Unorientable Surfaces

Unorientable surfaces are commonly referred to in mathematical popularizations as one-sided surfaces. The simplest example is the Möbius strip, and you're often asked to imagine what would happen if you tried to paint it different colors on opposite sides:


Alternatively you are asked to imagine what it would be like to be an ant crawling upon the surface of a Möbius strip, as in this famous work of the Dutch artist Maurits Escher:


However all of these views of the Möbius strip are extrinsinsic: they look upon the Möbius strip as viewed from outside from surrounding 3-dimensional space. In contrast the point of view that modern topology takes with respect to the Möbius strip is the point of view of a mythical 2-dimensional being imbedded entirely within the Möbius strip. Think of a 2-dimensional crab swimming in a 2-dimensional aquarium having the shape of a Möbius strip:


Notice that as the crab completes a circuit around the Möbius strip, its right and left sides gets interchanged. This is the crucial feature of norientable surfaces which distinguishes them from the more familiar orientable "two-sided" surfaces.

Once we adopt this point of view, we can use a much more convenient model of the Möbius strip:


This "gluing diagram" asks us to take a rectangle and abstractly glue the left hand side to the right hand side with a twist. We emphasize the word "abstractly", ie. we are not supposed to move the rectangle around, physically twist it in three dimensional space and glue it. Rather we should imagine this from the point of view of our 2-dimensional crab moving within this abstractly glued rectangular strip. As the crab moves rightwards and encounters the right edge, it starts to emerge at the left edge, but with a twist, because of the abstract gluing. A similar thing happens when the crab moves leftwards and encounters the left edge: it then reemerges at the right edge, again with a twist. (Note that the crab doesn't notice the right or left edge as it crosses. From its point of view these edges aren't there: it's just the same to the crab as moving around in the middle of the rectangle. Nor does the crab notice its left and right sides being interchanged, until it returns to its starting point.) However when the crab encounters the top or bottom edge, it bounces back, since these edges aren't glued to anything else -- they form the boundary of the Möbius strip. (Technically this process of abstract gluing is called forming a quotient space.) We can easily simulate this using computer graphics:

You need a JAVA enabled browser to view this, such as Netscape Navigator v3.0 for the Macintosh (or v2.0 for Windows)



The Möbius strip is named after the German mathematician August Möbius (1790-1868).



We can now use the insights we have acquired here to study another more exotic example of an unorientable surface, the Klein bottle.



Here is the Java source code for the above animation.


For an alternative take on some of these topics, check out this innovative multi-disciplinary project by a group of Yale undergraduates: