This book contains selected topics from the history of geometry, with "modern" proofs of some of the results, as well as a fully modern treatment of selected basic issues in geometry. The book aims at future teachers of mathematics. All too often the geometry which goes into the syllabus for teacher-students presents the material as pedantic and formalistic, suppressing its dynamic character and its role as part of the foundation for our common cultural heritage. The motivation for the book is to open up these aspects of the field. Another motivation is to provide an invitation to mathematics in general. It is an unfortunate fact that today, at a time when mathematics and knowledge of mathematics are more important than ever, phrases like math avoidance and math anxiety are very much in the public vocabulary. An important task is seriously attempting to heal these ills. Thus the book also aims at an informed public, interested in making a new beginning in math.
Table of Contents:
Part I. A Cultural Heritage: Chapter 1. Early Beginnings 1.1 Prehistory 1.2 Geometry in the New Stone Age 1.3 Early Mathematics and Ethnomathematics Chapter 2. The Great River Civilizations 2.1 Civilizations long dead -- and yet alive 2.2 Birth of Geometry as we know it 2.3 Geometry in the Land of the Pharaoh 2.4 Babylonian Geometry Chapter 3. Greek and Hellenic Geometry 3.1 Early Greek Geometry. Thales of Miletus 3.2 The story of Pythagoras and the Pythagoreans 3.3 The Geometry of the Pythagoreans 3.4 The Discovery of Irrational Numbers 3.5 Origin of the Classical Problems 3.6 Constructions by Compass and Straightedge 3.7 Squaring the Circle 3.8 Doubling the Cube 3.9 Trisecting any Angle 3.10 Plato and the Platonic Solids 3.11 Archytas and the Doubling of the Cube Chapter 4. Geometry in the Hellenistic Era 4.1 Euclid and Euclids Elements 4.2 The Books of Euclids Elements 4.3 The Roman Empire 4.4 Archimedes 4.5 Erathostenes and the Duplication of the Cube 4.6 Nicomedes and his Conchoid 4.7 Apollonius and the Conic Sections 4.8 Caesar and the End of the Republic in Rome 4.9 The Murder of Hypatia 4.10 The Decline and Fall of the Roman Empire Chapter 5. The Geometry of Yesterday and Today 5.1 The Dark Middle Ages 5.2 Geometry Reawakening: A new Dawn in Europe 5.3 Elementary Geometry and Higher Geometry 5.4 Desargues and the two Pascals 5.5 Descartes and Analytic Geometry 5.6 Geometry in the 18th. Century 5.6.1 Cramers theorem 5.7 Some Features of Modern Geometry Chapter 6. Geometry and the Real World 6.1 Mathematics and Predicting Catastrophes 6.2 Catastrophe Theory 6.3 Geometric Shapes in Nature 6.4 Fractal Structures in Nature Part II. Introduction to Geometry Chapter 7. Axiomatic Geometry 7.1 The Postulates of Euclid and Hilbert's Explanation 7.2 Non-Euclidian Geometry 7.3 Logic and Intuitive Set Theory 7.4 Axioms, Axiomatic Theories and Models 7.5 General Theory of Axiomatic Systems Chapter 8. Axiomatic Projective Geometry 8.1 Plane Projective Geometry 8.2 An Unsolved Geometric Problem 8.3 The Real Projective Plane Chapter 9. Models for non-Euclidian Geometry 9.1 Three Types of Geometry 9.2 Hyperbolic Geometry 9.3 Elliptic Geometry 9.4 Euclidian and non-Euclidian Geometry in Space 9.5 Riemannian Geometry Chapter 10. Making Things Precise 10.1 Relations and Their Uses 10.2 Identification of Points 10.3 Our Number System 10.3.1 The integers 10.3.2 The rational numbers 10.3.3 The real numbers 10.3.4 The complex numbers Chapter 11. Projective Space 11.1 Coordinates in the Projective Plane 11.2 Projective n-Space 11.3 Affine and Projective Coordinate Systems Chapter 12. Geometry in the Affine and the Projective plane 12.1 The Theorem of Desargues 12.2 Duality for the projective plane 12.3 Naive Definition and First Examples of Affine Plane Curves 12.4 Straight Lines 12.5 Conic Sections in the Affine plane 12.6 Constructing Points on Conic Sections by Compass and Straightedge 12.7 Further Properties of Conic Sections 12.8 Conic Sections in the Projective Plane 12.9 The Theorems of Pappus and Pascal Chapter 13. Algebraic Curves of Higher Degrees in the Affine Plane 13.1 The Cubical Curves in the affine plane 13.1.1 Cubical parabolas 13.1.2 Semi cubical parabolas 13.1.3 Degenerate curves and degeneration of a family of curves 13.1.4 Folium of Descartes 13.1.5 Elliptic curves 13.1.6 Trisectrix of MacLaurin and the Clover leaf curve 13.2 Curves of Degree Higher than Three 13.3 Affine Algebraic Curves 13.4 Singularities and Multiplicities 13.5 Tangency Chapter 14. Higher Geometry in the Projective plane 14.1 Projective Curves 14.2 Projective Closure and Affine Restriction 14.3 Smooth and Singular Points on Affine and Projective Curves 14.4 The Tangent of a Projective Curve 14.5 Projective Equivalence 14.6 Asymptotes 14.7 General Conchoids 14.8 The Dual Curve 14.9 The Dual of Pappus'Theorem 14.10 Pascals Mysterium Hexagrammicum Chapter 15. Sharpening the Sword of Algebra 15.1 On Rational Polynomials 15.2 The Minimal Polynomial 15.3 The Euclidian Algorithm 15.4 Number Fields and Field Extensions 15.5 More on Field Extensions Chapter 16. Constructions with Straightedge and Compass 16.1 Review of Legal Constructions 16.2 Constructible Points 16.3 What is Possible? 