Logical Structure of Chapter 4

Math 403, Spring 2008

This chapter deals with isometries. Isometries are defined in Section 4.1 as transformations which preserve distance. It turns out that isometries also preserve all the other objects and relationships of Euclidean geometry: angle measure, orthogonality, straight lines, circles, parallel lines, vector length, scalar product, etc.

Intuitively, you can think of isometries as ways you can map an infinite sheet of paper onto itself without stretching or tearing.

The chapter describes 4 types of isometries: translation (same as in Chpt. 3), reflection over a line, rotation around a point, and glide reflection (which is a reflection followed by a translation parallel to the line of reflection).

The main goal of the chapter is to prove Theorem 4.36, which classifies isometries by saying that the above 4 types comprise all possible isometries.

The proof of Theorem 4.36 is the most complex proof we have encountered in this class and is actually spread throughout the chapter. There are several major steps:

1) Analyze fixed points of isometries. Show that an isometry must have either no fixed points, exactly one fixed point, a line of fixed points, or all points fixed (identity). See Section 4.2.

2) Use the above information about fixed points to show that every isometry can be written as the composition of either 0, 1, 2, or 3 reflections over lines. See Theorem 4.20

3) Analyze compositions of reflections:
(a) The composition of 0 reflections means the identity transformation.
(b) The composition of 1 reflection is a reflection, of course.
(c) The composition of 2 reflections is either a rotation (Theorem 4.28) or a translation (Theorem 4.24), depending on whether the lines of reflection intersect or are parallel.
(d) The composition of 3 reflections is either another reflection (Theorems 4.26, Theorem 4.29) or a glide reflection (Theorem 4.37), depending on whether the 3 lines of reflection are concurrent, are all parallel, or are neither.

Why is the classification of isometries of so much interest? Besides the inherent interest and beauty of this result, it provides a powerful tool for the study of Euclidean geometry. Read page 48 again. Because isometries form a group, concepts from group theory can be brought to bear on geometric questions.