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Never HIGHLIGHT a Book Again!Virtually all testable terms, concepts, persons, places, and events are included. Cram101 Textbook Outlines gives all of the outlines, highlights, notes for your textbook with optional online practice tests. Only Cram101 Out ... Read more

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This book should be of interest to undergraduate degree students taking introductory geometry courses. ... Read more

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An Unabridged Printing, To Include All Exercises: Foundations - Elementary Theorems On Order - The Affine Group In The Plane - Euclidean Plane Geometry - Ordinal And Metric Properties Of Conics - Inversion Geometry And Related Topics - Comprehensive Index ... Read more

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This graduate level text covers an exciting and active area of research at the crossroads of several different fields in Mathematics and Physics. In Mathematics it involves Differential Geometry, Complex Algebraic Geometry, Symplectic Geometry, and in Physics String Theory and Mirror Symmetry. Drawing extensively on the author's previous work, the text explains the advanced mathematics involved simply and clearly to both mathematicians and physicists. Starting with the basic geometry of connections, curvature, complex and Kähler structures suitable for beginning graduate students, the text covers seminal results such as Yau's proof of the Calabi Conjecture, and takes the reader all the way to the frontiers of current research in calibrated geometry, giving many open problems. ... Read more

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An Unabridged, Digitally Enlarged Printing To Include Updated Typeface And All Figures: Introduction - Pangeometry - Hyperbolic Geometry - The Elliptic Geometry - Analytic Non-Euclidian Geometry - Historical Note ... Read more

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This book is an extensive monograph on Sasakian manifolds, focusing on the intricate relationship between K er and Sasakian geometries. The subject is introduced by discussion of several background topics, including the theory of Riemannian foliations, compact complex and K er orbifolds, and the existence and obstruction theory of K er-Einstein metrics on complex compact orbifolds. There is then a discussion of contact and almost contact structures in the Riemannian setting, in which compact quasi-regular Sasakian manifolds emerge as algebraic objects. There is an extensive discussion of the symmetries of Sasakian manifolds, leading to the study of Sasakian structures on links of isolated hypersurface singularities. This is followed by an in-depth study of compact Sasakian manifolds in dimensions three and five. The final section of the book deals with the existence of Sasaki-Einstein metrics. 3-Sasakian manifolds and the role of sasakian-Einstein geometry in String Theory are discussed separately. ... Read more

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Customer Reviews (1)

A set of complex problems in real analysis are solved in detail
The following definition of convexity is used in this book.

A subset X of R2 or R3 is convex if whenever x and y are in X the segment connecting them is also in X.

Using this definition, a series of problems are presented and solved. Organized by chapter, they are

Chapter 1) Problems in which convexity is used either by analogy or for subsidiary arguments
*) The intersection of connected open sets
*) Approximations to homomorphisms of R2 onto itself
*) On the projection of a plane set of finite linear measure

Chapter 2) Problems which can be reduced to problems on convex sets
*) Covering a three-dimensional set with sets of smaller diameter

Chapter 3) Problems on convex sets
*) Approximation to plane convex sets
*) Geometrical properties for which triangles are the extremal convex curves

Chapter 4) Problems concerned with the structure of subclasses of the class of convex sets
*) The asymmetry of curves of constant width
*) Sets of constant width contained in a set of given minimal width
*) Extremal properties of triangles circumscribing plane convex sets
*) On the closest packing by equilateral triangles

Each of the problems is dealt with in detail with theorems followed by proofs. A remark summarizing the problem occurs after every problem is resolved. In terms of difficulty, some real analysis background is necessary to understand the work.
... Read more

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This book is an investigation of the mathematical and philosophical factors underlying the discovery of the concept of noneuclidean geometries, and the subsequent extension of the concept of space. Chapters one through five are devoted to the evolution of the concept of space, leading up to chapter six which describes the discovery of noneuclidean geometry, and the corresponding broadening of the concept of space. The author goes on to discuss concepts such as multidimensional spaces and curvature, and transformation groups. The book ends with a chapter describing the applications of nonassociative algebras to geometry. ... Read more

Customer Reviews (1)

A compendium of facts
This is not a book at all; it is a compendium of facts. Without the slightest hint of a continuous narrative or of relating things to the avowed theme, one work after another is discussed, with the discussion consisting entirely of quotations or mild paraphrases of some highlights of the original works. Thus no one will read this book from cover to cover. It can be used as a reference only. As such, its usefulness is of course directly proportional to its reliability. On this matter I have some doubts.