16.4 Trisecting an Angle 16.5 Doubling the Cube 16.6 Squaring the Circle 16.7 Regular Polygons 16.8 Constructions by Folding 16.9 Concluding Remarks on Constructions Chapter 17. Fractal Geometry 17.1 Fractals and their Dimensions 17.2 The von Koch Snowflake Curve 17.3 Fractal Shapes in Nature 17.4 The Sierpinski Triangles 17.5 A Cantor Set Chapter 18. Catastrophe Theory 18.1 The Cusp Catastrophe: Geometry of a Cubic Surface 18.2 Rudiments of Control Theory
Review :
From the reviews: .,." The book can be recommended as an excellent textbook for several possible courses (such as Historical topics in geometry, or Introduction to modern geometry). It can be recommended to historians of mathematics, and to professional mathematicians, teachers and students wanting to understand the origins of modern geometry."
European Mathematical Society Newsletter, p. 30, June 03
.,." As the author states, the book aims at working resp. future teachers but also at the "informed public," and this readership will really profit from it."
G.Kawohl, Monatshefte fA1/4r Mathematik 140, Heft 4 (2003)
"This book, or at least the first half of it, should really be compulsory reading for everyone who ever teaches mathematics. It contains a wealth of (his)stories that would make a course in mathematics, in particular in geometry, interesting and exciting. ...In conclusion I would say that the book is really about mathematical culture; interesting things that every mathematical teacher should know are well explained. ..."
Hendrik Van Maldeghem, Belgian Mathematical Society - Simon Stevin Bulletin, 2003, p. 159-160
"This is a wonderful book based on lectures on geometry given by the author to undergraduate students at the University of Bergen, Norway. ...
Summarizing: I think that this book is a beautiful treatment of geometry. ... I strongly recommend this book to instructors who want to find a textbook for a comprehensive undergraduate geometry course, or for lecturers considering a presentation on geometry for the general public. The book will be very useful also for high-school and undergraduate interested in studying geometry. Areceptive and informed general public interested in mathematics will find in the first (and, maybe, second) part of the book a very readable and fun approach to geometry."
Mihaela Poplicher, The MAA ONLINE book review column, for complete review see http: //www.maa.org/reviews/holmegeom.html
"Holme .. offers a highly readable survey of geometry. ..."
D.P.Turner, Choice October 2002
.,." To my mind this book delivers both a broad survey of the history of geometry and - at a few well-chosen points - a deeper insight. It impressively works out the importance of geometry as one of the gresat achievements of human culture. The way this book is written makes it capable of conveying this idea to many culturally interested people who do not shy away from getting involved into mathematics. I especially recommend this book to students and teachers in mathematics and related fields."
J.Lang, Internationale Mathematische Nachrichten 2002, Vol. 56, Issue 191
.,." Im Gegensatz zu vielen anderen GeometriebA1/4chern wird in dem vorliegenden Band kein Aufgebot an speziellen Bezeichnungen verwandt. So kann man eigentlich in jedem Kapitel anfangen zu lesen, bzw. das Buch auch als Nachschlagewerk verwenden. Damit wird es auch dem vom Autor sich selbst gesetzten Ziel gerecht, die Leser nicht durch eine pedantische und formalistische Presentation von der Dynamik und SchAnheit der Geometrie abzulenken. Wie Holme in seiner Einleitung schreibt, will er insbesondere unseren kA1/4nftigen Lehrern und der Mathematik interessierten Gemeinschaft in seinem Buch ein umfangreiches und abgerundetes Bild der Geometrie prAsentieren. Gleichzeitig soll die Monographie auch als GrundlagefA1/4r Vorlesungen A1/4ber Geometrie dienen. Meiner Ansicht nach ist der Autor seinem Ziel gerecht geworden."
