the exterior calculus. As a consequence, Oprea restricts himself to the Serret-Frenet equations whereas O'Neill introduces Cartan's structural equations -- of which Serret-Frenet is simply a special case -- as the method of moving frames in full generality. The structural equations are then used (by O'Neill) in both curve and surface theory.

4 comments about Basic Mathematics.

1. Serge Lang's Basic Mathematics is an excellent overview of algebra and geometry. If you are in high school needing a tutorial primer, or an adult continuing their education after some years, this book will provide through its clarity, examples, and exercises (selected answers are in the back of the book)the refresher course you need for more advanced mathematics, such as calculus and linear algebra.



2. Lang's Basic Mathematics is a famous mathematician's look at everything a well-prepared high school student ought to know about math before starting calculus. The exercises are thought-provoking and the solutions are enlightening. There is just enough but not too much drill on each point before moving on, and throughout there is a wonderfully mathematical attitude about the material. Recommended for anyone who has had algebra once and wants to know a lot more about what mathematics is really all about.



3. Serge Lang's text presents the topics that he feels students should understand before commencing their study of college mathematics. As such, working through this text is a good way for you to supplement what you learned in high school with material that will aid you in studying mathematics in college. Therefore, I particularly recommend it for prospective mathematics majors.
   
   The material in the text is well motivated and clearly presented. While Lang explains how to perform routine calculations, he focuses on the underlying structure of the mathematics. The material is developed logically and results are proved throughout the text. However, the presentation of the material is marred by numerous errors, most, but not all, of which are typographical.
   
   The problems range from routine calculations to proofs. Many of the problems are challenging and some require considerable ingenuity to solve. Answers to some of the exercises are presented in the back of the text. I should warn you that if you are used to artificial textbook problems in which the correct solution is a "nice" number, you will find that is not the case here. Also, it is useful to read through the problem sets before you begin solving them so that you can do related problems at the same time.
   
   The first section of the book covers algebra. Properties of the integers, rational numbers, and real numbers are examined and compared. There is also more routine material on linear equations, systems of linear equations, powers and roots, inequalities, and quadratic equations.
   
   A brief discussion of logic precedes a section on geometry. Basic assumptions about distance, angles, and right triangles are used as a starting point rather than Euclid's postulates. This leads to a discussion of isometries, including reflections, translations, and rotations. Area is discussed in terms of dilations. The treatment here is different from that in the high school text Geometry which Lang wrote with Gene Murrow. I found the material on isometries quite interesting. Be aware that the notation and some of the terminology in this section is not standard.
   
   The third section of the book covers coordinate geometry. Distance is interpreted in terms of coordinates. This leads to a discussion of circles. Transformations are reinterpreted using coordinates. Segments, rays, and lines are presented using parametric equations. A chapter on trigonometry covers standard topics, but also includes a section on rotations. The section concludes with a chapter on conic sections. Of particular interest is a proof that all Pythagorean triples can be generated from points on the unit circle with rational coordinates.
   
   The final section of miscellaneous topics addresses functions, more generalized mappings, complex numbers, proofs by mathematical induction, summations, geometric series, and determinants. The text concludes by demonstrating how determinants can be used to solve systems of linear equations.
   
   The eminent mathematicians I. M. Gelfand and Kunihiko Kodaira have also contributed to books intended for high school students. Those of you planning to study mathematics in college would benefit from working through their texts as well.



4. Serge Lang died September 2005, and it was a great loss for many people; he has been a prominent mathematician, who has published many book and articles. He had a very good memory, and it is said that he wrote a book in the course of one weekend on a bet. I don’t know if that’s true, but you can sense that he feels at home writing about mathematics.
   
   Basic Mathematics is suited both for the younger readers who hasn’t begun high school yet, and for older readers who needs to refresh their skills. I believe that many people would benefit working through this book before starting in high school, as it will ease and speed up things. The book is structured in a way that it clearly brings the most important of the mathematics which later is to be used.
   
   The book has four parts: Algebra, Intuitive Geometry, Coordinate Geometry, and Miscellanous. There are 17 chapter spread over these four parts, which each deals with an important mathematical subject. Of mention are “Functions”, “Operations on Points”, “Distance and Angles”, and “Linear Equations”. It’s mainly basic mathematical subjects, which are dealt with in an “advanced way”, so the author doesn’t look down on his reader. Nothing is dwelt upon, but nothing important isn’t absent either.
   