Let us takes as a random sample the brief discussion of Pascal's poster Essay pour les coniques on p. 140. First of all the publication year of this essay is given as 1604 even though we are informed that Pascal lived 1623-1662 and that he published it at age 16, from which the correct publication year (1639) can be inferred. One cannot help but think that while this misprint was innocent enough, if it is indicative of others then these are bound to be worse. It also makes little sense to remark that "An English translation appeared in 1987" (referring to the book on Desargues by Field & Gray) when an English translation appeared in 1928 (Isis X.1) and was made widely available through inclusion in Smith's famous source book. But of course the mathematics is the most important thing. Here it is claimed of this poster that "in it Pascal proved Pascal's theorem." This is plainly false. Pascal's explicitly states his theorem (in convoluted form) without proof.

Now, on the very next page we read that "Newton shows" (p. 141) his theorem on the projective classification of cubics in the Enumeratio. Again this is plainly false, as again the theorem is stated without proof. Perhaps this can be excused since earlier there is a footnote that says: "Newton's notes on this work, containing the proofs omitted from the printed text, have been published by Derek Thomas Whiteside [602, vol. 2, pp. 10-89]." Fine, but by following this reference we will not find the proof in question, nor in fact anything relating to projective reasoning, as this was unknown to Newton at the time the preliminary treatise referred to was written in the 60s. In fact, Whiteside himself writes right in the introduction to the work referred to (vol. 2, p. 6) that Newton "abandoned completely" the 60s algebraic approach in favour of an approach which "in effect affords a geometric interpretation." Instead of a reference to this "completely abandoned" approach, the correct reference for background notes to Newton's projective classification is vol. VII, pp. 410-433. In conclusion, in the space of only two pages, Rosenfeld has sent us on two wild goose chases for proofs that do not in fact exist. That is unacceptable for a reference work. ... Read more

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Essentials of Geometry to include chapters covering the elements of geometry and the 5 axioms’. Euclidean and Non-Euclidean Geometry, Proportion, Planes, Desargue’s Theorem, Pascal’s Theorem, and Geometrical Construction. ... Read more

Customer Reviews (7)

The Foundations of Geometry - Forgotten Books edition

The FORGOTTEN BOOKS edition of Hilbert's Foundations of Geometry isn't Hilbert's Geometry. Notice the number of pages (which I didn't when ordering it). This publication contains ONLY the diagrams in large format (with a very few absent) from the text of Hilbert's Geometry. There is no title page or author listed, but this is in fact what the content is from. It is clearly a scan from an old book, so there must be some historical context for it. Maybe someone can clarify the mystery. I give it 5 stars because these comments will probably show up among the reviews of Hilbert's full text and I don't want to skew the star rating of the book, but this particular reprint I don't find of any actual value, except that it's from Hilbert and there may be some interesting reason why it occurs as an independent publication.

Along with this reprint, I also ordered the FB Classic Reprint of Elements of Geometry and Trigonometry by Charles Davies. These two books are the first reprints I've purchased from any of the reprint publishers selling on amazon. For more on the quality of Forgotten Books reprints, see my review of Davies' book. The mysterious Hilbert-diagrams text they sell under the title of Hilbert's Foundations of Geometry is, I suspect, an anomaly. Besides, their honest page-count should raise questions about the content. Now you know what that content is.

Buy the Open Court edition - a better translation into English
For the best translation into English (not Townshend's translation) see Hilbert's "Foundations of Geometry" as extended by Paul Bernays, Open Court Publishing Co, second edition in English, 1971.(This should be a translation of the 10th and final edition in German by Bernays, which dates from 1968.)Bernays was Hilbert's assistant at Göttingen beginning in 1917 and his co-author of "Grundlagen der Mathematik" (1934-39).
Hardcover ISBN: 0875481639
Paperback ISBN: 0875481647

Incomplete
This is the first book ever to present the axiomatic foundations of euclidean geometry. The first edition appeared in the nineties of the nineteenth century.

Most of the book can be read and appreciated by someone who is mature in elementary euclidean geometry (in fact the material was originally conceived to be used in a summer school for mathematics teachers in Germany). If you expect to find a treatment that will fill up all the gaps in the elementary books you will be disappointed, it does not. If you are looking for a text that does fill all the gaps try to get a copy Forders' book The foundations of Euclidean geometry,.

This edition is not based on the last German edition that is available and does not contain the appendices by Hilbert and thesupplements by Paul Bernays, so as a text on the foundations of euclidean geometry it is not useless but it is surely crippled.

I do not dare to give a book with Hilberts name on it less than five stars.

