section 9.6 topology

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

Section 9.6

Topology


What You Will Learn

TopologyMöbius StripKlein BottleMapsJordan CurvesTopological Equivalent

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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.

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Möbius BandA 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 twoends together.

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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!

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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!


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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.)


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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 Möbiusband interlocked with one larger band with two (half) twists!


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Klein BottleThe punctured Klein bottle resembles a bottle but only has one edge and one side.

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Klein Bottle

Klein bottle, a one-sided surface, blown in glass by Alan Bennett.

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Klein BottleImagine 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 isnot there, so youcontinue to followthe neck, which isnow inside the bulb.

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Klein BottleAs 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 isone-sided.

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MapsMapmakers 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.

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Maps

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MapsMathematicians have show that on different surfaces, more colors may be needed. A Möbius band requires a maximum of six, while a torus (doughnut) requires amaximum of seven.

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Jordan CurvesA 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 anoutside.

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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.

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Topological EquivalenceThe doughnut and coffee cup are topologically equivalent.

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Topological EquivalenceIn 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).

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Topological Equivalence

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section 9.6 topology

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