Geometry Books

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Book SynopsisEuclidean and other geometries are distinguished by the transformations that preserve their essential properties. Using linear algebra and transformation groups, this book provides a readable exposition of how these classical geometries are both differentiated and connected. Following Cayley and Klein, the book builds on projective and inversive geometry to construct ''linear'' and ''circular'' geometries, including classical real metric spaces like Euclidean, hyperbolic, elliptic, and spherical, as well as their unitary counterparts. The first part of the book deals with the foundations and general properties of the various kinds of geometries. The latter part studies discrete-geometric structures and their symmetries in various spaces. Written for graduate students, the book includes numerous exercises and covers both classical results and new research in the field. An understanding of analytic geometry, linear algebra, and elementary group theory is assumed.Trade Review'This extremely valuable book tells the story about classical geometries - euclidean, spherical, hyperbolic, elliptic, unitary, affine, projective - and how they all fit together. At the center are geometric transformation groups, both continuous groups such as isometry or collineation groups, and their discrete subgroups occurring as symmetry groups of polytopes, tessellations, or patterns, including reflection groups. I highly recommend the book!' Egon Schulte, Northeastern University, Massachusetts'This is a book written with a passion for geometry, for complete lists, for consistent notation, for telling the history of a concept, and a passion to give an insight into a situation before going into the details.' Erich W. Ellers, zbMATHTable of ContentsIntroduction; 1. Homogenous spaces; 2. Linear geometries; 3. Circular geometries; 4. Real collineation groups; 5. Equiareal collineations; 6. Real isometry groups; 7. Complex spaces; 8. Complex collineation groups; 9. Circularities and concatenations; 10. Unitary isometry groups; 11. Finite symmetry groups; 12. Euclidean symmetry groups; 13. Hyperbolic coxeter groups; 14. Modular transformations; 15. Quaternionic modular groups.

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Book SynopsisStraightedge and compass.- Euclid's approach to geometry.- Coordinates.- Vectors and Euclidean spaces.- Perspective.- Projective planes.- Transformations.- Non-Euclidean geometry.Trade ReviewFrom the reviews:"This is an introductory book on geometry, easy to read, written in an engaging style. The author’s goal is … to increase one’s overall understanding and appreciation of the subject. … Along the way, he presents elegant proofs of well-known theorems … . The advantage of the author’s approach is clear: in a short space he gives a brief introduction to many sides of geometry and includes many beautiful results, each explained from a perspective that makes it easy to understand." (Robin Hartshorne, SIAM Review, Vol. 48 (2), 2006)"The pillars of the title are … Euclidean construction and axioms, coordinates and vectors, projective geometry, and transformations and non-Euclidean geometry. … The writing style is both student-friendly and deeply informed. The pleasing brevity of the book … makes the book especially suitable as an instruction to geometry for the large and critically important population of undergraduate mathematics majors … . Each chapter concludes with a well-written discussion section that combines history with glances at further results. There is a good selection of thought-provoking exercises." (R. J. Bumcrot, Mathematical Reviews, Issue 2006 e)"The author acts on the assumption of four approaches to geometry: The axiomatic way, using linear Algebra, projective geometry and transformation groups. … Each of the chapters closes with a discussion giving hints on further aspects and historical remarks. … The book can be recommended to be used in undergraduate courses on geometry … ." (F. Manhart, Internationale Mathematische Nachrichten, Issue 203, 2006)"Any new mathematics textbook by John Stillwell is worth a serious look. Stillwell is the prolific author of more than half a dozen textbooks … . I would not hesitate to recommend this text to any professor teaching a course in geometry who is more interested in providing a rapid survey of topics rather than an in-depth, semester-long, examination of any particular one." (Mark Hunacek, The Mathematical Gazette, Vol. 91 (521), 2007)"The title refers to four different approaches to elementary geometry which according to the author only together show this field in all its splendor: via straightedge and compass constructions, linear algebra, projective geometry and transformation groups. … the book can be recommended warmly to undergraduates to get in touch with geometric thinking." (G. Kowol, Monatshefte für Mathematik, Vol. 150 (3), 2007)"This book presents a tour on various approaches to a notion of geometry and the relationship between these approaches. … The book shows clearly how useful it is to use various tools in a description of basic geometrical questions to find the simplest and the most intuitive arguments for different problems. The book is a very useful source of ideas for high school teachers." (EMS Newsletter, March, 2007)“The four pillars of geometry approaches geometry in four different ways, devoting two chapters to each, the first chapter being concrete and introductory, the second more abstract. … The content is quite elementary and is based on lectures given by the author at the University of San Francisco in 2004. … The book of Stillwell is a very good first introduction to geometry especially for the axiomatic and the projective point of view.” (Yves Félix, Bulletin of the Belgian Mathematical Society, Vol. 15 (1), 2008)Table of ContentsStraightedge and compass.- Euclid’s approach to geometry.- Coordinates.- Vectors and Euclidean spaces.- Perspective.- Projective planes.- Transformations.- Non-Euclidean geometry.

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Book Synopsis1 Axiomatic Systems and Finite Geometries.- 2 Non-Euclidean Geometry.- 3 Geometric Transformations of the Euclidean Plane.- 4 Projective Geometry.- 5 Chaos to Symmetry: An Introduction to Fractal Geometry.- Appendices.- B Hilbert's Axioms for Plane Geometry.- C Birkhoff's Postulates for Euclidean Plane Geometry.- D The SMSG Postulates for Euclidean Geometry.- E Some SMSG Definitions for Euclidean Geometry.- F The ASA Theorem.- References.Trade ReviewSecond Edition J. N. Cederberg A Course in Modern Geometries "Cederberg’s book has the virtue of exceptionally clear exposition and at the same time, it is brief enough not to exhaust one’s patience . . . I have seen hundreds of college-level texts on geometry and this is one of the handful I like the most." —THE UMAP JOURNALTable of Contents1 Axiomatic Systems and Finite Geometries.- 2 Non-Euclidean Geometry.- 3 Geometric Transformations of the Euclidean Plane.- 4 Projective Geometry.- 5 Chaos to Symmetry: An Introduction to Fractal Geometry.- Appendices.- B Hilbert’s Axioms for Plane Geometry.- C Birkhoff’s Postulates for Euclidean Plane Geometry.- D The SMSG Postulates for Euclidean Geometry.- E Some SMSG Definitions for Euclidean Geometry.- F The ASA Theorem.- References.

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