''Geometry of Complex Numbers: Circle Geometry, Moebius Transformation, Non-Euclidean Geometry'' is an undergraduate textbook on

geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, whose topics include

circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...

s, the

complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...

inversive geometry Inversive activities are processes which self internalise the action concerned. For example, a person who has an Inversive personality internalises his emotions from any exterior source. An inversive heat source would be a heat source where all th ...

, and

non-Euclidean geometry In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geo ...

. It was written by

Hans Schwerdtfeger Hans Wilhelm Eduard Schwerdtfeger (9 December 1902 – 26 June 1990) was a German-Canadian-Australian mathematician who worked in Galois theory, matrix theory, theory of groups and their geometries, and complex analysis Complex analysis, ...

, and originally published in 1962 as Volume 13 of the Mathematical Expositions series of the

University of Toronto Press The University of Toronto Press is a Canadian university press founded in 1901. Although it was founded in 1901, the press did not actually publish any books until 1911. The press originally printed only examination books and the university calen ...

. A corrected edition was published in 1979 in the Dover Books on Advanced Mathematics series of

Dover Publications Dover Publications, also known as Dover Books, is an American book publisher founded in 1941 by Hayward and Blanche Cirker. It primarily reissues books that are out of print from their original publishers. These are often, but not always, books ...

(). The Basic Library List Committee of the

Mathematical Association of America The Mathematical Association of America (MAA) is a professional society that focuses on mathematics accessible at the undergraduate level. Members include university, college, and high school teachers; graduate and undergraduate students; pure a ...

has suggested its inclusion in undergraduate mathematics libraries.

Topics

The book is divided into three chapters, corresponding to the three parts of its subtitle: circle geometry,

Möbius transformation In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form f(z) = \frac of one complex variable ''z''; here the coefficients ''a'', ''b'', ''c'', ''d'' are complex numbers satisfying ''ad'' ...

s, and non-Euclidean geometry. Each of these is further divided into sections (which in other books would be called chapters) and sub-sections. An underlying theme of the book is the representation of the

Euclidean plane In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions of ...

as the plane of complex numbers, and the use of complex numbers as coordinates to describe geometric objects and their transformations. The chapter on circles covers the

analytic geometry In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry. Analytic geometry is used in physics and engineerin ...

of circles in the complex plane. It describes the representation of circles by

2\times 2

Hermitian matrices In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the -th row and -th column is equal to the complex conjugate of the element in the -th ...

, the inversion of circles,

stereographic projection In mathematics, a stereographic projection is a perspective projection of the sphere, through a specific point on the sphere (the ''pole'' or ''center of projection''), onto a plane (geometry), plane (the ''projection plane'') perpendicular to ...

, pencils of circles (certain one-parameter families of circles) and their two-parameter analogue, bundles of circles, and the

cross-ratio In geometry, the cross-ratio, also called the double ratio and anharmonic ratio, is a number associated with a list of four collinear points, particularly points on a projective line. Given four points ''A'', ''B'', ''C'' and ''D'' on a line, the ...

of four complex numbers. The chapter on Möbius transformations is the central part of the book, and defines these transformations as the fractional linear transformations of the complex plane (one of several standard ways of defining them). It includes material on the classification of these transformations, on the characteristic parallelograms of these transformations, on the

subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...

s of the group of transformations, on iterated transformations that either return to the identity (forming a periodic sequence) or produce an infinite sequence of transformations, and a geometric characterization of these transformations as the circle-preserving transformations of the complex plane. This chapter also briefly discusses applications of Möbius transformations in understanding the projectivities and perspectivities of

projective geometry In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, pro ...

. In the chapter on non-Euclidean geometry, the topics include the

Poincaré disk model In geometry, the Poincaré disk model, also called the conformal disk model, is a model of 2-dimensional hyperbolic geometry in which all points are inside the unit disk, and straight lines are either circular arcs contained within the disk th ...

of the

hyperbolic plane In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P'' ...

elliptic geometry Elliptic geometry is an example of a geometry in which Euclid's parallel postulate does not hold. Instead, as in spherical geometry, there are no parallel lines since any two lines must intersect. However, unlike in spherical geometry, two lines a ...

spherical geometry 300px, A sphere with a spherical triangle on it. Spherical geometry is the geometry of the two-dimensional surface of a sphere. In this context the word "sphere" refers only to the 2-dimensional surface and other terms like "ball" or "solid sp ...

, and (in line with

Felix Klein Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and group ...

Erlangen program In mathematics, the Erlangen program is a method of characterizing geometries based on group theory and projective geometry. It was published by Felix Klein in 1872 as ''Vergleichende Betrachtungen über neuere geometrische Forschungen.'' It is nam ...

