Groups and Symmetry: A Guide to Discovering Mathematics (Mathematical World, Vol. 5) [Paperback]

David W. Farmer (Author)

Book Description

ISBN-10: 0821804502 | ISBN-13: 978-0821804506 | Publication Date: November 15, 1995

In most mathematics textbooks, the most exciting part of mathematics--the process of invention and discovery--is completely hidden from the reader. The aim of Groups and Symmetry is to change all that. By means of a series of carefully selected tasks, this book leads readers to discover some real mathematics. There are no formulas to memorize; no procedures to follow. The book is a guide: Its job is to start you in the right direction and to bring you back if you stray too far. Discovery is left to you. <P>Suitable for a one-semester course at the beginning undergraduate level, there are no prerequisites for understanding the text. Any college student interested in discovering the beauty of mathematics will enjoy a course taught from this book. The book has also been used successfully with nonscience students who want to fulfill a science requirement.

Editorial Reviews

Review

"Nicely produced and concentrates on the informal analysis of geometrical patterns with the emphasis on informality ... could serve as a useful collection of activities to precede a formal course and would provide a range of intuitive experiences to which the more formal treatment could refer." ---- The Mathematical Gazette

"On the basis of this book it is possible to tailor a good course for high school students to really discover mathematics ... for anyone who is working with high school students in an advanced level the book is really recommended." ---- Zentralblatt MATH

"Written in a lively conversational style ... entertaining, and sometimes provoking, and will doubtlessly prove useful to its intended audience." ---- Mathematical Reviews

Product Details

• Paperback: 102 pages
• Publisher: American Mathematical Society (November 15, 1995)
• Language: English
• ISBN-10: 0821804502
• ISBN-13: 978-0821804506
• Product Dimensions: 10.2 x 7 x 0.5 inches
• Shipping Weight: 5.6 ounces (View shipping rates and policies)
• Average Customer Review: 4.5 out of 5 stars  See all reviews (2 customer reviews)
• Amazon Best Sellers Rank: #241,264 in Books (See Top 100 in Books)

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Customer Reviews

Most Helpful Customer Reviews

27 of 27 people found the following review helpful:

5.0 out of 5 stars the best math textbook I've ever had, February 9, 2006

By 

Sparrow R. Jones "student" (Pocatello, ID USA) - See all my reviews
(REAL NAME)   

This review is from: Groups and Symmetry: A Guide to Discovering Mathematics (Mathematical World, Vol. 5) (Paperback)

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.

19 of 19 people found the following review helpful:

4.0 out of 5 stars An excellent primer on abstract algebra, September 22, 2002

By 

Charles Ashbacher (Marion, Iowa United States) - See all my reviews
(TOP 500 REVIEWER)    (VINE VOICE)    (HALL OF FAME REVIEWER)   

This review is from: Groups and Symmetry: A Guide to Discovering Mathematics (Mathematical World, Vol. 5) (Paperback)

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.