   Exercises is included in nearly all sections, so that the reader can train himself in both a manipulative and a theoretical level. Som sections has many exercises (which can be tough at times), while some has only three or four. The difficulty is raised, of course, but if you just do the exercises, you’ll notice how well the book is structured in that basic techniques are used later in more advanced subjects.
   
   Recommended.

No comments about Extremal Graph Theory.

1 comments about Algebraic Topology: A First Course (Mathematics Lecture Note Series).

1. This text is suitable for students of mathematics without prior knowledge of algebraic topology. The best thing with this is Part 2 which treats singular homology theory. However, you may want to resort to Maunder for an effeective introductin to elelmentary homotopy theory, and to Dold for and intruduction to orientation and duality.

No comments about Geometry: Tchrs'.

1 comments about Methods of Homological Algebra.

1. Homological algebra is one of those subjects that in order to understand, you need to know already. Category theory wouldn't hurt either, nor some algebraic geometry and algebraic topology. Unfortunately, you need to know homological algebra to do some of these things as well. The great strength of Gelfand and Manin's work is that it ties together examples from all of these areas and coherently integrates them into some of the best mathematical prose I've ever read. The book is recent enough that its authors write from a position of vast perspective on fifty years of research, and the subject as they present it is about as up-to-date as possible, yet cleanly developed and not overwhelming. Unlike many books whose subject matter was influenced by modern algebraic geometry, this one does not merely pay lip service to standard references on its vast prerequisites, but systematically develops them (specifically, the ideas of category theory and abelian categories) in an entire, large chapter.
   
   The book's only tangible drawback is the presence of errors, despite the revision. The previous edition was said to be riddled with them, and the authors have indeed brought the count down to a nearly respectable level, with those remaining relatively minor. The remaining errors are more jarring than confusing, however, and this is not a sticking point.
   
   Finally, I would like to emphasize that neither this book nor any other is suitable for beginners in homological algebra. This is an aspect of the field, and its remedy is to study the applications, algebraic geometry and algebraic topology most of all. The ideas of homological algebra are derived not from first principles but from mathematicians' experiences doing mathematics, and both the subject matter and the many excellent examples in the book will resonate more with a student whose knowledge they cast in a new light.

5 comments about The Colours of Infinity: The Beauty, The Power and the Sense of Fractals.

1. Just a perfect combination of top-level science outreach with the
   fantastic music of David Gilmour.
   
   It's a journey into the wonders of the fractal geometry explained in
   a clear and easy-to-follow way.
   
   Buy it now !!! You won't regret !!!



2. What an illuminating, thought provoking book and dvd! I have watched it several times and each time it has opened my eyes more to the amazing possibilities in almost every aspect of our existance here on earth. And then who doesn't like David Gilmour's music ?



3. Depending upon one's education, it is possible to gain either an art appreciation of the fractal geometry of the Mandelbott Set or a realization of how all life is ordered and the universe structured. Life most certainly would exist somewhere out there and it would resemble what we have here on Earth. While viewing it, I felt that this was at least fifty percent of the riddle of the universe explained in at least its basis. The other fifty percent would be what Stephen Hawkin called the other formulae that will reverse this one. He hopes we will find this one before it begins to act in its turn. That would mark the collapse of the universe and he seems to think that we might prevent that. But the collapse may be inevitable and part of the eternal operation of two formulae.
   
   The Mystery remains; was this ordered and if so, by whom or what? We may never know, but for this devastating mystery, we have David Gilmour's compelling score to propel us along through an eternal race toward infinity.
   
   My only complaint is that the film needs re-mastering so that the fractal geometry can be expressed better. Ironic that the documentary that defines the detail of the universe is so fuzzy! Also, there is a second section of fractal art expression on the DVD that will only play on a computer. That ought to be fixed. I think it is fitting that Arthur C. Clarke is the narrator for this film, so I hope this original can be cleaned up and not trashed for an updated version with all new presentation and cast.
   
   This documentary should be shown in all high-school science classes. In fact, I think it ought to be shown to everyone regardless of partticipation in science curriculum because it also assists the refinement of questions like evolution and religion.



