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Perspectives on Projective Geometry • Jürgen Richter-Gebert Perspectives on Projective Geometry • Jürgen Richter-Gebert Perspectives on Projective Geometry A Guided Tour Through Real and Complex Geometry 123 Jürgen Richter-Gebert TU München Zentrum Mathematik (M10) LS Geometrie Boltzmannstr. 3 85748 Garching Germany [email protected] ISBN 978-3-642-17285-4 e-ISBN 978-3-642-17286-1 DOI 10.1007/978-3-642-17286-1 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2011921702 Mathematics Subject Classification (2010): 51A05, 51A25, 51M05, 51M10 c Springer-Verlag Berlin Heidelberg 2011 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation,reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: deblik, Berlin Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) About This Book Let no one ignorant of geometry enter here! Entrance to Plato’s academy Once or twice she had peeped into the book her sister was reading, but it had no pictures or conversations in it, “and what is the use of a book,” thought Alice, “without pictures or conversations?” Lewis Carroll, Alice’s Adventures in Wonderland Geometry is the mathematical discipline that deals with the interrelations of objects in the plane, in space, or even in higher dimensions. Practicing geometry comes in very different flavors. More than any other mathematical discipline, the field of geometry ranges from the very concrete and visual to the very abstract and fundamental. In the one extreme, geometry deals with very concrete objects such as points, lines, circles, and planes and studies the interrelations between them. On the other side, geometry is a benchmark for logical rigor, the elegance of axiom systems, and logical chains of proof. There is a third way of thinking about geometry that stands alongside the visual and the logic-based approaches: the algebraic treatment. Here algebraic structures such as vectors, matrices, and equations are used to form a kind of parallel world, in which each geometric object and relation has an algebraic manifestation. In this parallel world, too, the considerations may be very concrete and algorithmic or very abstract and functorial. This book explains v vi About This Book how to treat the fundamental objects of geometry using appropriate algebraic methods. Many of the techniques presented in this book have their roots in the work of the great geometers of the nineteenth century like Pl¨ucker, Grassmann, M¨obius, Klein, and Poincar´e (to mention only a few). The algebraic representations are, however, more by far than a way to express geometric objects by numbers. Very often, finding the right algebraic structure unveils the “true” nature of a geometric concept. It may open new perspectives on and deep insights into matters that seemed elementary at first sight and help to generalize, connect, interpret, visualize, and under- stand. This is what this book is about. Its ultimate aim is to present the beauty that lies in the rich interplay of geometric structures and their al- gebraic counterparts. A warning should be issued right at the beginning. It is relatively easy to transform geometric objects into algebraic ones. For in- stance, points in the plane may be easily represented by their xy-coordinates. However, these “naive” approaches to representing geometric objects are very often not the ones that lead to far-reaching conclusions. Often it is useful to introduce more sophisticated algebraic methods that may seem more abstract at first sight but are ultimately more powerful and elegant. Guided by these more abstract and elegant structures, it may even happen that one is willing to modify the original concept of the geometric first-class citizens (say points or lines) and for instance add some new type of (more abstract) objects. When we talk about homogeneous coordinates, one of the most fundamen- tal concepts of this book, exactly this will happen. We will first see that in the plane very elementary operations such as computing the line through two points and computing the intersection of two lines can be very elegantly expressed if lines as well as points are represented by three-dimensional coor- dinates (where nonzero scalar multiples are identified). Taking a closer look at the relation of planar points and their three-dimensional representing vec- tors, we will observe that certain vectors do not represent points in the real Euclidean plane. This motivates the search for a geometric interpretation of these nonexistent points. It turns out that they may be interpreted as “points that are infinitely far away.” We will then extend the usual two-dimensional plane by these new points at infinity and obtain a richer and more elegant geometric system: the system of projective geometry. In a certain sense this way of thinking is quite similar to the work of chemists at the time when the periodic table of the elements was about to be discovered. Based on the elements known so far, they looked for ways to explain their behavior. At some point they spotted a structure and certain symmetry principles into which all the known chemical elements could be fit- ted (the periodic table of the elements). However, some places in the periodic table did not correspond to known elements. It soon became more reasonable to claim the existence of these undiscovered elements than to give up the inner beauty and explanatory power of the periodic table. Later on, all ele- ments whose existence had been conjectured were indeed discovered. The role About This Book vii of “discovering elements” in mathematics is played by the “interpretation of concepts.” We will meet such situations quite often in this book. The spirit of this book. In a sense, this book is much more about the “how” than about the “what” of geometry. The reader will recognize that very often we will study very simple objects and their relations. Elementary objects such as points, lines, circles, conics, angles, and distances are the real first-class citizens in our approach. Also, the operations we study will be quite elementary: intersecting two lines, intersecting a line and a conic, calculating tangents, etc. Most of these operations may in principle be per- formed with some advanced high-school mathematics. Regardless of that, our emphasis will be on structures that at the same time allow us to express the fundamental objects as well as the operations on them in a most elegant way. So the algebraic representation of an object never stands alone; it is always related to the operations that should be performed with the object. As mentioned before, these advanced representations often lead to new in- sights and broaden our understanding of the seemingly well-known objects. In this respect our philosophy here is very close to Felix Klein’s famous book “Elementary Mathematics From an Advanced Standpoint.” While reading this book, the reader will find that the definitions and con- cepts are more important than the theorems. Very often the same (sometimes elementary) theorems are re-proved with different approaches. A topic that will show up over and over again is the question of how elegantly and gen- eralizably these proofs can be performed with the various methods. I hope that the reader will find these multiple perspectives on related topics a good way to gain a deeper understanding of what is going on. A little history. As mentioned before, many of the techniques in this book go back to what could be called the golden age of geometry, the hundred years between 1790 and 1890. In this period, starting with Gaspard Monge many fundamental geometric concepts were discovered that went far beyond Eu- clid’s Elements (which until then had dominated geometric thinking). Many of these new concepts were intimately related to the underlying algebraic structures. In that period, algebra and geometry underwent a kind of coevo- lution, inspiring and enhancing each other. Projective geometry turned out to be one of the most fundamental structures that at the same time had the most elegant algebraic representation. The concepts of linear and multilinear algebra were developed in close connection to their geometric significance. The development culminated in the revolutionary discovery of what now is called “hyperbolic geometry”: a geometric structure that violates the fifth postulate of Euclid and is still logically on an equal footing with his geom- etry. At its time, this discovery was so revolutionary that C.F. Gauss, who was one of the main protagonists in this discovery and at the same time one of the world’s leading mathematicians, kept it a secret and never published anything on that topic. (We will dedicate several chapters to this topic.) The viii About This Book key to an elegant treatment of hyperbolic geometry again lies in projective approaches. Nowadays, hyperbolic geometry is a well-established, amazingly rich mathematical subject with flourishing connections to many other fields, such as topology, group theory, number theory, combinatorics, numerics, and many more. Unfortunately, in the nineteenth century the field of geometry grew per- haps a bit too fast.
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