Enjoyable
Hilbert gives his new system of axioms and studies their consistency, independence and necessity. Consider for example the theorem that the angle sum in any triangle cannot be greater than two right angles. We can prove it as follows. Consider a triangle ABC with the angles labelled so that ABC<=ACB. Let D be the midpoint of BC. Draw AD and extend it to E so that AD=DE. By SAS, ACD=BDE, so that angle CAD=angle BAE and angle DBE=angle ACB. Thus ABC has the same angle sum as ABE. ABC<=ACB means that AC=BE<=AB, so angle BAE<=angle AEB, so angle BAE<=angle BAC/2. In other words: for any angle A in any triangle we can construct a new triangle with equal angle sum that has as one of its angles A/2. By repeating this process we can make the angle A as small as we like. Thus, if the angle sum of some triangle was greater than two right angles, and we applied this procedure, we would get a new triangle where two of the angles are greater than two right angles, which is impossible. The "as small as we like" part gives away the fact that we are relying on Archimedes' axiom, which is necessary. "The investigation of this matter which [Max] Dehn has undertaken at my urging led to a complete clarification of this problem. ... If Archimedes' axiom is dropped then from the assumption of infinitely many parallels through a point it does not follow that the sum of the angles in a triangle is less than two right angles. Moreover, there exists a geometry (the non-Legendrian geometry) in which it is possible to draw through a point infinitely many parallels to a line and in which nevertheless the theorems of Riemannian (elliptic) geometry hold. On the other hand there exists a geometry (the semi-Euclidean geometry) in which there exists infinitely many parallels toline through a point and in which the theorems of Euclidean geometry still hold. From the assumption that there exist no parallels it always follows that the sum of the angles in a triangle is greater than two right angles." Another interesting topic is the connection between laws of algebra and the theorems of Pappus (which Hilbert calls Pascal's) and Desargues. Geometrically, we can multiply two numbers a and b using only the axioms of projective geometry as follows. We choose a line to be the "x-axis" and call one of its points the origin O and another of its points the unit 1. Mark Oa and Ob on this line. Draw another line, the "y-axis", through O. Pick some point i on the y-axis. Connect 1 and i, and draw the parallel to this line through b, meeting the y-axis at b' (as usual, "parallel to l" means: meets l at an arbitrarily designated line called the line at infinity). Connect a and 1 and draw the parallel to this line through b'. In Euclidean geometry this line cuts the x-axis at ab. In general, then, we may define multiplication in this way. The algebraic identity ab=ba now becomes a geometric theorem. This is the beautiful part: ab=ba is not just any old geometric theorem, it is in fact equivalent to Pappus's theorem: the construction of ab consisted of the line connecting 1 and i and three more lines, the construction of ba consists of the line connecting 1 and i and three more lines, each of which is parallel to one of the lines from the ab construction. Therefore, deleting the line connecting 1 and i, Pappus applies and says ab=ba. Similarly, Desargues is equivalent to a(bc)=(ab)c.

Available for Free
This historic book is available for free from Project Gutenberg http://www.gutenberg.org. Search for Geometry. This book is one of a few books available. This is the complete Open Court text. It is available both as a pdf file and a TeX file. ... Read more

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The geometry of two and three dimensional space has long been studied for its own sake, but its results also underlie modern developments in fields as diverse as linear algebra, quantum physics, and number theory. This text is a careful introduction to Euclidean geometry that emphasizes its connections with other subjects. Glimpses of more advanced topics in pure mathematics are balanced by a straightforward treatment of the geometry needed for mechanics and classical applied mathematics. The exposition is based on vector methods; an introductory chapter relates these methods to the more classical axiomatic approach. The text is suitable for undergraduate courses in geometry and will be useful supplementary reading for students of mechanics and mathematical methods. ... Read more

Customer Reviews (1)

Brief, British
Beautiful introductory book about Basic Geometry, written in a brief british style, for beginning undergraduate students.The book can be read by (and can be taught to) anyone with some background in traditional high school Mathematics. The book begins with Euclidean Geometry and ends with a brief introduction to Differential Geometry of curves and surfaces. It is an interesting read (and a pleasant teach, unlike most undergraduate text books) with nice historical threads and good classical problems.The author provides hints and short solutions to almost all the problems at the end of the book. The problems are an essential ingredient of the book, so don't miss them! The author probably did not quite intend to write a yellow "Geometry for Dummies" so he does not dwell (not for too long at least) on the trivial stuff.Geometry was never part of the Trivium! You will feel good aboutyourself and about Geometry, one of the oldest disciplines of science, after reading this book. I did. ... Read more

