EAN: 9781441920638

UNDERGRADUATE TEXTS IN MATHEMATICS THE FOUR P...

Many people think there is only one right way to teach geometry. For two millennia  the right
way was Euclid's way  and it is still good in many respects. But in the 1950s the cry Down with
triangles! was heard in France and new geometry books appeared  packed with linear algebra but
with no diagrams. Was this the new right way  or was the right 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.

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