Geometry of Complex Numbers: Difference between revisions

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{{Short description|Maths textbook}}

{{about|the book|the area of mathematics|complex geometry}}

{{italic title}}

[[File:Geometry of Complex Numbers.jpg|thumb|1979 edition]]

'''''Geometry of Complex Numbers: Circle Geometry, Moebius Transformation, Non-Euclidean Geometry''''' is an undergraduate textbook on [[geometry]], whose topics include [[circle]]s, 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]] ({{ISBN|0-486-63830-8}}), 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.{{r|hunacek}}

==Topics==

The book is divided into three chapters, corresponding to the three parts of its subtitle: circle geometry, [[Möbius transformationstransformation]]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]] as the [[Complex plane|plane of complex number]]s, and the use of [[complex numbersnumber]]s as coordinates to describe geometric objects and their transformations.{{r|hunacek}}

The chapter on circles covers the [[analytic geometry]] of circles in the complex plane.{{r|monk}} It describes the representation of circles by <math>2\times 2</math> [[Hermitian matrix|Hermitian matrices]],{{r|goodman|crowe}} the [[Inversive geometry|inversion of circles]], [[stereographic projection]], [[Apollonian circles|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.{{r|goodman}}

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==Audience and reception==

''Geometry of Complex Numbers'' is written for advanced undergraduates{{r|eves}}

and its many exercises[[exercise (mathematics)|exercise]]s (called "examples") extend the material in its sections rather than merely checking what the reader has learned.{{r|crowe|eves}} Reviewing the original publication, A. W. Goodman and [[Howard Eves]] recommended its use as secondary reading for classes in [[complex analysis]],{{r|goodman|eves}} and Goodman adds that "every expert in classical function theory should be familiar with this material".{{r|goodman}} 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.{{r|monk}}

By the time of itshis 19792015 republicationreview, reviewer 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.{{r|hunacek}} Reviewer R. P. Burn echoesshares 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".{{r|burn}} Nevertheless Hunacek repeats Goodman's and EveEves's recommendation for its use "as supplemental reading in a course on complex analysis",{{r|hunacek}} and Burn concludes that "the republication is welcome".{{r|burn}}

==Related reading==

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]].{{r|burn}}

Other books onusing thecomplex geometrynumbers offor complex[[analytic numbersgeometry]] 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]].{{r|hunacek}}

==References==

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| jstor = 3617961

| pages = 68–69

| title = noneReview of ''Geometry of Complex Numbers''

| volume = 65}}</ref>

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| doi = 10.1017/S000843950002693X

| issue = 1

| journal = [[Canadian Mathematical Bulletin]]

| pages = 155–156

| title = noneReview of ''Geometry of Complex Numbers''

| volume = 7}}</ref>| doi-access = free

}}</ref>

<ref name=eves>{{citation

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| jstor = 2313225

| page = 1021

| title = noneReview of ''Geometry of Complex Numbers''

| volume = 69}}</ref>

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| journal = [[Mathematical Reviews]]

| mr = 0133044

| title = noneReview of ''Geometry of Complex Numbers''}}</ref>

<ref name=hunacek>{{citation

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| journal = MAA Reviews

| publisher = [[Mathematical Association of America]]

| 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}}</ref>

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| journal = Proceedings of the Edinburgh Mathematical Society

| pages = 258–259

| title = noneReview of ''Geometry of Complex Numbers''

| volume = 13}}</ref>| doi-access = free

}}</ref>

<ref name=primrose>{{citation

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| issue = 360

| journal = [[The Mathematical Gazette]]

| pages = 170–170170

| title = noneReview of ''Geometry of Complex Numbers''

| volume = 47}}</ref>| s2cid = 125530808

}}</ref>

}}

==External links==

*''[https://archive.org/details/geometryofcomple0000schw Geometry of Complex Numbers]'' (1979 edition) at the [[Internet Archive]]

[[Category:Circles]]