Geometry of Complex Numbers

From Wikipedia, the free encyclopedia

Template:Short description

Script error: No such module "about".Page Module:Infobox/styles.css has no content.Page Template:Infobox book/styles.css has no content.

Geometry of Complex Numbers
Lua error in package.lua at line 80: module 'Module:InfoboxImage/data' not found.
1979 edition
AuthorHans Schwerdtfeger
LanguageEnglish
SubjectGeometry
GenreMathematics
PublisherUniversity of Toronto Press
Publication date
1962
Publication placeCanada

Script error: No such module "Check for conflicting parameters".

Geometry of Complex Numbers 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 1962 as Volume 13 of the Mathematical Expositions series of the University of Toronto Press. A corrected edition was published in 1979 in the Dover Books on Advanced Mathematics series of Dover Publications (Template:ISBN), including the subtitle Circle Geometry, Moebius Transformation, Non-Euclidean Geometry. The Basic Library List Committee of the Mathematical Association of America has suggested its inclusion in undergraduate mathematics libraries.[1]Template:R/superscript

Topics

The book is divided into three chapters, corresponding to the three parts of its subtitle: circle geometry, Möbius transformations, 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 as the plane of complex numbers, and the use of complex numbers as coordinates to describe geometric objects and their transformations.[1]Template:R/superscript

The chapter on circles covers the analytic geometry of circles in the complex plane.[2]Template:R/superscript It describes the representation of circles by 2×2 Hermitian matrices,[3]Template:R/superscript[4]Template:R/superscript the inversion of circles, stereographic projection, pencils of circles (certain one-parameter families of circles) and their two-parameter analogue, bundles of circles, and the cross-ratio of four complex numbers.[3]Template:R/superscript

The chapter on Möbius transformations is the central part of the book,[4]Template:R/superscript and defines these transformations as the fractional linear transformations of the complex plane (one of several standard ways of defining them).[1]Template:R/superscript It includes material on the classification of these transformations,[2]Template:R/superscript on the characteristic parallelograms of these transformations,[4]Template:R/superscript on the subgroups 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.[3]Template:R/superscript This chapter also briefly discusses applications of Möbius transformations in understanding the projectivities and perspectivities of projective geometry.[1]Template:R/superscript

In the chapter on non-Euclidean geometry, the topics include the Poincaré disk model of the hyperbolic plane, elliptic geometry, spherical geometry, and (in line with Felix Klein's Erlangen program) the transformation groups of these geometries as subgroups of Möbious transformations.[1]Template:R/superscript

This work brings together multiple areas of mathematics, with the intent of broadening the connections between abstract algebra, the theory of complex numbers, the theory of matrices, and geometry.[2]Template:R/superscript[5]Template:R/superscript Reviewer Howard Eves writes that, in its selection of material and its formulation of geometry, the book "largely reflects work of C. Caratheodory and E. Cartan".[6]Template:R/superscript

Audience and reception

Geometry of Complex Numbers is written for advanced undergraduates[6]Template:R/superscript and its many exercises (called "examples") extend the material in its sections rather than merely checking what the reader has learned.[4]Template:R/superscript[6]Template:R/superscript Reviewing the original publication, A. W. Goodman and Howard Eves recommended its use as secondary reading for classes in complex analysis,[3]Template:R/superscript[6]Template:R/superscript and Goodman adds that "every expert in classical function theory should be familiar with this material".[3]Template:R/superscript 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.[2]Template:R/superscript

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.[1]Template:R/superscript 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".[7]Template:R/superscript Nevertheless Hunacek repeats Goodman's and Eves's recommendation for its use "as supplemental reading in a course on complex analysis",[1]Template:R/superscript and Burn concludes that "the republication is welcome".[7]Template:R/superscript

As background on the geometry covered in this book, reviewer R. P. Burn suggests two other books, Modern Geometry: The Straight Line and Circle by C. V. Durell, and Geometry: A Comprehensive Course by Daniel Pedoe.[7]Template:R/superscript

Other books using complex numbers for analytic geometry include Complex Numbers and Geometry by Liang-shin Hahn, or Complex Numbers from A to...Z by Titu Andreescu and Dorin Andrica. However, Geometry of Complex Numbers differs from these books in avoiding elementary constructions in Euclidean geometry and instead applying this approach to higher-level concepts such as circle inversion and non-Euclidean geometry. Another related book, one of a small number that treat the Möbius transformations in as much detail as Geometry of Complex Numbers does, is Visual Complex Analysis by Tristan Needham.[1]Template:R/superscript

References

  1. ^ a b c d e f g h Page Module:Citation/CS1/styles.css has no content.Hunacek, Mark (May 2015), "Review of Geometry of Complex Numbers", MAA Reviews, Mathematical Association of America
  2. ^ a b c d Page Module:Citation/CS1/styles.css has no content.Monk, D. (June 1963), "Review of Geometry of Complex Numbers", Proceedings of the Edinburgh Mathematical Society, 13 (3): 258–259, doi:10.1017/s0013091500010956
  3. ^ a b c d e Page Module:Citation/CS1/styles.css has no content.Goodman, A. W., "Review of Geometry of Complex Numbers", Mathematical Reviews, MR 0133044
  4. ^ a b c d Page Module:Citation/CS1/styles.css has no content.Crowe, D. W. (March 1964), "Review of Geometry of Complex Numbers", Canadian Mathematical Bulletin, 7 (1): 155–156, doi:10.1017/S000843950002693X
  5. ^ Page Module:Citation/CS1/styles.css has no content.Primrose, E. J. F. (May 1963), "Review of Geometry of Complex Numbers", The Mathematical Gazette, 47 (360): 170, doi:10.1017/s0025557200049524, S2CID 125530808
  6. ^ a b c d Page Module:Citation/CS1/styles.css has no content.Eves, Howard (December 1962), "Review of Geometry of Complex Numbers", American Mathematical Monthly, 69 (10): 1021, doi:10.2307/2313225, JSTOR 2313225
  7. ^ a b c Page Module:Citation/CS1/styles.css has no content.Burn, R. P. (March 1981), "Review of Geometry of Complex Numbers", The Mathematical Gazette, 65 (431): 68–69, doi:10.2307/3617961, JSTOR 3617961
Retrieved from "http://cosmopedia.co/index.php?title=Geometry_of_Complex_Numbers&oldid=909870"