4. The book and the DVD are truly amazing and beautiful. It has introduced me to a wonderful world which is a marriage of art and mathematics. The price is almost too low for such a masterpiece.



5. I agree with pretty much all the other reviewers here. As a novice to fractals (I am 50 years old, but never could get into math...), I found the book most satisfyingly informative without, as another reviewer put it, dumbing down the concepts. I also liked the anthology approach, with authors in different specialties writing the different chapters. Yes the DVD is of pretty poor quality (probably made worse by watching in on a high-rez monitor?), but it is more than 20 years old--and made for TV.
   
   It should be emphasized that this book is NOT a reissue. It was produced specifically for the TWENTIETH ANNIVERSARY of the video documentary's release. The book is, indeed, published in 2004 and the accompanying color plates are suitably gorgeous. I watched the DVD first and recommend other people new to fractals do so. Much of the book repeats or is an expansion of what's on the DVD, but rather than seeming repetitive, it made the topic more comfortably familiar to me as I got into reading the more scientifically specific explanations. The writers/interviewees were all impressively warm and personable as well as infectiously enthusiastic about their subject. Like Carl Sagan and astronomy, books like this are great PR for science and math. Live appearances by Arthur C. Clarke were particularly poignant, given that he just died last month.

2 comments about Using Algebraic Geometry (Graduate Texts in Mathematics).

1. I just completed a course that used this book as a...reference. Granted, it is a first edition, but it reads like a rough draft. The presence of three authors is all too obvious in the inconsistent writing of proofs, paragraphs, and even exercises. Some proofs are just plain wrong, and many have gaping holes in them. Notation is confusing, and changes without warning or explanation. I will say this much in its favor: many important results are presented, although the proofs are absent. It makes a good source for named theorems, but that's about it.



2. Once thought to be high-brow estoeric mathematics, algebraic geometry is now finding applications in a myriad of different areas, such as cryptography, coding algorithms, and computer graphics. This book gives an overview of some of the techniques involved when applying algebraic geometry. The authors gear the discussion to those who are attempting to write computer code to solve polynomial equations and thus the first few chapters cover the algebraic structure of ideals in polynomial rings and Grobner basis algorithms. The reader is expected to have a fairly good background in undergraduate algebra in order to read this book, but the authors do give an introduction to algebra in the first chapter. Many exercises permeate the text, some of which are quite useful in testing the reader's understanding. The Maple symbolic programming language is used to illustrate the main algorithms, and I think effectively so. The authors do mention other packages such as Axiom, Mathematica, Macauley, and REDUCE to do the calculations. The chapter on local rings is the most well-written in the book, as the idea of a local ring is made very concrete in their discussion and in the examples. The strategy of studying properties of a variety via the study of functions on the variety is illustrated nicely with an example of a circle of radius one. Later, in a chapter on free resolutions, the authors discuss the Hilbert function and give a very instructive example of its calculation, that of a twisted cubic in three-dimensional space. They mention the conjecture on graded resolutions of ideals of canonical curves and refer the reader to the literature for more information. Particularly interesting is the chapter on polytopes, where toric varieties are introduced. The authors motivate nicely how some of the more abstract constructions in this subject, such as the Chow ring and the Veronese map, arise. The important subject of homotopy continuation methods is discussed, and this is helpful since these methods have taken on major applications in recent years. In optimization theory, they serve as a kind of generalization of the gradient methods, but do not have the convergence to local minima problems so characteristic of these methods. In addition, one can use homotopy continuation methods to get a computational handle on the Schubert calculus, namely, the problem of finding explicity the number of m-planes that meet a set of linear subspaces in general position. There are some software packages developed in the academic environment that deal with homotopy continuation, such as "Continuum", which is a projective approach based on Bezout's theorem; and "PHC", which is based on Bernstein's theorem, the latter of which the authors treat in detail in the book. My primary reason for purchasing the book was mainly the last chapter on algebraic coding theory. The authors do give an effective presentation of the concepts, including error-correcting codes, but I was disappointed in not finding a treatment of the soft-decision problem in Reed-Solomon codes.

   In general this is a good book and worth reading, if one needs an introduction to the areas covered. Students could definitely benefit from its perusal.