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This volume is a collection of refereed expository and research articles in discrete and computational geometry written by leaders in the field. Articles are based on invited talks presented at the AMS-IMS-SIAM Summer Research Conference, "Discrete and Computational Geometry: Ten Years Later", held in 1996 at Mt. Holyoke College (So. Hadley, MA). Topics addressed range from tilings, polyhedra, and arrangements to computational topology and visibility problems. Included are papers on the interaction between real algebraic geometry and discrete and computational geometry, as well as on linear programming and geometric discrepancy theory. ... Read more

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This self-contained book offers a new and direct approach to the theories of special functions with emphasis on spherical symmetry in Euclidean spaces of arbitrary dimensions. Written after many years of lecturing to mathematicians, physicists and engineers in scientific research institutions in Europe and the USA, it uses elementary concepts to present the spherical harmonics in a theory of invariants of the orthogonal group. One of the highlights of this book is the extension of the classical results of the spherical harmonics into the complex. This is particularly important for the complexification of the Funk-Hecke formula which successfully leads to new integrals for Bessel- and Hankel functions with many applications of Fourier integrals and Radon transforms. Exercises have been included to stimulate mathematical ingenuity and to bridge the gap between well known elementary results and their appearance in the new formations. ... Read more

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This monograph describes the stochastic behavior of the solutions to the classic problems of Euclidean combinatorial optimization, computational geometry, and operations research. Using two-sided additivity and isoperimetry, it formulates general methods describing the total edge length of random graphs in Euclidean space. The approach furnishes strong laws of large numbers, large deviations, and rates of convergence for solutions to the random versions of various classic optimization problems, including the traveling salesman, minimal spanning tree, minimal matching, minimal triangulation, two-factor, and k-median problems. Essentially self-contained, this monograph may be read by probabilists, combinatorialists, graph theorists, and theoretical computer scientists. ... Read more

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This book presents the foundations of Euclidean geometry from the point of view of mathematics, taking advantage of all the developments since the appearance of Hilbert's classic work. Here, real affine space is characterized by a small number of axioms involving points and line segments making the treatment self-contained and thorough. This treatment is accessible for final year undergraduates and graduate students, and can also serve as an introduction to other areas of mathematics such as matroids and antimatroids, combinatorial convexity, the theory of polytopes, projective geometry and functional analysis. ... Read more

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Convex Analysis is the calculus of inequalities while Convex Optimization is its application. Analysis is inherently the domain of the mathematician while Optimization belongs to the engineer. In layman's terms, the mathematical science of Optimization is the study of how to make a good choice when confronted with conflicting requirements. The qualifier Convex means: when an optimal solution is found, then it is guaranteed to be a best solution; there is no better choice. Any Convex Optimization problem has geometric interpretation. Conversely, recent advances in geometry and in graph theory hold Convex Optimization within their proofs' core. This book is about Convex Optimization, convex geometry (with particular attention to distance geometry), and nonconvex, combinatorial, and geometrical problems that can be relaxed or transformed into convex problems. A virtual flood of new applications follows by epiphany that many problems, presumed nonconvex, can be so transformed. International Edition II ... Read more

Customer Reviews (3)

Epiphany after Epiphany
Convex Optimization & Euclidean Distance Geometry

I thought I'd use this book as a reference since the unusually large Index is a good place to locate the definitions.Dattorro starts from the basic premises and works through the algebra with many examples and many good illustrations.

I've found that Dattorro's perspective on each subject (optimization and distance geometry) is both algebraic and geometric.He bridges those unexpectedly well.His approach to rank minimization, for example, is how I would have thought of doing it, in terms of eigenvalues.It feels right to me.

Dattorro's notation is "progressive."A vector is represented by a single letter, say x, with no embellishment to distingush it from a real variable.That makes the presentation simple, but takes some getting used to as does his style of "missing articles" (e.g. the) and replacement everywhere of "i.e." with latin "id est."

The book is organized by convex optimzation first then distance geometry second,
three chapters devoted to each.The appendices support seven chapters total and take half the book!It's a big book.

Dattorro's treatment of distance geometry is the book's main strength.The main result is a new expression for the relationship between the semidefinite positive and Euclidean distance cones, and takes a long time to get there.Along the way, he goes back to 1935 and integrates the results of Schoenberg (before modern linear algebra), Cayley and Menger, Critchley, Gower, then augments that with some later results like Hayden, Wells, Liu, & Tarazaga, and then more contemporary results like Deza & Laurent, Wolkowicz, Saul and Weinberger to name only a few.Then, of course he shows how that all relates to optimization.I particularly liked the geographical map reconstruction examples where only distance ordering was known.