Ch.Birkenhake, Jahresberichte der DMV, JB 106. Band (2004), Heft 3
.,." Wer sich einen im historischen Kontext verankerten Aoeberblick A1/4ber die Geometrie verschaffen mAchte, ist mit diesem Buch sehr gut bedient. ... es kann durchaus als Klassiker gelten."
H.Walser, Elemente der Mathematik 2004, Vol. 59
"This book discusses central themes from the history of classical and modern geometry a ] . The book can be recommended as an excellent textbook for several possible courses (such as Historical topics in geometry, or Introduction to modern geometry). It can be recommended to historians of mathematics, and to professional mathematicians, teachers and students wanting to understand the origins of modern geometry." (EMS, June 2003)
"Audun Holme has written an elementary book on geometry a ] aimed primarily at American mathematics teachers in both elementary and high school, as well as at 'an informed public interested in making a new beginning in mathematics.' a ] As a popular account of one area of mathematics, [Holmes] has some attractive new features. a ] it provides a sense of the drama of mathematical developments and their role in world culture. This could be a valuable contribution." (H. Wu, Notices of the AMS, Vol. 51 (5), 2004)
"This is an introduction in selected parts of geometry together with a detailed view on the importance of geometry in the history of mathematics. a ] As the author states, the book aims at working resp. future teachers but also at the 'informed public', and this readership will really profit a lot from it." (G. Kowol, MonatsheftefA1/4r Mathematik, Vol. 140 (4), 2003)
"Holme (Univ. of Bergen, Norway) offers a highly readable survey of geometry. a ] Readability is enhanced by the inclusion of more than 160 figures. a ] Upper-division undergraduates through faculty." (D. P. Turner, CHOICE, October 2002)
"This book a ] should really be compulsory reading for everyone who ever teaches mathematics. It contains a wealth of (his)stories that would make a course in mathematics, in particular in geometry, interesting and exciting. a ] I must admit I learned a lot reading these chapters. a ] In conclusion I would say that the book is really about mathematical culture; interesting things that every mathematical teacher should know are well explained." (Hendrik van Maldeghem, Belgian Mathematical Society, 2003)
"This is a wonderful book based on lectures on geometry given by the author to undergraduate students at the University of Bergen, Norway. a ] I think that this book is a beautiful treatment of geometry. It might very well serve as textbook for geometry courses a ] . I strongly recommend this book to instructors who want to find a textbook for a comprehensive undergraduate geometry course a ] . The book will be very useful also for high-school and undergraduate interested in studying geometry." (Mihaela Poplicher, MAA, 25.01.2003)
"To my mind this book delivers both a broad survey of the history of geometry and a "at a few well chosen points a" a deeper insight. It impressively works out the importance of geometry as one of the great achievements of human culture. The way this book is written makes it capable of conveying this idea to many culturally interested people who do not shy away from getting involved into mathematics. I especially recommend this book to students and teachers in mathematics and related fields." (J. Lang, Internationale Mathematische Nachrichten, Vol. 56 (191), 2002)
From the reviews: ..." The book can be recommended as an excellent textbook for several possible courses (such as Historical topics in geometry, or Introduction to modern geometry). It can be recommended to historians of mathematics, and to professional mathematicians, teachers and students wanting to understand the origins of modern geometry."
European Mathematical Society Newsletter, p. 30, June 03
..." As the author states, the book aims at working resp. future teachers but also at the "informed public," and this readership will really profit from it."
G.Kawohl, Monatshefte fr Mathematik 140, Heft 4 (2003)
"This book, or at least the first half of it, should really be compulsory reading for everyone who ever teaches mathematics. It contains a wealth of (his)stories that would make a course in mathematics, in particular in geometry, interesting and exciting. ...In conclusion I would say that the book is really about mathematical culture; interesting things that every mathematical teacher should know are well explained. ..."
Hendrik Van Maldeghem, Belgian Mathematical Society - Simon Stevin Bulletin, 2003, p. 159-160
"This is a wonderful book based on lectures on geometry given by the author to undergraduate students at the University of Bergen, Norway. ...
Summarizing: I think that this book is a beautiful treatment of geometry. ... I strongly recommend this book to instructors who want to find a textbook for a comprehensive undergraduate geometry course, or for lecturers considering a presentation on geometry for the general public. The book will be very useful also for high-school and undergraduate interested in studying geometry. A receptive andinformed general public interested in mathematics will find in the first (and, maybe, second) part of the book a very readable and fun approach to geometry."
