Mathematical Reflections: In a Room with Many Mirrors

by Peter Hilton, Derek Holton, Jean Pedersen

A relaxed and informal presentation conveying the joy of mathematical discovery and insight. Frequent questions lead readers to see mathematics as an accessible world of thought, where understanding can turn opaque formulae into beautiful and meaningful ideas. The text presents eight topics that illustrate the unity of mathematical thought as well as the diversity of mathematical ideas. Drawn from both "pure" and "applied" mathematics, they include: spirals in nature and in mathematics; the modern topic of fractals and the ancient topic of Fibonacci numbers; Pascals Triangle and paper folding; modular arithmetic and the arithmetic of the infinite. The final chapter presents some ideas about how mathematics should be done, and hence, how it should be taught. Presenting many recent discoveries that lead to interesting open questions, the book can serve as the main text in courses dealing with contemporary mathematical topics or as enrichment for other courses. It can also be read with pleasure by anyone interested in the intellectually intriguing aspects of mathematics.

Genres: Mathematics

368 pages, Hardcover

First published December 13, 1996

Reviews

[Yofish · Rating 4 out of 5]

Really more like 4.5 stars.

This was a really neat idea for a book. And it hasn’t aged poorly. They aim for smart HS math students, or maybe early college, or even post-college but math-curious (or so it seems to me). If you’re one of those, you should really give this a shot.

The authors present a bunch of mathematical concepts that are easy to present at that level. (They seem to go out of their way to make sure that calculus is not necessary.) They want to show how a mathematician thinks and works, and they also want to make math fun. They (mostly) succeed. There is a very conversational tone. Even a professional mathematician like me found some topics I hadn’t seen presented before.

This is not a book you can read casually---you really need to concentrate and have pencil/paper handy. There are certainly places where they breeze by some things too quickly, but not more than really should be expected. Each chapter is (basically) independent of the others; but within each chapter, things build up.

The topics/chapters (overly simplified here) are: spirals and polar coordinates, modular arithmetic, Fibonacci (and Lucas and....) numbers, folding strips of paper into (approximate) polygons, tiling, Pascal’s triangle, pigeonhole principle and infinite arithmetic, fractals and non-integral dimension.