I recommend this book to anyone who wants both a good introduction to convex optimization and a reference to some latest techniques, a few of which Dattorro may have invented.There is a good review of semidefinite programming, and what he writes about distance geometry refreshes old math with new.

Masterly
"Dattorro has given a masterly account of the properties of convex squared-distance cones."

-Casper J. Albers, Frank Critchley, & John C. Gower, Group Average Representations in Euclidean Distance Cones, in Brito, Bertrand, Cucumel, and de Carvalho, editors, Selected Contributions in Data Analysis and Classification (Studies in Classification, Data Analysis, and Knowledge Organization), Springer, pp.445-454, 2007.

Get inspired for your research
I have read many books in optimization and in engineering, this is really one of the most exceptional books in my library. Dattorro is a talented writer. The book has plenty of illustrations that make the content of the book very easy to understand. The author is obviously influenced byBoyd Convex Optimization and this is actually a very good thing. In my opinion he has extended Boyd's book in the best possible way, by providing exciting applications and also a different perspective for understanding convexity, polytopes etc. The book is very useful for understanding Semidefinite {rogramming (SDP). To my knowledge this is the only book that gets so deeply in SDP (theory and mostly applications). The chapters on Euclidean Distance Geometry are unique in the sense that the only other books that I found in the literature on that topic date back to 1930. I think the biggest hit of the book is the algorithm for rank minimization.

Unfortunately the book hasn't gotten enough attention from the scientists and engineers probably because you can also get a free pdf copy for the author's website.Although I can get a free online version, I bought a copy becauseit is worth it (as it is for Boyd's book). I have recommended this book to may PhD students that got inspired for their research. ... Read more

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The book conveys a distinctive approach, stimulating readers to develop a broader, deeper understanding of mathematics through active participation—including discovery, discussion, and writing about fundamental ideas.It provides a series of interesting,challenging problems, then encourages readers to gather their reasoningsand understandings of each problem. ... Read more

Customer Reviews (2)

First-principle arguments: a great intro to theoretical geo.
Geometry has always come as a difficult subject for me.For some reason, it's hard for me to "see" the answers and proofs like some people can.But, that's where a book like this came in handy.

Basically, this book is good for two things:first-principle arguments and model building.The problems in here almost require you to make models and draw pictures to see for sure what's going on.So, for a person raised on the calculator like myself, having to make models is a good thing (although very challenging!).Also, the first principle arguments are good for thought and frustration:Almost every problem in this book takes a very long time to solve, with combined thinking time, study time with your notes, and actually writing out your solution.Only on a couple of the problems that my class worked on was I able to get away with using less than one sheet of paper to write out my answer.Basically, you start from the very beginning and look at concepts you know already, like straight lines, and try to explain them as clearly and concisely as you can, which isn't always easy.But, Henderson gives hints for almost every problem, gives background on each problem, and sometimes provides solutions to the problems (but rarely!); he also provides motivation for studying the concepts rather than to pass a math course, which is very good.He also does an excellent job of introducing the concept of hyperbolic geometry by first discussing what properties stay the same or differ between the sphere and plane, which is excellent because hyperbolic geometry is not as easy to grasp as the other two; this instructional method is actually carried out through the book, an excellent way to introduce two types of geometry that many people (including myself) may have never seen before.

Honestly, my only gripe is that the book is SO based on first principle arguments that, while it provides excellent framework for prospective elementary and high-school teachers, it won't do much for the applied mathematician who wants to see some computational examples because there are barely any!It forces a student to be held accountable for abstract thought and proof, but there are many aspects of geometry that can be handled with computations like arclength, areas of polygons, etc., which are not discussed in the book.

Basically, here's the short version:It's a great book for the student who wants to work through arguments based on first principles, but be prepared to work hard!Because almost none of the exercises come with answers, I don't find this book suitable for independent study, so I hope you have a good teacher!Also, get a ball, some rubberbands, and plenty of paper with some good erasers; you'll need them!

Geometry Teacher Likes It!!
I found this book caused me to rethink much of how I approach Geometry personally and in my classroom.From the first problem, "What is Straight?" which had me thinking about my own assumptions straight lines, I have been thinking of ways to approach Geometry differently with my students.

I enjoy the problem-centered exposition, but at times, I wish I had a little more direction.The emphasis on INTUITIVELY understanding what is going on in these different spaces, and on working with physical models (the hyperbolic models are cool), is a refreshing change from an algebraic/matrix approach.This book is all about DOING geometry, and formulating convincing arguements to your "Why?" questions. ... Read more