Term Project Description

Title – “Constructions of non-orientable surfaces such as the Klein Bottle”

Synopsis – The Klein bottle is a mathematical object called a ‘compact non-orientable 2-manifold’. What this means is that, despite being closed, it does not have an interior and exterior (compare this to a closed sphere, which has an interior and exterior). There are several ways to construct a Klein bottle. For example, we can take a square, glue an opposite pair of sides (resulting in a cylinder), and then glue the resulting edges together but in opposing orientation. Well, in fact, the construction described above is actually not possible to do in real life – at a certain point, the material would have to pass through itself. However, in the virtual world of animation, there is no problem with allowing material to pass through itself – this is why I think animation would be a great medium for visualizing this construction of the Klein bottle. Another construction of the Klein bottle is obtained by gluing two Mobius bands along their boundary – once again, it cannot be done in real life, but it can be done with animation.

Objective – To use animation to help students interested in topology (which includes the study of shapes such as the Klein bottle) better understand such topological constructions that cannot be physically achieved in our world. When I first read about such constructions, I had a somewhat difficult time understanding how they worked, because I didn’t have a good visual representation of them. This is why I think creating an animation of this process could be a very useful endeavour. Audio with a narrator describing the construction as it is happening can also be used.

Audience – This would be primarily intended for undergraduate math students (or more generally anyone with a serious interest in math), specifically, those interested in topology.

References – Klein bottle on Wikipedia: https://en.wikipedia.org/wiki/Klein_bottle
– Klein bottle at Wolfram: http://mathworld.wolfram.com/KleinBottle.html
– Mobius strip on Wikipedia: https://en.wikipedia.org/wiki/M%C3%B6bius_strip
– Nonorientable surface at Wolfram: http://mathworld.wolfram.com/NonorientableSurface.html
– Chapter 12 (specifically sections 74-77) of the textbook “Topology” by James Munkres (second edition)