Updated 7/5/01

The Project
The Basics

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An Exploration of Surfaces

This web site was created as a mathematical resource and is meant to serve as a teaching tool to give some insight into the world of rubbersheet geometry and topological manipulations. By creating a variety of 3D animations, we hope to provide an easier way to visualize some of these manipulations, and thereby encourage a better understanding of topology. (More detailed information on the history and future of the project can be found on The Project page.)

Though this site includes a number of links to the best topology sites we could find on the web (see our References section), the sort of constructions and transformations we have created on this site seem to be rather uncommon. Most of the internet resources we found dealt with parameterized equations, and therefore focused only on the most standard representations of the surfaces.

This site, on the other hand, focuses mostly on "physical" manipulations. For anyone unfamiliar with some of the basics of topology, we provide The Basics where the essential moves and operations are introduced. In the Constructions section, animations show the constructions of various surfaces (obtained by manipulating a two dimensional sheet). In the Transformations section, we provide some visual translations from one surface to another "equivalent" surface.

Our main hope for this site is to provide a useful resource for visualizing some of the basics of topology. Any comments and/or suggestions on how to improve the site are always welcome.

About viewing the Animations
Read this before viewing the site!

All of the animations here have been rendered into Windows Media Player and Quicktime files. If you do not have Media Player or Quicktime on your computer, you will have to download one of them. Windows Media Player may be downloaded at http://windowsmedia.com/download/, and Quicktime may be downloaded at http://www.apple.com/quicktime/download/.

When viewing an animation, we suggest that you play the animation several times to observe what happens to the surface. If you use Quicktime (or some of the older versions of Media Player), you can also use the scroll bar to smoothly scroll forward and backward in the movie, a feature which is especially helpful for some of the less intuitive spots in the transformations. Don't close down an animation until you're sure you understand it, and be sure to read the textual description of the animation too.