Non-Orientable Surfaces Gallery

A non-orientable surface is one for which it is impossible to make a consistent choice of "front" and "back" everywhere on the surface. A small enough patch of a surface is similar to a piece of the plane and is certainly two-sided; it is easy enough to designate one side of the patch to be the front and the other side to be the back. Suppose that the whole surface is covered with such patches. To be consistent, wherever two such patches overlap, the choice of front and back selected for the two patches must agree. If a choice of front and back can be made that is consistent over the entire surface, then the surface is orientable. For a non-orientable surface, such a consistent choice is impossible.

Front and back can be defined in terms of normal vectors to the surface. A vector of length one that is perpendicular to a surface is called a unit normal to the surface. For a surface that is embedded in three-dimensional space, there is just one direction at each non-singular point that is perpendicular the surface, and there are two unit normals at that point, pointing in opposite directions. Selecting one of these two vectors at a given point amounts to deciding which side of the surface is considered to be the front side near that point.

To help see the impossibility of making a consistent choice of front and back for an entire non-orientable surface, set the "Surface Coloration" in the "Action" menu to "Distinctly Colored Sides Default," and select "Normal Orientation" in the "View" menu. (The default orientation for non-orientable surfaces is "No Orientation.") In this view, each surface patch has a front side, which is colored blue, and a back side, which is colored yellow. You will see that there are "seams" in the surface where the coloration changes abruptly.

The Non-Orientable Surfaces gallery is one of several sub-galleries of the Surfaces Gallery. For more information on surfaces on on parametric surfaces in particular, you should see the documentation for the Surfaces Gallery.