) the transformation groups of these geometries as subgroups of Möbious transformations. This work brings together multiple areas of mathematics, with the intent of broadening the connections between

abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''a ...

, the theory of complex numbers, the theory of matrices, and geometry. Reviewer

Howard Eves Howard Whitley Eves (10 January 1911, New Jersey – 6 June 2004) was an American mathematician, known for his work in geometry and the history of mathematics. Eves received his B.S. from the University of Virginia, an M.A. from Harvard Universi ...

writes that, in its selection of material and its formulation of geometry, the book "largely reflects work of C. Caratheodory and E. Cartan".

Audience and reception

''Geometry of Complex Numbers'' is written for advanced undergraduates and its many exercises (called "examples") extend the material in its sections rather than merely checking what the reader has learned. Reviewing the original publication, A. W. Goodman and

recommended its use as secondary reading for classes in

complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...

, and Goodman adds that "every expert in classical function theory should be familiar with this material". However, 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. By the time of his 2015 review, Mark Hunacek 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 unlikely to be usable as the main text for a course. Reviewer R. P. Burn shares Hunacek's concerns about readability, and also complains that Schwerdtfeger "consistently lets geometrical interpretation follow algebraic proof, rather than allowing geometry to play a motivating role". Nevertheless Hunacek repeats Goodman's and Eves's recommendation for its use "as supplemental reading in a course on complex analysis", and Burn concludes that "the republication is welcome".

References

{{reflist, refs= {{citation , last = Burn , first = R. P. , date = March 1981 , doi = 10.2307/3617961 , issue = 431 , journal = The Mathematical Gazette , jstor = 3617961 , pages = 68–69 , title = Review of ''Geometry of Complex Numbers'' , volume = 65 {{citation , last = Crowe , first = D. W. , date = March 1964 , doi = 10.1017/S000843950002693X , issue = 1 , journal =

Canadian Mathematical Bulletin The ''Canadian Mathematical Bulletin'' (french: Bulletin Canadien de Mathématiques) is a mathematics journal, established in 1958 and published quarterly by the Canadian Mathematical Society. The current editors-in-chief of the journal are Antoni ...

, pages = 155–156 , title = Review of ''Geometry of Complex Numbers'' , volume = 7, doi-access = free {{citation , last = Eves , first = Howard , authorlink = Howard Eves , date = December 1962 , doi = 10.2307/2313225 , issue = 10 , journal =

American Mathematical Monthly ''The American Mathematical Monthly'' is a mathematical journal founded by Benjamin Finkel in 1894. It is published ten times each year by Taylor & Francis for the Mathematical Association of America. The ''American Mathematical Monthly'' is an e ...

, jstor = 2313225 , page = 1021 , title = Review of ''Geometry of Complex Numbers'' , volume = 69 {{citation , last = Goodman , first = A. W. , journal =

Mathematical Reviews ''Mathematical Reviews'' is a journal published by the American Mathematical Society (AMS) that contains brief synopses, and in some cases evaluations, of many articles in mathematics, statistics, and theoretical computer science. The AMS also pu ...

, mr = 0133044 , title = Review of ''Geometry of Complex Numbers'' {{citation , last = Hunacek , first = Mark , date = May 2015 , journal = MAA Reviews , publisher =

, title = Review of ''Geometry of Complex Numbers'' , url = https://www.maa.org/press/maa-reviews/geometry-of-complex-numbers-circle-geometry-moebius-transformation-non-euclidean-geometry {{citation , last = Monk , first = D. , date = June 1963 , doi = 10.1017/s0013091500010956 , issue = 3 , journal = Proceedings of the Edinburgh Mathematical Society , pages = 258–259 , title = Review of ''Geometry of Complex Numbers'' , volume = 13, doi-access = free {{citation , last = Primrose , first = E. J. F. , date = May 1963 , doi = 10.1017/s0025557200049524 , issue = 360 , journal =

The Mathematical Gazette ''The Mathematical Gazette'' is an academic journal of mathematics education, published three times yearly, that publishes "articles about the teaching and learning of mathematics with a focus on the 15–20 age range and expositions of attractive ...

, pages = 170 , title = Review of ''Geometry of Complex Numbers'' , volume = 47, s2cid = 125530808

External links

*
Geometry of Complex Numbers
' (1979 edition) at the

Internet Archive The Internet Archive is an American digital library with the stated mission of "universal access to all knowledge". It provides free public access to collections of digitized materials, including websites, software applications/games, music, ...

Circles Inversive geometry Non-Euclidean geometry Mathematics textbooks 1962 non-fiction books 1979 non-fiction books

Topics

Audience and reception

Related reading

References

External links