A Topological Picturebook

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A Topological Picturebook 2006

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Find Antiquarian Books Book Value. They help the viewer guess correctly where the artist meant to place things relative to each other. For example, even a modest amount of perspective convergence prevents you from mistaking a three-dimensional picture for a two-dimensional diagram. This print convincingly depicts an over and undershot waterwheel perpetually driven by a closed stream.

The water flows from the base of the wheel down a zigzag millrace which is supported by two towers, and spills from it again, two stories above the wheel.

A Topological Picturebook

A hemispherical bowl has a single, circular edge. So does a disc. If you sew the disc to the hemisphere you obtain a closed, two-sided surface; it has an inside and an outside. Topologists still call this a sphere, and it could be inflated back to a geometrical sphere if you like. Now take a long, narrow strip of paper, give it a half twist and glue the ends together.

The resulting surface is not only one sided, it has but a single, closed edge. What happens if this edge is sewn to the rim of a disc?

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The story of everting spheres in 3-space by regular homotopies is the case history of a nontrivial visualization problem of remarkable complexity and compelling beauty. The task is to show the motion of a spherical surface through itself in space so that, without tearing or creasing, the surface is turned inside out. That so many different graphical methods were applied to the same problem, when it is more in the nature of mathematics to display the versatility of one method by applying it to a variety of different problems, makes it a paradigm for descriptive topology. It is premature to formulate explicit ground rules for comparing the many solutions to this graphical problem.

Topology -- from Wolfram MathWorld

Instead let me tell a very short story about golden rectangles instead. There are two important groups associated with a closed surface: the fundamental group and the mapping class group.

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The former, also known as the first homotopy group of the surface considered as a topological space , is probably the more familiar of the two. In this chapter I would like to revive a few of the traditional images that go with the second, more picturesque group.