Foundations of Incidence Geometry Projective and Polar Spaces
Incidence geometry is a central part of modern mathematics that has an impressive tradition. The main topics of incidence geometry are projective and affine geometry and, in more recent times, the theory of buildings and polar spaces.Embedded into th
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Johannes Ueberberg
Foundations of Incidence Geometry Projective and Polar Spaces
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Johannes Ueberberg SRC Security Research & Consulting GmbH Graurheindorfer Strasse 149A 53117 Bonn Germany [email protected]
ISSN 1439-7382 ISBN 978-3-642-20971-0 e-ISBN 978-3-642-20972-7 DOI 10.1007/978-3-642-20972-7 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2011936532 Mathematics Subject Classification (2010): 51E24, 51A50, 51A05 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)
To Monika, Vera and Philipp
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Preface
Incidence geometry has a long tradition starting with Euclid [25] and his predecessors and is at the same time a central part of modern mathematics. The main progress in incidence geometry until about 1900 are the axioms of affine and projective geometries and the two fundamental theorems of affine and projective geometries as described in David Hilbert’s “Grundlagen der Geometrie” [27]. The specific value of the fundamental theorems is due to the fact that a few geometric axioms give rise to algebraic structures as vector spaces and groups. Starting from these results, modern incidence geometry has been developed under the strong influence of Jacques Tits and Francis Buekenhout. Jacques Tits developed the fascinating theory of buildings and classified the buildings of spherical types. The main representatives of buildings of spherical type are projective spaces and polar spaces. Starting from the theory of buildings, Francis Buekenhout laid the foundations for modern incidence geometry, also called diagram geometry or Buekenhout– Tits-geometry, and contributed a number of outstanding results such as the classification of quadratic sets or (together with Ernest Shult) the Theorem of Buekenhout–Shult about polar spaces. The objective of this book is to give a comprehensible insight into this fascinating and complex matter. The book provides an introduction into affine and projective geometry, into polar spaces and into quadratic sets and quadrics.
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