Experiments in Topology [Paperback]

Stephen Barr (Author)

Book Description

Publication Date: March 1, 1989

With this book and a square sheet of paper, the reader can make paper Klein bottles; then by intersecting or cutting the bottle, make Moebius strips. Conical Moebius strips, projective planes, the principle of map coloring, the classic problem of the Koenigsberg bridges and other aspects of topology are clearly explained.

Product Details

• Paperback: 210 pages
• Publisher: Dover Publications (March 1, 1989)
• Language: English
• ISBN-10: 0486259331
• ISBN-13: 978-0486259338
• Product Dimensions: 7.9 x 5.4 x 0.5 inches
• Shipping Weight: 5.6 ounces (View shipping rates and policies)
• Average Customer Review: 3.5 out of 5 stars  See all reviews (2 customer reviews)
• Amazon Best Sellers Rank: #1,134,984 in Books (See Top 100 in Books)

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Most Helpful Customer Reviews

14 of 14 people found the following review helpful:

4.0 out of 5 stars Great book for high school teachers and the popular audience, November 16, 2002

By 

Dr. Lee D. Carlson (Baltimore, Maryland USA) - See all my reviews
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This review is from: Experiments in Topology (Paperback)

Written for the beginner in topology but still presupposing a certain amount of mathematical maturity, this book is great fun and should be useful to high school teachers, and those who might be giving public talks on mathematics to audiences who are very interested in mathematics, but who don't have a substantial background in the subject. It is always helpful in those scenarios to have concrete examples that will illustrate some of the more difficult constructions in topology.

In chapter 1, the author attempts to give an intuitive definition of topology. The author uses various pictures and handwaving arguments to explain various notions in topology, such as homeomorphism ("coffee cup = donut"), simply connected (object with no holes), homology (two circles can intersect at one point only), Jordan curves, and Euler's theorem.

In chapter 2, the author uses paper models to illustrate the topology of surfaces, both orientable and nonorientable. The author gives instructions on how to make a paper model of a Klein bottle, but cautions the reader that such a model is not an exact representation of the mathematical object rigorously defined in topology, since the surface passes through itself in the paper construction.

Chapter 3 is a set of instructions on how to make a "shortest" Moebius strip. The procedures for doing this are interesting and fun, for the author constructs a Moebius strip whose length is less than its width, by a factor of 1 over the square root of 3. He devotes an appendix for an improvement due to Martin Gardner of Scientific American fame.

In chapter 4 the author constructs what he calls a "conical Moebius strip". The author asks the reader to consider an annulus with a radial slit, with which of course one can construct a Moebius strip by twisting the ends and joining them. But he asks how large the hole must be in relation to the outside diameter. The answer to this is that one does not need any hole at all in order to carry out the construction. In fact, an angular segment can be cut out instead of the radial cut, and this leads him to construct the conical Moebius strip.

The author returns to the Klein bottle in chapter 5, and shows first what happens if the usual construction of the Klein bottle is cut down the center symmetrically: two Moebius strips are obtained. But to construct this model is difficult, so he gives alternate constructions for making the Klein bottle. He then shows what happens to the various models when the pieces are cut.

But how do you make a projective plane using scissors and paper? Intuitively one can imagine this would be very difficult, but the author shows ways to do it in chapter 6. His strategy for making these models is to teach the concept of symmetry in topology, and he pulls this off very well.

The famous 4-color problem for maps, i.e. that one needs only 4 colors for a map, is considered in chapter 7. At the time of publication, the 4-color problem was still open, so the author attempts to try and explain it using various diagrams and subdivisions thereof. The 4-color problem was proved using computer algorithms by the mathematicians K. Appel and W. Haken in 1976.

Network topology is considered in chapter 8, with the famous Koenigsberg bridge problem leading off the discussion. The author also introduces the very important Betti numbers, these having far-reaching ramifications in topology. And interestingly, the author is able, via a consideration of loop-cuts and cross-cuts in paper models of the Klein bottle and projective plane, to introduce the very important concept of "duality". The theory of knots makes its appearance here, although the discussion is very short. The author is well-aware of the difficulties in finding a classification theory of knots, but more could possibly be done here in the lines of the rest of the book to illustrate some of the peculiarities of knots.

Chapter 9 is really fun, for it concerns the torus with a puncture, and how to turn it inside out. The diagrams are helpful and the intuition gained valuable. The mathematician Steven Smale found a way of turning the sphere inside out in the early 1060s, but the author does not tackle Smale's method! This is unfortunate, since it is very difficult to follow the steps in Smale's method, at least for me.

The author does not want to leave the reader with the impression that topology is all scissors, paper, and tape, so he devotes the last two chapters of the book to point-set topology. Concepts such as continuity, limit points, and neighborhoods are discussed. It is quite difficult to explain to beginning readers and students of topology what a neighborhood actually does without having the notion of a metric or distance, but the author does a fairly good job here.

15 of 23 people found the following review helpful:

3.0 out of 5 stars useful but not broad enough, December 22, 1998

By 

Kevin M. Iga (Pepperdine University (Malibu, CA)) - See all my reviews
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This review is from: Experiments in Topology (Paperback)

The subject of topology lends itself to many different kinds of experimentation for undergraduate students. But this book spends a disproportionate amount of space on the Mobius strip and related non-orientability issues when it could deal with more knot theory and homotopy theory than it does, and it could introduce finite topologies and Morse theory which abound in self-exploration.