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Publisher:

Springer Verlag

Publication Date:

1998

Number of Pages:

291

Format:

Hardcover

Series:

Universitext

Price:

69.95

ISBN:

978-0387984377

Category:

General

[Reviewed by , on ]

Elena Anne Marchisotto

02/11/1999

This source book for models of basic incidence geometries, including some found in the theories of codes, groups, graphs, and statistical designs, can serve as a good supplementary textbook for undergraduate and beginning graduate geometry courses. The author first treats finite geometries with a small number of points, and then topological geometries whose point sets are surfaces and whose lines are curves drawn on those surfaces. The book has many pictures, almost on every page. Most of these were created with the program *xfig*. There are also stereograms (which can be viewed with both the parallel and the cross-eyed techniques), produced with the programs *MetaPost* and *Mathematica*, and color inserts made with the programs *Adobe Illustrator* and *Adobe Photoshop*.

The Fano plane plays an important role in part one of the book. The author makes excellent use of this smallest projective plane to demonstrate how to interpret pictures and draw appropriate conclusions from them. He uses it to illustrate the numerous "construction principles" he develops for drawing pictures, noting the different kinds of objects and substructures that can be represented by diagrams.

If we trace the author's path a bit using the Fano plane, we can acquire a sense of how the book unfolds. He begins with axioms for a projective plane and shows how the Fano plane satisfies them. Then he explores some of the characteristics of the Fano plane, noting similarities with other geometries that occur later in the book. For example, since the Fano geometry is homogeneous, as are most of the geometries he treats, there is an automorphism that carries one point to the other. So even if line pencils of the geometry appear to be different in pictures (which he demonstrates), they are actually indistinguishable. We see how a rotation of 120 degrees around the middle point of the traditional picture of the Fano plane corresponds to an automorphism of order 3. Reflections in the 3 symmetry axes of the diagram correspond to involutory automorphisms and so all the symmetries of the regular triangle that underlie the diagram correspond to the automorphisms of the Fano geometry. We can then generalize to a geometry that admits an automorphism of order m and use the author's first "construction principle" for modeling such a geometry: try to draw a picture of the geometry on a regular m-gon, such that all or most of the symmetries of the m-gon translate into automorphisms of the geometry. A model of the Fano plane on the tetrahedron introduces another construction priniciple: modeling geometries on highly regular spatial objects like the Platonic solids. The author continues to develop construction principles throughout part one. He discusses various techniques for viewing stereographic images, and explores a veritable wealth of pictures including planar and spatial models of three-dimensional projective space with their associated symmetric designs, models of projective planes and biplanes of various orders, models and star diagrams of affine planes, Benz planes and generalized polygons. There are beautiful color inserts of models and a nice summary of how to build them. Part one closes with some games and puzzles that would make interesting activities for a mathematics club.

Part two of the book treats infinite geometries. The author gives an overview of the most important kinds of geometries on surfaces that have topological circles as lines. He uses flat affine and projective planes to explore geometries on surfaces, constructs different kinds of flat circle planes, and describes various subgeometries and Lie geometries associated with such circle planes of order 3.

*A Geometric Picture Book* effectively portrays what many of us find beautiful in geometric objects. I recommend it highly for instructors and students of geometry, and for a general audience with some level of mathematical sophistication.

Elena Anne Marchisotto (emarchisotto@csun.edu) is professor of mathematics at California State University, Northridge. Her primary research interest is foundations of geometry. She is currently writing (with Francisco Rodriguez-Consuegra) a book entitled *Mario Pieri's Philosophy of Geometry*.

**I. FINITE GEOMETRIES**.- Introduction via the Fano plane.- Designs.- Configurations.- Generalized Quadrangles.- The Smallest Three-Dimensional Projective Space.- The Projective Plane of Order 3.- The Projective Plane of Order 4. - The Projective Plane of Order 5. - Star Gazing in Affine Planes up to Order 8. - Biplanes.- Semibiplanes.- The Smallest Benz Planes.- Generalized Polygons.- Color Pictures and Building the Models.- Some Fun Games and Puzzles.- **II. GEOMETRIES ON** **SURFACES**.- Introduction via Flat Affine Planes.- Flat Circle Planes - An Overview.- Flat Projective Planes.- Spherical Circle Planes.- Cylindrical Circle Planes of Rank 3.- Toroidal Circle Planes.- Appendix A: Models on Regular Solids.- Appendx B: Mirror Technique Stereograms.

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