8.5 Transformational Geometry, Symmetry, and Tessellations 501 Consider the capital letters of the alphabet. Now consider a kindergarten-age child who is practicing to write capital letters. Usually, children will write some of the letters “backwards.” However, some letters cannot be made backwards. For example, the capital letter A would look the same backwards as it does forwards, but the capital letters B and C would look different backwards than they do forwards. The capital letters that appear the same forwards as they do backwards have a property called symmetry that we will define and study in this section. Symmetry is part of another branch of geometry known as transformational geometry. SECTION 8.5 LEARNING GOALS Upon completion of this section, you will be able to: 7 Perform reflections of geometric figures. 7 Perform translations of geometric figures. 7 Perform rotations of geometric figures. 7 Perform glide reflections of geometric figures. 7 Identify symmetries of geometric figures. 7 Identify tessellations of geometric figures. Why This Is Important Transformational geometry is used to describe movement of geometric figures. For this reason, transformational geometry is used to model many problems in physics, chemistry, biology, and other scientific areas. Furthermore, transformational geometry plays an important role in the branch of mathematics called group theory, which we will study in Chapter 9. When taken together, group theory and transformational geometry can be used to understand any physical phenomenon that involves symmetry and patterning. Thus far we have discussed Euclidean geometry. We will now introduce a second type of geometry called transformational geometry. In transformational geometry, we study various ways to move a geometric figure without altering the shape or size of the figure. When discussing transformational geometry, we often use the term rigid motion. Transformational Geometry, Symmetry, and Tessellations Consider trapezoid ABCD in Fig. 8.47. If we move each point on this trapezoid 4 units to the right and 3 units up, the trapezoid is in the location specified by trapezoid ′ ′ ′ ′ A BC D. This figure illustrates one type of rigid motion. When studying rigid motions, we are concerned only about the starting and ending positions of the figure. When discussing rigid motions of two-dimensional figures, we note there are four basic types of rigid motions: reflections, translations, rotations, and glide reflections. We call these four types of rigid motions the basic rigid motions in a plane. After we discuss the four rigid motions, we will discuss symmetry of geometric figures and tessellations. Definition: Rigid Motion or Transformation The act of moving a geometric figure from some starting position to some ending position without altering its shape or size is called a rigid motion (or transformation). D C A B D9 C9 A9 B9 Figure 8.47 Reflections In our everyday life, we are quite familiar with the concept of reflection. In transformational geometry, a reflection is an image of a geometric figure that appears on the opposite side of a designated line. Stefan Dahl Langstrup/ Uber Images/Alamy Stock Photo
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