520 CHAPTER 8 Geometry Topology The branch of mathematics called topology is the study of geometric properties and spatial relations unaffected by the continuous change of shape or size of figures. Topology is sometimes referred to as “rubber sheet geometry” because it deals with the bending and stretching of geometric figures. In this section, we will study several interesting objects and how topology is used to study these objects. Möbius Strip One of the first pioneers of topology was the German astronomer and mathematician August Ferdinand Möbius (1790–1866). A student of Gauss (see Profile in Mathematics on page 276), Möbius was the director of the University of Leipzig’s observatory. He spent a great deal of time studying geometry and he played an essential part in the systematic development of projective geometry. He is best known for his studies of the properties of one-sided surfaces, including the one called the Möbius strip. Examine the outline of the map of the continental United States shown on the left. Now suppose you were given four crayons, each of a different color. Could you color this map with the four crayons in a way so that no two bordering states have the same color? In this section, we will discuss this question and many other questions that are relevant to the branch of mathematics known as topology. Topology SECTION 8.6 LEARNING GOALS Upon completion of this section, you will be able to: 7 Learn about topological objects including Möbius strips, Klein bottles, maps, and Jordan curves. 7 Understand the topological equivalence of two or more objects. Why This Is Important When combined with other branches of mathematics, topology plays an important role in mathematical and scientific research. Advanced research in probability, statistics, measurement theory, game theory, and mathematical computing relies on the basic concepts of topology. Research in computer science, biology, genetics, and robotics depends on topology to connect many abstract concepts. Some of the products in our world that were developed with the aid of topology include Global Positioning Systems (GPS), Facebook, YouTube, Snapchat, and many of the other applications that we use with our smartphones. Research Activities 56. Bilateral and Radial Symmetry In the study of biology, reflective symmetry is called bilateral symmetry and rotational symmetry is called radial symmetry. Write a report on the role symmetry plays in the study of biology. 57. M. C. Escher Write a paper on the mathematics displayed in the artwork of M. C. Escher. Include such topics as tessellations, optical illusions, perspective, and non-Euclidean geometry. 54. Reflective Symmetry Examine each capital letter in the alphabet and determine which letters have reflective symmetry about a vertical line through the center of the letter. * 55. Rotational Symmetry Examine each capital letter in the alphabet and determine which letters have 180° rotational symmetry about a point in the center of the letter. * *See Instructor Answer Appendix
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