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Number. Shape, and Symmetry: An Introduction to Number Theory, Geometry, and Group Theory

Diane L. Hermann and Paul J. Sally, Jr.
Chapman & Hall/CRC
Publication Date: 
Number of Pages: 
[Reviewed by
Peter Olszewski
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Well-rounded approaches to logic and proofs have been achieved in Number, Shape, and Symmetry: An Introduction to Number Theory, Geometry, and Group Theory. Proofs are, by far, the most challenging thing for any mathematics student to master. The proofs in this book guide the student from simple ideas such as proving that \(a \times 0=0\) to more advanced ventures, as in

if \(n\) is a positive integer, show the highest power of a prime \(p\) dividing \(n!\) is \(p^k\), where \(k=\left\lfloor\frac{n}{p}\right\rfloor + \left\lfloor\frac{n}{p^2}\right\rfloor + \left\lfloor\frac{n}{p^3}\right\rfloor + \cdots\).

It is good to see the arithmetic developed in detail from the fundamental axioms so that students have a clear understanding of each consequence. It is also good that the authors do not take for granted how to solve equations as \(2x + 1 = 7\).

The book is divided into two parts. Part I deals with the key fundamental ideas of Number Theory and the rules for arithmetic, presented by way of all the axioms. Part II talks about Geometry, Symmetry, Groups, Graph Theory, Tessellations, and Connections.

Throughout the book, the notion of a group is woven into the text. Although the terms and ideas are used starting in Chapter 2, the precise definitions of groups, abelian groups, and commutative rings are not given until Section 6.2. Students may wonder why these terms are used early in the text if they have never been exposed to them before.

The text has a nice, natural build-up in difficulty of problems. It is clearly seen that both Diane L. Herrmann and Paul J. Sally, Jr., have dedicated a great deal of time to writing the text. The book clearly mirrors their years of experience teaching this material to high school students, students not majoring in mathematics, and even mathematics students. Each section is written to be manageable for students to learn, with just the correct amount of content. When I was reading the text, I thought it was my own personal professor who was not only teaching and presenting material, but was guiding me through each step of the lesson through clear examples, as if presented in a face-to-face class.

My comments and suggestions for the text are as follows:

  1. On page 12, I would like to see the classic diagram for the various number sets. I believe this will be a good visual for all students to see.
  2. On page 15, arrow diagrams should be used in the discussion of functions.
  3. I feel as though Chapter 6 should be split into two sections beyond labeling the sections as 6.1 and 6.2. I believe more can be said with congruences mod m; I would include such topics as the Chinese Reminder Theorem. Since section 6.2 presents groups, I feel as though there should be a break between these sections.
  4. The applications in Chapter 7 are well done. Could there be others to include?
  5. Chapter 9, Section 10.2, and Chapter 13 could be further enhanced by having a computer program accompany the book to give students a chance to interact and experiment with rotations, translations, the various constructions, and to have a three dimensional view of the various polyhedra. Perhaps color should be used for the images in Chapter 13.
  6. On page 220, there is no formal definition of subgroup given but the term is used as an example.
  7. On page 221, I would like to see a complete treatment of the Klein-4 Group.
  8. On page 245, a fully worked out example of disjoint cycles should be presented.
  9. On page 302, a larger picture for the Königsberg Bridge problem is recommended.

My other general suggestions for the book are to perhaps include a section or chapter on Cayley graphs and group presentations. This will make a nice connection to groups and graph theory. For the Challenge Problems, a supplemental guide can be given along with the book that helps students think about the solutions. This could be a very similar concept to an “Alive” book. By having an interactive software package with the book can only be a further help to students.

My one concern for the next is the limited audience the book may receive in high schools. Most high schools in the United States teach mathematics courses up to AP Calculus and Statistics. In terms of high school students, I see the book being used in a special independent study setting or in a special gifted students setting. On the college level, this is a great book to use as either a primary or supplementary book for a Number Theory class.

Peter Olszewski is a Mathematics Lecturer at Penn State Erie, The Behrend College and is also an editor for Larson Texts, Inc. in Erie, PA. He can be reached at Webpage: Outside of teaching and textbook editing, he enjoys playing golf, playing guitar, reading, gardening, traveling, and painting landscapes.

The Triangle Game


The Beginnings of Number Theory
Setting the Table: Numbers, Sets and Functions
Rules of Arithmetic
A New System
One's Digit Arithmetic

Axioms in Number Theory
Consequences of the Rules of Arithmetic
Inequalities and Order

Divisibility and Primes
Greatest Common Divisor

The Division and Euclidean Algorithms
The Division Algorithm
The Euclidean Algorithm and the Greatest Common Divisor
The Fundamental Theorem of Arithmetic

Variations on a Theme
Applications of Divisibility
More Algorithms

Congruences and Groups
Congruences and Arithmetic of Residue Classes
Groups and Other Structures

Applications of Congruences
Divisibility Tests
Days of the Week
Check Digits

Rational Numbers and Real Numbers
Fractions to Decimals
Decimals to Fractions
Rational Numbers
Irrational Numbers
How Many Real Numbers?

Introduction to Geometry and Symmetry


 Polygons and Their Construction
Polygons and Their Angles

Symmetry Groups
Symmetric Motions of the Triangle
Symmetric Motions of the Square
Symmetries of Regular n-gons

Symmetric Motions as Permutations
Counting Permutations and Symmetric Groups
Even More Economy of Notation

Regular Polyhedra
Euler’s Formula
Symmetries of Regular Polyhedra
Reections and Rotations
Variations on a Theme: Other Polyhedra

Graph Theory
The Königsberg Bridge Problem
Colorability and Planarity
Graphs and Their Complements

Tessellating with a Single Shape
Tessellations with Multiple Shapes
Variations on a Theme: Polyominoes
Frieze Patterns
Infinite Patterns in Two and Three Dimensions

The Golden Ratio and Fibonacci Numbers
Constructible Numbers and Polygons

Appendix: Euclidean Geometry Review