By Hans Schwerdtfeger. The author has performed a distinct service by making this material so conveniently accessible in a single book. Its focus lies in the intersection of geometry, analysis, and algebra, with the exposition generally taking place on a moderately advanced level. Much emphasis, however, has been given to the careful exposition of details and to the development of an adequate algebraic technique.

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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition. One of the nice things about geometry is that it can be approached in several different ways. You can do it synthetically, from axioms, either quite rigorously Axiomatic Geometry by Lee or not-so-rigorously, allowing appeal to pictures Geometry for College Students , by Isaacs.

Or, you can do what the book under review does, and study it from the perspective of complex numbers. After all, a point in the Euclidean plane can be represented by an ordered pair of real numbers, which on the one hand is just a vector, but which on the other hand can be identified with a complex number.

All that additional multiplicative structure that the field of complex numbers enjoys can then be put to good use to study geometry, an approach which results in all sorts of interesting things. The idea is fruitful enough, in fact, that not only does the book under review exploit it, but so do other books see, e. It turns out that many interesting theorems of plane Euclidean geometry can be proved using the complex numbers, including both fairly well-known results e.

The book under review also exploits the properties of the complex numbers to obtain facts about geometry, but is not really comparable to the references cited in the previous paragraph. It eschews the basic theorems of basic plane geometry in favor of more sophisticated ideas, including non-Euclidean geometry.

It is written at a somewhat more demanding level than these other references, and the prerequisites for reading it are not trivial: a student should certainly have had a previous course in linear algebra and be comfortable dealing with matrices, should know what a group is, and be familiar as well with the basic facts of real analysis in the plane.

The book is divided into three large chapters, each chapter further divided into sections, which are themselves further divided into subsections. These mappings have strong geometric content it can be shown, for example, that the set of all circles and lines is invariant under any such transformation and most books on complex analysis spend at least some time discussing them; few, however, address them in the kind of detail that Schwerdtfeger does.

Schwerdtfeger addresses a number of topics connected to Moebius transformations that are not found in the average text, including iterated transformations and applications to projective geometry specifically, projectivities and perspectivities ; projective geometry is referred to briefly but not developed in depth.

The original text was published in For this Dover edition, the author added a supplemental bibliography and some appendices bringing things up to date, at least as of The bibliography is pretty extensive but still, of course, quite dated, and a number of entries are to papers that are not in English.

For one thing, the exposition is terse. While many topics treated in the book are definitely valuable things for an undergraduate to know, there are also some topics presented here e.

For all these reasons, I doubt this book will find much use as a primary textbook for an undergraduate course; however, there is a lot of interesting mathematics in this book that is not readily found elsewhere, so as a reference for instructors, or perhaps as supplemental reading in a course on complex analysis, it might prove quite useful.

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## Geometry of Complex Numbers

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition. One of the nice things about geometry is that it can be approached in several different ways. You can do it synthetically, from axioms, either quite rigorously Axiomatic Geometry by Lee or not-so-rigorously, allowing appeal to pictures Geometry for College Students , by Isaacs. Or, you can do what the book under review does, and study it from the perspective of complex numbers. After all, a point in the Euclidean plane can be represented by an ordered pair of real numbers, which on the one hand is just a vector, but which on the other hand can be identified with a complex number.

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## Geometry of Complex Numbers: Circle Geometry, Moebius Transformation, Non-Euclidean Geometry

The author has performed a distinct service by making this material so conveniently accessible in a single book. Its focus lies in the intersection of geometry, analysis, and algebra, with the exposition generally taking place on a moderately advanced level. Much emphasis, however, has been given to the careful exposition of details and to the development of an adequate algebraic technique. In three broad chapters, the author clearly and elegantly approaches his subject. The first chapter, Analytic Geometry of Circles, treats such topics as representation of circles by Hermitian matrices, inversion, stereographic projection, and the cross ratio. The second chapter considers in depth the Moebius transformation: its elementary properties, real one-dimensional projectivities, similarity and classification of various kinds, anti-homographies, iteration, and geometrical characterization.

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Geometry of Complex Numbers: Circle Geometry, Moebius Transformation, Non-Euclidean Geometry is an undergraduate textbook on geometry , whose topics include circles , the complex plane , inversive geometry , and non-Euclidean geometry. It was written by Hans Schwerdtfeger , and originally published in as Volume 13 of the Mathematical Expositions series of the University of Toronto Press. The Basic Library List Committee of the Mathematical Association of America has suggested its inclusion in undergraduate mathematics libraries. Each of these is further divided into sections which in other books would be called chapters and sub-sections.