Copyright 2013, 2010, 2007, Pearson, Education, Inc. Section 9.6 Topology

Copyright 2013, 2010, 2007, Pearson, Education, Inc. What You Will Learn Topology Möbius Strip Klein Bottle Maps Jordan Curves Topological Equivalent 9.6-2

Copyright 2013, 2010, 2007, Pearson, Education, Inc. Definitions The branch of mathematics called topology is sometimes referred to as “rubber sheet geometry” because it deals with bending and stretching of geometric figures

Copyright 2013, 2010, 2007, Pearson, Education, Inc. Möbius Band A Möbius strip, also called a Möbius band, is a one-sided, one- edged surface. Construct a Möbius band by taking a strip of paper, giving one end a half twist, and taping the two ends together

Copyright 2013, 2010, 2007, Pearson, Education, Inc. Properties of a Möbius Band It is one-edged. Place a felt marker on the edge and without removing the marker trace along the edge. Remarkably, the marker travels around the entire “edge” and ends where it begins! 9.6-5

Copyright 2013, 2010, 2007, Pearson, Education, Inc. It is one-sided. Place a felt marker on the surface and without removing the marker trace along the surface. Remarkably, the marker travels around the entire “surface” and ends where it begins! Properties of a Möbius Band 9.6-6

Copyright 2013, 2010, 2007, Pearson, Education, Inc. Using scissors cut along the center of the length of the band. Remarkably, you end up with one larger band with three (half) twists! (Topologically the same as a Möbius band.) Properties of a Möbius Band 9.6-7

Copyright 2013, 2010, 2007, Pearson, Education, Inc. Make a small slit at a point about one- third of the width of the band. Cut along the strip, keeping the same distance from the edge. Remarkably, you end up with one small Möbius band interlocked with one larger band with two (half) twists! Properties of a Möbius Band 9.6-8

Copyright 2013, 2010, 2007, Pearson, Education, Inc. Klein Bottle The punctured Klein bottle resembles a bottle but only has one edge and one side

Copyright 2013, 2010, 2007, Pearson, Education, Inc. Klein Bottle Klein bottle, a one-sided surface, blown in glass by Alan Bennett

Copyright 2013, 2010, 2007, Pearson, Education, Inc. Klein Bottle Imagine trying to paint a Klein bottle. You start on the “outside” of the large part and work your way down the narrowing neck. When you cross the self-intersection, you have to pretend temporarily that it is not there, so you continue to follow the neck, which is now inside the bulb

Copyright 2013, 2010, 2007, Pearson, Education, Inc. Klein Bottle As the neck opens up, to rejoin the bulb, you find that you are now painting the inside of the bulb! What appear to be the inside and outside of a Klein bottle connect together seamlessly since it is one-sided

Copyright 2013, 2010, 2007, Pearson, Education, Inc. Maps Mapmakers have known for a long time that regardless of the complexity of the map and whether it is drawn on a flat surface or a sphere, only four colors are needed to differentiate each country (or state) from its immediate neighbors. Thus, every map can be drawn by using only four colors, and no two countries with a common border will have the same color

Copyright 2013, 2010, 2007, Pearson, Education, Inc. Maps

Copyright 2013, 2010, 2007, Pearson, Education, Inc. Maps Mathematicians have show that on different surfaces, more colors may be needed. A Möbius band requires a maximum of six, while a torus (doughnut) requires a maximum of seven

Copyright 2013, 2010, 2007, Pearson, Education, Inc. Jordan Curves A Jordan curve is a topological object that can be thought of as a circle twisted out of shape. Like a circle it has an inside and an outside

Copyright 2013, 2010, 2007, Pearson, Education, Inc. Topological Equivalence Two geometric figures are said to be topologically equivalent if one figure can be elastically twisted, stretched, bent, or shrunk into the other figure without puncturing or ripping the original figure

Copyright 2013, 2010, 2007, Pearson, Education, Inc. Topological Equivalence The doughnut and coffee cup are topologically equivalent

Copyright 2013, 2010, 2007, Pearson, Education, Inc. Topological Equivalence In topology, figures are classified according to their genus. The genus of an object is determined by the number of holes that go through the object. A cup and a doughnut each have one hole and are of genus 1 (and are therefore topologically equivalent)

Copyright 2013, 2010, 2007, Pearson, Education, Inc. Topological Equivalence