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Geometries and Transformations


  • Date Published: June 2018
  • availability: In stock
  • format: Hardback
  • isbn: 9781107103405

£ 62.99

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About the Authors
  • Euclidean 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.

    • Provides a unified framework for describing Euclidean and non-Euclidean geometries
    • Demonstrates the interconnectedness of different geometries using linear algebra
    • Includes both classical results and contemporary research, as well as numerous exercises
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    Reviews & endorsements

    '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, zbMATH

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    Product details

    • Date Published: June 2018
    • format: Hardback
    • isbn: 9781107103405
    • length: 452 pages
    • dimensions: 241 x 162 x 27 mm
    • weight: 0.81kg
    • availability: In stock
  • Table of Contents

    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.

  • Author

    Norman W. Johnson, Wheaton College, Massachusetts
    Norman W. Johnson was Professor Emeritus of Mathematics at Wheaton College, Massachusetts. Johnson authored and co-authored numerous journal articles on geometry and algebra, and his 1966 paper 'Convex Polyhedra with Regular Faces' enumerated what have come to be called the Johnson solids. He was a frequent participant in international conferences and a member of the American Mathematical Society and the Mathematical Association of America.

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