Explorations in topology: map coloring, surfaces, and knots

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About fifty of them are listed, along with suggested references. Although the choice of topics presented here is relatively standard, the method of presentation is not. He used the same approach in a previous book, Geometry by Discovery. Another omission that struck me as quite striking was the total lack of an Index.

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Because of the discovery approach adopted by this book, any instructor using it must be one who is enthusiastic about this approach, and who has the kind of specialized skill to make it work in a classroom; being able to deliver a clear, interesting lecture, for example, does not necessarily translate into the ability to tease mathematics out of students in the way this book is structured to do.

I have never taught any discovery-based courses, partly because the traditional lecture approach seems more natural to me and partly because I have some reservations about the method in general. I think it is entirely reasonable to believe that information is retained in a better, deeper, way if the student discovers it on his or her own, so to that extent I think the method has something to recommend it.

Gay, Explorations in Topology: Map Coloring, Surfaces and Knots, 2e

But at the same time I think that there is a trade off in the amount of information that can be presented in a given time period, and so on balance the benefits of this approach may be outweighed by the detriments. This is, of course, a matter of individual taste, and there are certainly quite a few people who think very highly of this method.

The book does not consist entirely of problems and projects for the student, however. In the chapter on Moebius strips, for example, the author discusses their history, gives some indication of how they appear in art and literature, and briefly looks at their applications in chemistry. Another unusual feature of the book is the way in which these mathematical problems are posed.

The author does so by means of conversations among the employees of an imaginary company called Acme Maps. They are written as though they were scripts of plays, complete with stage direction:. The book's innovative story-line style models the problem-solving process, presents the development of concepts in a natural way, and engages students in meaningful encounters with the material. Back to search. Journalen skapades , och modifierades senast Because I haven't worked with clay since high-school art classes, I made this mug at a paint-your-own-pottery studio.

In order to produce sharp boundaries, I surrounded each color region with masking tape and cut the shape out with a craft knife before applying the glaze.


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  • I have since discovered that it is much easier to paint the regions carefully by hand, applying the glazes in order from lightest to darkest. In topology, a standard method of conceptually constructing a torus is to take a rectangle and glue together the pairs of opposite edges. Usually, one imagines stretching the rectangle before gluing to make a traditional rounded doughnut shape, but in this model, an actual rectangle of fabric is sewn together to make this nearly flat torus.

    Knot Theory 3: Surfaces & Genus

    The rectangle is made up of seven congruent hexagons, each of which borders all of the others. It is not easy to see all at once how all of the colors in this torus fit together, but the model can be flattened out around each of the seven hexagons separately. In the photo array below, each hexagon in turn is shown surrounded by the other six.

    Explorations in Topology: Map Coloring, Surfaces and Knots

    To make your own version of this model, print out the hexagon below, enlarge it to whatever size you desire, and cut seven of these hexagons out of fleece or some other fabric that will not unravel when you cut it. Stitch the hexagons together as shown in the colored diagram. The hexagon has three different angles, even though two of them are very close to each other. Be careful not to flip a hexagon and swap these two angles.

    The first time I tried to make this model, I inadvertently introduced a full twist in the rectangle before stitching it shut.

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    While the twisted model, pictured above, is not as well-behaved as the flat model, I kind of like the sense of roundedness and fullness the twist gives the torus. Amazingly, there is a polyhedron, called the Szilassi polyhedron, that has seven hexagonal faces, each of which touches each of the others. The polyhedron, which Szilassi discovered in the 's, is a topological torus that is incredibly hard to photograph clearly but really fun to make.