Section 9.6
Topology
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
INB Table of Contents
2.3-2
Date
Topic
November 19, 2014
Map Color Project
60
November 19, 2014
Section 9.6 Notes
61
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Page #
What You Will Learn

Topology

Möbius Strip

Klein Bottle

Maps

Jordan Curves

Topological Equivalent
9.6-3
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.
9.6-4
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.
9.6-5
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!
9.6-6
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Properties of a Möbius Band

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!
9.6-7
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Properties of a Möbius Band
9.6-8

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.)
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Properties of a Möbius Band

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!
9.6-9
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Klein Bottle
The punctured Klein bottle resembles a
bottle but only has one edge and one side.
9.6-10
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Klein Bottle
Klein bottle, a
one-sided surface,
blown in glass by
Alan Bennett.
9.6-11
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.
9.6-12
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.
9.6-13
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.
9.6-14
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Maps
9.6-15
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.
9.6-16
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.
9.6-17
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.
9.6-18
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Topological Equivalence
The doughnut and coffee
cup are topologically
equivalent.
9.6-19
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).
9.6-20
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Topological Equivalence
9.6-21
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Map Color Project
9.6-22
Copyright 2013, 2010, 2007, Pearson, Education, Inc.