Mathematics in Art and Architecture Gek1518 Symmetry and Patterns

In this chapter, we are interested in the underlying structure of aesthetically pleasing plane figures. By a plane figure, we mean any subset of the plane. To understand the underlying structure, we examine the symmetries of the figure. An isometry of the plane is a distance-preserving transformation of the plane. This means that for any pair of points P and Q, the distance between the images under the transformation is the same as the distance between P and Q. A symmetry of a figure is an isometry that maps the figure back onto itself. There are four types of isometries, namely translation, reflection, rotation and glide reflection. A short description of each isometry of the plane is given in the next section. We will classify a figure in terms of its symmetry group, which is the set of all symmetries of the figure. If a figure admits a symmetry, it is said to be symmetrical.