Euclidean-Geometry

I have some 47 books in the geometry section of my shelves. If I had to discard 40 of these, Geometry and the Imagination would be among the 7 remaining.

Geometry is the study of relationships between shapes, and this book helps you see how shapes fit together. Ultimately, you must make the connections in your mind using your mind's eye. The illustrations and text help you make these connections. This is a book that requires effort and delivers rewards.

The leading mathematician of the 20th century, David Hilbert liked to quote "an old French mathematician" saying "A mathematical theory should not be considered complete until you have made it so clear that you can explain it to the first man you meet on the street". By that standard, this book by Hilbert was the first to complete several branches of geometry: for example, plane projective geometry and projective duality, regular polyhedra in 4 dimensions, elliptic and hyperbolic non-Euclidean geometries, topology of surfaces, curves in space, Gaussian curvature of surfaces (esp. that fact that you cannot bend a sphere without stretching some part of it, but you can if there is just one hole however small), and how lattices in the plane relate to number theory.

It is beautiful geometry, beautifully described. Besides the relatively recent topics he handles classics like conic sections, ruled surfaces, crystal groups, and 3 dimensional polyhedra. In line with Hilbert's thinking, the results and the descriptions are beautiful because they are so clear.

More than that, this book is an accessible look at how Hilbert saw mathematics. In the preface he denounces "the superstition that mathematics is but a continuation ... of juggling with numbers". Ironically, some people today will tell you Hilbert thought math was precisely juggling with formal symbols. That is a misunderstanding of Hilbert's logical strategy of "formalism" which he created to avoid various criticisms of set theory. This book is the only written work where Hilbert actually applied that strategy by dividing proofs up into intuitive and infinitary/set-theoretic parts. Alongside many thoroughly intuitive proofs, Hilbert gives several extensively intuitive proofs which also require detailed calculation with the infinite sets of real of complex numbers. In those cases Hilbert says "we would use analysis to show ..." and then he wraps up the proof without actually giving the analytic part.

If you find it terribly easy to absorb Hilbert's THEORY OF ALGEBRAIC NUMBER FIELDS and also Hilbert and Courant METHODS OF MATHEMATICAL PHYSICS, then of course you'll get a fuller idea of his math by reading them--but only if you find it very easy. Hilbert did. And that ease is a part of how he saw the subject. I do not mean he found the results easily but he easily grasped them once found. And you'll have to read both, and a lot more, to see the sweep of his view. For Hilbert the lectures in GEOMETRY AND THE IMAGINATION were among the crowns of his career. He showed the wide scope of geometry and finally completed the proofs of recent, advanced results from all around it. He made them so clear he could explain them to you or me.

There might be less than 10 mathematics books in the world that I am glad to put under my pillow when I go to sleep. And this book is one of the top three.

This book conclusively reveals that fourth dimension theories and related spiritual quests constitute, at the very least, a key pillar in the development of modern art, if not its foundation. The book gives the lie to strictly formalist interpretations of the historic impetus for modern art. Perhaps more importantly, it highlights that vast amounts of art history omits the intended metaphysical content of early modern painting. If the record were set straight sooner, one wonders whether the trajectory of art would have led to contemporary art that is more substantive and enriching than it so often is. Artists such as Brice Marden, Agnes Martin, Astrid Colomar, Anne Truitt and others whose work is infused with spiritual depth would be seen as those whose work is most linked to, rather than most divorced from, the artistic heritage of modernism. With even out-of-print paperback copies of this ground-breaking book selling for well over $100, a new printing would be as welcomed as it is deserved.

One should not practice so-called fine arts without reading this book!!! One should be able to fill the b l a n k in his/her understanding of the contemporary relationship between arts & science! Worthy of your valuable time!

i highly recommend this book to anyone interested in turn of the century art and/or concepts of higher space. very insightful on a subject once widespread and now obscure, i.e. the fourth dimension.

As someone who wasn't a math major but who has been trying to get up to speed on lebesgue measure and integration, I found this book to be truly accessible. Unlike other "introductory" texts (such as Kopp's "Measure, Integral and Probability") I could follow the reasoning in this book without much difficulty.

