MAT 539 Algebraic Topology

The Crosscap is an image of the real projective plane. It has a segment of double points, which terminates in two "pinch points", or "Whitney singularities". Click here to view a VRML 1.0 model. Use the mouse to navigate around, turn on/off the light sources, and so on... Click here to see a table of immersions and embeddings of real projective spaces.

The Steiner surface is also an image of the real projective plane. It contains three segments of double points each of which terminates in two "pinch points", or "Whitney singularities". A triple point is created where the three double point segments intersect. Click here to view a VRML 1.0 model.

Boy's surface is also image of a map from the real projective plane to R^3. It contains a continuous double point curve, which meets itself in a triple point. Click here to view a VRML 1.0 model. Here is another page with other pictures (some in the Geomview ".oogl" format) of the Boy surface. Here is an OOGL model of the Boy surface; open it in Geomview. There is also a nice steel model in front of the library of the Mathematisches Forschungsinstitut Oberwolfach.

To view the above VRML files you will need to download first a VRML plug-in for your browser. Here are several choices: You may try the older and unsupported Cosmo player, which works reasonably well with Internet Explorer and Netscape on older windows platforms, or The newer WorldView, or Better, you may install the Cortona plug-in. All these plug-ins are freely available, but the last plug-in provides better graphics support on Windows XP/2000, or Mac OSX platforms. Finally if on a Linux/Unix platform, download either vrmlview or the Java based browser vrwave. Alternatively here is a list of VRML viewers available for Linux platforms.