xiv Preface mathematical culture, but certainly not a violation of what happens in most sciences. I see nothing wrong with this, and, indeed, it might inspire some students to try to learn the proofs of those assumed results. Other chapters, such as Chapter 2 on Regular Polyhedra, present a beautiful piece of mathematics that, while accessible to undergraduates, seldom finds its way into the undergraduate curriculum. Since this is shown using the Euler characteristic, I think you can still say it gives a relationship between different areas. In fact, the theme of connections is renewed when the Euler characteristic is used to discuss map coloring and tessellations of the plane. Chapter 1 on Trisecting Angles is, for many, the quintessential instance of the combination of two areas of mathematics algebra and geometry. This is frequently seen in standard courses on algebra but only in an ab- breviated manner. I was quite surprised that the solution of the trisection problem was so accessible. I had been raised to think of it as a consequence of Galois theory. Anyone who knows Galois theory will certainly see its ele- ments in Chapter 1 but nothing close to its full force is required. (Actually Chapter 1 as well as Chapter 2 require minimal mathematical sophistication and background and in a simplified form could be made accessible to bright freshmen. It is, at least partly, why I placed them at the beginning of the book.) Another topic, like Chapter 4 on the Spectral Theorem, takes a result in the undergraduate curriculum and looks at it from several different points of view. In this case we look at the Spectral Theorem as the diagonalization of hermitian matrices, as a characterization of the unitary equivalence classes of hermitian linear transformations, and as the Principal Axis Theorem for quadratic forms. We then return to the connections theme and apply this result to a study of the critical points of a function of several variables. On the other hand, Chapter 6 on Modules stays within algebra but shows how a discussion of an abstract concept such as a module leads to more concrete results like the Spectral Theorem, Jordan forms for matrices, and the structure of finitely generated abelian groups. There are many other connections in mathematics that could be made accessible to the same audience. For example, many things in applied math- ematics deserve a place in this book but I lack the expertise and perspective to do them justice and be anything more than a scribe. I decided to leave them until their champion appears. ***** No chapter by itself contains enough material for a single course. I could see a course using this book to cover between two and four chapters, depend- ing on the backgrounds of the students and the chapters chosen. If any of Chapters 4 through 6 are covered, then the students have to understand
Previous Page Next Page