Mihaela Poplicher, The MAA ONLINE book review column, for complete review see http: //www.maa.org/reviews/holmegeom.html
"Holme .. offers a highly readable survey of geometry. ..."
D.P.Turner, Choice October 2002
..." To my mind this book delivers both a broad survey of the history of geometry and - at a few well-chosen points - a deeper insight. It impressively works out the importance of geometry as one of the gresat achievements of human culture. The way this book is written makes it capable of conveying this idea to many culturally interested people who do not shy away from getting involved into mathematics. I especially recommend this book to students and teachers in mathematics and related fields."
J.Lang, Internationale Mathematische Nachrichten 2002, Vol. 56, Issue 191
..." Im Gegensatz zu vielen anderen Geometriebchern wird in dem vorliegenden Band kein Aufgebot an speziellen Bezeichnungen verwandt. So kann man eigentlich in jedem Kapitel anfangen zu lesen, bzw. das Buch auch als Nachschlagewerk verwenden. Damit wird es auch dem vom Autor sich selbst gesetzten Ziel gerecht, die Leser nicht durch eine pedantische und formalistische Presentation von der Dynamik und Schnheit der Geometrie abzulenken. Wie Holme in seiner Einleitung schreibt, will er insbesondere unseren knftigen Lehrern und der Mathematik interessierten Gemeinschaft in seinem Buch ein umfangreiches und abgerundetes Bild der Geometrie prsentieren. Gleichzeitig soll die Monographie auch als Grundlage fr Vorlesungen berGeometrie dienen. Meiner Ansicht nach ist der Autor seinem Ziel gerecht geworden."
Ch.Birkenhake, Jahresberichte der DMV, JB 106. Band (2004), Heft 3
..." Wer sich einen im historischen Kontext verankerten berblick ber die Geometrie verschaffen mchte, ist mit diesem Buch sehr gut bedient. ... es kann durchaus als Klassiker gelten."
H.Walser, Elemente der Mathematik 2004, Vol. 59
"This book discusses central themes from the history of classical and modern geometry . The book can be recommended as an excellent textbook for several possible courses (such as Historical topics in geometry, or Introduction to modern geometry). It can be recommended to historians of mathematics, and to professional mathematicians, teachers and students wanting to understand the origins of modern geometry." (EMS, June 2003)
"Audun Holme has written an elementary book on geometry aimed primarily at American mathematics teachers in both elementary and high school, as well as at 'an informed public interested in making a new beginning in mathematics.' As a popular account of one area of mathematics, ÝHolmes¨ has some attractive new features. it provides a sense of the drama of mathematical developments and their role in world culture. This could be a valuable contribution." (H. Wu, Notices of the AMS, Vol. 51 (5), 2004)
"This is an introduction in selected parts of geometry together with a detailed view on the importance of geometry in the history of mathematics. As the author states, the book aims at working resp. future teachers but also at the 'informed public', and this readership will really profit a lot from it." (G. Kowol, Monatshefte fr Mathematik, Vol. 140 (4), 2003)
"Holme(Univ. of Bergen, Norway) offers a highly readable survey of geometry. Readability is enhanced by the inclusion of more than 160 figures. Upper-division undergraduates through faculty." (D. P. Turner, CHOICE, October 2002)
"This book should really be compulsory reading for everyone who ever teaches mathematics. It contains a wealth of (his)stories that would make a course in mathematics, in particular in geometry, interesting and exciting. I must admit I learned a lot reading these chapters. In conclusion I would say that the book is really about mathematical culture; interesting things that every mathematical teacher should know are well explained." (Hendrik van Maldeghem, Belgian Mathematical Society, 2003)
"This is a wonderful book based on lectures on geometry given by the author to undergraduate students at the University of Bergen, Norway. I think that this book is a beautiful treatment of geometry. It might very well serve as textbook for geometry courses . I strongly recommend this book to instructors who want to find a textbook for a comprehensive undergraduate geometry course . The book will be very useful also for high-school and undergraduate interested in studying geometry." (Mihaela Poplicher, MAA, 25.01.2003)
"To my mind this book delivers both a broad survey of the history of geometry and at a few well chosen points a deeper insight. It impressively works out the importance of geometry as one of the great achievements of human culture. The way this book is written makes it capable of conveying this idea to many culturally interested people who do not shy away from getting involved into mathematics. I especially recommend this book to students and teachers inmathematics and related fields." (J. Lang, Internationale Mathematische Nachrichten, Vol. 56 (191), 2002)
