AlgTop7b: The Klein bottle and projective plane (cont.)

The Klein bottle and the projective plane are the basic non-orientable surfaces. The Klein bottle, obtained by gluing together two Mobius bands, is similar in some ways to the torus, and is something of a curiosity. The projective plane, obtained by gluing a disk to a Mobius band, is one of the most fundamental of all mathematical objects. Of all the surfaces, it most closely resembles the sphere.

This is the second video of the seventh lecture in this beginner's course on Algebraic Topology, given by Assoc Prof N J Wildberger of the School of Mathematics and Statistics at UNSW.

Category:

Science & Technology

Tags:

• AlgTop7b
• mathematics
• algebraic
• topology
• Klein
• bottle
• projective
• plane
• course
• UNSW
• Wildberger
• non-orientable
• Mobius
• band

[njwildberger]

Hi@Toxie207 It is a subtle issue. Our space M in question is the space of all lines in the plane. It is not the plane itself. Each line passing through the given point represents a distinct point in M. So as we rotate those lines about the fixed point, we get a loop of distinct points in M. The trick is in realizing that the space of all lines M is itself a two dimensional surface, and we are trying to figure out what it looks like.

[Toxie207]

1:50 But isn't the loop the union of all the lines intersecting at that point? Maybe I misunderstand what a 'loop' is?