Four Pillars of Geometry (eBook)

This new textbook demonstrates that geometry can be developed in four fundamentally different ways, and that all should be used if the subject is to be shown in all its splendor. Euclid-style construction and axiomatics seem the best way to start, but linear algebra smooths the later stages by replacing some tortuous arguments by simple calculations. And how can one avoid projective geometry? It not only explains why objects look the way they do; it also explains why geometry is entangled with algebra. Finally, one needs to know that there is not one geometry, but many, and transformation groups are the best way to distinguish between them. In this book, two chapters are devoted to each approach, the first being concrete and introductory, while the second is more abstract.

Geometry, of all subjects, should be about taking different viewpoints, and geometry is unique among mathematical disciplines in its ability to look different from different angles. Some students prefer to visualize, while others prefer to reason or to calculate. Geometry has something for everyone, and students will find themselves building on their strengths at times, and working to overcome weaknesses at other times. This book will be suitable for a second course in geometry and contains more than 100 figures and a large selection of exercises in each chapter.

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Many people think there is only one "e;right"e; way to teach geometry. For two millennia, the "e;right"e; way was Euclid's way, and it is still good in many respects. But in the 1950s the cry "e;Down with triangles!"e; was heard in France and new geometry books appeared, packed with linear algebra but with no diagrams. Was this the new "e;right"e; way, or was the "e;right"e; way something else again, perhaps transformation groups? In this book, I wish to show that geometry can be developed in four fundamentally different ways, and that all should be used if the subject is to be shown in all its splendor. Euclid-style construction and axiomatics seem the best way to start, but linear algebra smooths the later stages by replacing some tortuous arguments by simple calculations. And how can one avoid projective geometry? It not only explains why objects look the way they do; it also explains why geometry is entangled with algebra. Finally, one needs to know that there is not one geometry, but many, and transformation groups are the best way to distinguish between them. Two chapters are devoted to each approach: The ?rst is concrete and introductory, whereas the second is more abstract. Thus, the ?rst chapter on Euclid is about straightedge and compass constructions; the second is about axioms and theorems. The ?rst chapter on linear algebra is about coordinates; the second is about vector spaces and the inner product.