The only criticism I have of the book has to do with the first chapter. Its purpose is to provide background mathematical material and given the author's clear ability to explain difficult concepts, I wish that it covered that material in greater detail.

For others who may be looking to build a foundational understanding of this material but who may not be mathematicians, I'd also recommend Pitt's "Measure and Integration for Use" (1985) or his "Integration, Measure and Probability" (1963) (both out of print but fairly easy to find). Those books, along with Jones', are well-used items in my library.

"Lebesgue Integration on Euclidean Space" is a nearly ideal introduction to Lebesgue measure, integration, and differentiation. Though he omits some crucial theory, such as Egorov's Theorem, Jones strengthens his book by offereing as examples subjects that others leave as exercises. The best example of this is his section on L^p spaces for 0 < p < 1.

The book's greatest strength, however, is its readability. Whereas Royden gives no hint as to how much work is needed between steps, Jones highlights important steps in proofs, not just the important proofs. It is this motivated style that makes his book useful.

Jones is so careful in his construction of the theory that differentiation does not appear until Chapter 15, and specific results for R^1 come only in Chapter 16. But the wait is worth it.

While Jones has written a great introduction, the book cannot be used for more advanced courses. As the title suggests, the discussion is restricted to Euclidean spaces. In addition, his direct jump to measure on R^n and the use of "special rectangles" therein make the development incongruous with other books. But what is sacrificed in depth is made up for in breadth, with Jones hinting at how the theory is used in other branches of math. There's even an entire chapter devoted to the Gamma function!

As a student, I have found Jones's book more instructive on basic theory than Royden, Rudin, and Wheeden & Zygmund. I highly recommend it as a first-semester introduction to Lebesgue theory or as a source of clean, fundamental presentations of proofs.

This book is a treasure trove of mathematical technique. It covers topics that are relevant to many broad areas of real and functional analysis including signal processing and approximation theory. The author takes the time not only to prove the results, but also to construct the proofs so that the technique is made explicit to the reader. The author also motivates definitions by breaking them into the successively more complicated pieces so as to build intuition in the reader.

I especially recommend this book to anyone who lacks formal training in mathematics or wishes to develop mathematical technique in the areas of real and functional analysis.

Geometry is such a sweet science and this book is a honeypot. Ross Honsberger is one of the best in creating crafty, delightful problems. In reading this book, you will be amazed at how hard the problems appear when stated and how simple and elegant the proofs are.
Some topics are relatively obscure. I am sure that Honsberger is quite correct in stating that few modern mathematicians have heard of the symmedian point of a triangle. I certainly had not. However, the level of the material is such that even teachers of high school geometry will be able to find some suitable exercises. College instructors will find it essential. Problems are given at the end of each chapter and detailed solutions are included in an appendix.
It is hard to read this book and not understand why the Greeks were so captivated by geometry. Properly presented, as is done here, it is addictive.

Published in Smarandache Notions Journal, reprinted with permission.

Ross Honsberger presents the articles of modern euclidean geometric properties in a way that is easy to follow and understand. The method of proofs and development of ideas illustrate different solving strategies which is valuable for self-studying young students. With the help of simple diagrams, I found no difficulties in understanding the articles. Problems are well-graded with solutions are well-presented at the back. A MUST-READ book for students participating maths contest!

This book is, as the author mentioned in the preface, a marriage of two parts. The first part provided more or less a self-contained introduction to the theory of Riemannian holonomy groups, which usually couldn't be found in differential geometry textbooks. The second part is a research monograph on exceptional holonomy groups, which is the subject that the author is famous at. This book contains lots of topics which are hard to be found in any other books. For example, it contains a proof of the Calabi conjecture, which I've never seen in anywhere else except Yau's original papers. It also has a concise introduction to Calabi-Yau manifolds, which includes lots of topics about CY manifolds that are hard to be found in just a single book. Overall, it's a great introduction to the theory of holonomy groups. And also provides a good start about differential geometric side of the theory of Calabi-Yau manifolds, together with a roughly complete list of further references.

This book about euclidean and non-euclidean geometry is great! A must for researh or math class!

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.

