31 of 31 people found the following review helpful
Sparrow R. Jones
- Published on Amazon.com
This book was the foundational textbook for a 100-level class in symmetry at my university. I recommend it highly to anyone who wants to get a better feel for what mathematicians actually do and think about and work with. Folks who never got into the higher math classes often have a different idea of what mathematics is all about than mathematicians. At the level of introductory algebra and geometry and even some calculus, math education often seems to be mainly about memorizing formulas and recognizing in which situations to apply them. That's an important thing to learn, but it is not useful for imparting an idea and a feel of the field of mathematics as a whole. Farmer's book brings home the understanding that mathematics is, at its heart, about patterns and that mathematics is not so much about memorization and application as it is about discovery.
The level of mathematical understanding required to get something useful out of this book is low. I believe the professor required beginning algebra as the prerequisite. If you can count to six, recognize the difference between a square and a pentagon, and understand that variables like n, m, or x can be used as substitutes for numbers then you probably have enough mathematical sophistication to work your way through this book and gain insights into the beauty of higher math.
21 of 21 people found the following review helpful
- Published on Amazon.com
Groups are the first structures encountered in abstract algebra and form the foundation for most of the others. Fortunately, they are also the easiest to physically represent, so in some sense they are the most concrete. In this book, groups are introduced as the motions and structures of geometric figures, so the presentation is largely by diagram rather than formula. Very little previous knowledge of mathematics is required and after reading the book, you will have a solid understanding of what a group is.
The first topic is the moving of a complete figure to a different location of the plane defined by a grid of points. By keeping the figure rigid and fixed in orientation, a set of legal moves is defined. After that, some of the rules are relaxed and that allows for additional moves to be added. Exercises and problems are put forward here and throughout the book, and with the accent on figures, often give the appearance of a game.
The next steps are then to allow for all possible rotations, translations and reflections of the objects, using these to explain the structure of a group. This is an effective way to introduce group theory, and is how I will do it if I teach abstract algebra again. Permutation and plane tiling symmetry groups are then introduced and examined, and their relationship to the previous groups discussed, which introduces the concept of isomorphism.
Basic group theory is something that everyone can understand, as humans have a natural affinity for patterns and recognizing them despite "trivial" alterations. This book is an excellent primer on group theory and I strongly recommend it to anyone either learning or teaching abstract algebra.
Published in Journal of Recreational Mathematics, reprinted with permission.