Geometry of Complex Numbers

Hans Schwerdtfeger wrote this book, and published it in the year 1962, as Volume 13 originally, of the Mathematical Expositions series of the University of Toronto Press. In the Dover Books on Advanced Mathematics series of Dover Publications a corrected edition was published in the year 1979. America’s Basic Library List Committee of the Mathematical Association has suggested its inclusion in undergraduate mathematics libraries.

One of the fundamental laws of algebra is the geometrical representation of a complex number. A complex number like z = α + iβ can be denoted as a point P(α, β) in a plane called Argand plane, where α is the real part and β is an imaginary part. The value of i = \[\sqrt{-1}\]. Here, students will learn about the representation of Z modulus on the Argand plane, section formula, polar form, and many more fundamental concepts of geometry of complex numbers. 

Topics 

Altogether the book is divided into three chapters, corresponding to the three parts of its subtitle: Circle Geometry, Non-Euclidean geometry, and Möbius transformations. Furthermore, each of these three chapters is divided into sections (which would be called chapters in other books ) and sub-sections. The representation of the Euclidean plane is an underlying theme of this book as the plane of complex numbers, and to describe geometric objects and their transformations, the complex numbers are used as coordinates.

The analytic geometry of circles in the complex plane is covered by the chapter on circles.

The inversion of circles, hermitian matrices, stereographic projection, pencils of circles (certain one-parameter families of circles), and their two-parameter analogs, bundles of circles, and the cross-ratio of four complex numbers are some of the most important concepts.

The central part of the book is the chapter on Möbius transformations. It defines these transformations as the fractional linear transformations of the complex plane. Materials are included by it on the classification of these transformations, on the characteristic parallelograms of these transformations, on the subgroups of the group of transformations, on iterated transformations. 

Applications of Möbius transformations are also briefly discussed in this chapter in understanding the perspectives of projective geometry. The topics include elliptic geometry, spherical geometry, and (in line with Felix Klein's Erlangen program) the transformation groups of these geometries as subgroups of Möbious transformations in the chapter on non-Euclidean geometry.

Multiple areas of mathematics are brought together by this work, with the intent of broadening the connections between abstract algebra, the theory of complex numbers, the theory of matrices, and geometry. Howard Eves, a reviewer writes, in its selection of material and its formulation of geometry, “a book largely reflects work of C. Caratheodory and E. Cartan". These factors are responsible for many phenomena in our daily life. There are many live examples also in our daily life which could justify this phenomenon. 

Reception and Audience 

For advanced undergraduates, Geometry of Complex Numbers is written and its many exercises (called "examples") extend the material in its sections rather than merely checking what the reader has learned. Its use is recommended by Howard Eves and A. W. Goodman while reviewing the original publication, as secondary reading for classes in complex analysis, and Goodman adds that "every expert in classical function theory should be familiar with this material". Whereas, reviewer Donald Monk wonders whether the material of the book is too specialized to fit into any class, and has some minor complaints about details that could have been covered more elegantly.

Mark Hunacek by the time of his 2015 review, wrote that "the book has a decidedly old-fashioned vibe" making it more difficult to read, and that the dated selection of topics made it uneven to be usable as the main text course. R. P. Burn, a reviewer shares Hunacek's concerns about readability, and also complains that Schwerdtfeger rather than allowing geometry to play a motivating role "consistently lets geometrical interpretation follow algebraic proof". Hunacek repeats Goodman's and Eve's recommendations for its use "as supplemental reading in a course on complex analysis", and Burn concludes that "the republication is welcome".

Related Readings 

Reviewer R. P. Burn suggests two other books, as background on the geometry covered in this book, Modern Geometry: The Circle and Straight Line by C. V. Durell, and Geometry: Daniel Pedoe’s a Comprehensive Course.

Other books include Complex Numbers and Geometry by Liang-shin Hahn using complex numbers for analytic geometry or Complex Numbers from A to Z by Dorin Andrica and Titu Andreescu.