The advent of non-Euclidean geometry resulted in many different areas of mathematics, some being specifically related to geometry, others being more general, such as proof theory and model theory. This book is an excellent overview of a particular branch of non-Euclidean geometry called hyperbolic geometry. There are good exercises in the book, and the author gives a detailed history of the subjects after the end of each chapter. After a brief review of Euclidean geometry in chapter 1, emphasizing the metric properties of Euclidean space, orthogonal transformations, and isometries, the author discusses spherical geometry in chapter 2. Spherical and hyperbolic geometries are dual to each other, in the sense that in spherical geometry, a line through a point outside a given line is never parallel to the given line; but in hyperbolic geometry there are infinitely many such lines. Also, the sum of the angles of a spherical triangle is always greater than 180 degrees ; but in hyperbolic geometry less than 180 degrees. Hyperbolic geometry is of crucial importance in physics, particularly in the theory of relativity, and the author begins a discussion of this kind of geometry in chapter 3. Hyperbolic n-space is viewed more as dual to elliptic geometry in the sense that it is modeled as a unit sphere of imaginary radius with only the positive sheet of this (disconnected) set retained. The author outlines in detail the important properties of hyperbolic geometry along with its trigonometry. This is followed in the next chapters by a model of hyperbolic n-space as a conformal ball and an upper half-space, and a consideration of the isometries of hyperbolic space. The Mobius transformations are given detailed treatment. The famous classical discrete groups are introduced, along with the crystallographic groups. The discussion gets more abstract in some parts here, for the author introduces some algebraic notions such as valuation rings, in order to prove Selberg's lemma. The author finally lays the groundwork for a theory of hyperbolic manifolds in chapter 8, by first introducing geometric spaces. These are defined by four axioms, which are generalizations of Euclid's first four axioms, and two of these axioms imply that any geometric manifold is an n-manifold. The discussion is specialized in the next chapter to geometric surfaces, where the famous Gauss-Bonnet theorem, relating the area of a surface to its Euler characteristic, is proved for spherical, Euclidean, or hyperbolic surfaces. The author studies the collection of similarity equivalence classes of complete structures for a geometric surface, namely the moduli space of such structures. Physicists, particularly string theorists, will appreciate the resulting discussion on Teichmuller space and the Dehn-Nielsen theorem. Considerations of a nature more familiar to geometric topologists follows in the next chapter, where it is shown how to explicitly construct hyperbolic 3-manifolds. Dehn surgery is employed to study the complement of the figure 8 knot. The discussion is very interesting, for it employs explicit detailed constructions that would take many hours to dig out of the literature. The general case of n-dimensional hyperbolic manifolds is the subject of chapter 11, with the constructions in chapter 10 generalized to deal with high dimensions. The author considers also the two closed, orientable, hyperbolic manifolds of the same homotopy type have the same volume by using the Gromov invariant, a quantity defined in terms of the singular homology on the manifold. The reader will get a taste of the Haar measure in the proof of the result, and later an overview of measure homology. The later is very interesting, as it brings in techniques from differential topology and the de Rham complex, and it also defines a notion of a "straightening" and smearing of a singular complex. Mostow rigidity, which says that for any two closed, connected, orientable, hyperbolic n-manifolds, with n greater than 2, a homotopy between these will also be an isometry, is also proven here. The next chapter is more involved than the rest, and deals with the case of geometrically finite n-manifolds. Dealing with cusps and "sharp corners" from the actions of discrete groups is given detailed and rigorous discussion here. The discussion leads naturally to a treatment of orbifolds in the next chapter. These objects have been extremely important in string theories in high energy physics, and the author does an excellent job of detailing their properties.

I got this book as a second hand and shortly its very very nice book.
The applications are very smart and clear ,
Its contexts and illustrations are adequate ,precise and really easy to read and understand.
I realy loved this book ,and i guess this is how the geometry Should be taught as rich ideas with apps not in abstract form.
You will find a nice proof for fermat's least time principle,
and lots lots more intersting ideas good for physics and computer
graphics programming.
This book really worth any price.

this is a really great introduction to hyperbolic geometry. especially if you want to study gammas acting on the upper half plane. it starts at a much lower level then any other text.