6.1 Introduction

"The Theory of Groups is a branch of mathematics in which one does something to something and then compares the result with the result obtained from doing the same thing to something else, or something else to the same thing..."

-James R. Newman

Why do we find butterflies appealing? What is it about a snowflake that can hold our attention? Why do we find some designs beautiful and others unattractive? Such subjective observations can hardly be thought to be within the realm of the objective explanatory power of mathematics, and yet the concept of "symmetry," an idea that underlies what humans consider to be beautiful, drives right to the heart of mathematical thinking.

Item 2875 / Uzbek, EMBROIDERED SUSANI (nineteenth-twentieth century). Courtesy of Kathleen Cohen.

Item 2925 /Tony Link Design, FIERY ORNAMENTAL PATTERN (2007). Courtesy of iStockphoto.com/Tony Link Design.

Item 2926 /Adam Mandoki, CUPOLA (2007). Courtesy of iStockphoto.com/Adam Mandoki.

The mathematical study of symmetry is the rigorous study of the commonalities between objects or situations. For example, a plain equilateral triangle with no distinguishing markings to differentiate one side or angle from another has six geometric symmetries. These result from actions, or transformations, such as flipping or rotating the triangle, that leave it looking the same as when we started. Furthermore, we can compose symmetries to make other symmetries.

Interestingly, the ways in which we can move an equilateral triangle and have it appear as it did when we started have much in common with the ways in which we can rearrange a stack of three objects, such as pancakes. This similarity is not accidental; rather, it is an indication of some deeper concept in action, one that transcends both equilateral triangles and stacks of pancakes. Mathematicians call objects that adhere to this deeper concept a group.

As we will see, the study of groups is one way mathematicians extend the notion of arithmetic to objects that are "beyond numbers." Group theory unites the spatial thinking of geometry with the symbolic realm of algebra. It is beautiful in its generalization and abstraction, providing deep insights into a wide array of phenomena. In addition to its theoretical power and usefulness, the study of groups also has many applications to the real world. The techniques of group theory can be used to study the solvability of certain kinds of equations, explain the existence of elementary physical particles, verify the fact that physical laws are the same everywhere on the planet, or determine when a deck of cards is sufficiently shuffled. These applications are, of course, in addition to the fascinating use of group concepts and techniques in the traditional art forms of various cultures.

Apart from its applications, the study of symmetry and groups reveals deep and surprising connections between different areas of mathematics itself. Group theory is part of a larger discipline known as abstract algebra. Abstract algebra takes the skills and techniques learned in high school algebra—related to how the system of numbers works in a general sense through the use of variables— and extends these tools into the realm of geometry and beyond. Abstract algebra shows the way in which incredibly interesting and complex structures can be created simply by putting a few rules in place as to how a small group of symbols can be moved around and related to one another. In this symbolic and abstract setting we find a unity among things that, on the surface, may seem quite different. Just as the concept of bilateral symmetry unites the forms of butterflies, airplanes, and humans, group theory and abstract algebra show a relationship between the fundamental structures of logic and mathematical reality.

Item 3078 /Andrey Prokhorov, 3D SCHEME (2007). Courtesy of iStockphoto.com/Andrey Prokhorov.

In this unit we will start by examining different types of visual symmetry, and we will see how the concept of groups enables us to classify different design motifs in both one and two dimensions. From there we will examine the symmetries inherent in permutations, such as the ways in which one can stack pancakes or shuffle cards. With these concepts in hand, we will catch a glimpse of one of the crown jewels of abstract algebra, Galois theory. We will then shift gears to study the role that symmetry plays in our understanding of the physical universe.

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Next: 6.2 Types of Symmetry Leading to Groups