"/'mə:bɪəs/ ► noun a surface with one continuous side formed by joining the ends of a rectangle after twisting one end through 180° ORIGIN early 20th cent.: named after August F. Möbius (1790-1868), German MathematicianThough it was independently discovered by Johann Benedict Listing at the same time.
One of the main practical uses for a möbius strip is for drive belts; but I think that the best use is in knitting (though perhaps I'm a bit biased).
There seems to be two main categories for making a möbius; flat and circular.
The first option is for the cast on edge to be the long horizontal, the piece is worked flat and then twisted and sewn along the vertical edge. This would create an obvious seam in the work.
The second option is when the cast on edge is the shorter vertical edge, again worked flat, then twisted and then the cast on / bind off edge is sewn together. This is very similar to the first option and again this would create a seam.
The third option is to provisionally cast on along the short vertical edge and then it is grafted together at the end.
All the following require circular needles.
This option is to cast on with circular needles and then make one twist in before joining in the round. So the bottom edge will be the cast on and the top will be the bind off.
The next three options all involve what I consider a little bit of magic. They are started at the middle of the strip and then the piece is worked on circular needles around the entire edge of the work. Therefore, all the outside edges will be the bind off edge.
The first one involves a regular cast on, such as a knitted or long tail cast on. Using the same circular needles that the stitches were cast on, the same number of stitches are picked up from the cast on edge.
The second and third options are relatively similar. They both require a provisional cast on. Cat Bordhi's Moebius Cast On is made using the needle tip and the cable of the same needle and the other is the Alternating Cast-On for Moebius from June Hemmons Hiatt’s “The Principles of Knitting”, which uses two circular needles of the same size.
I'll leave you with a joke from "The Big Bang Theory". Why did the chicken cross the möbius strip? To get to the same side.
The New Oxford English Dictionary (1999) Oxford University Press p.1187
Flynn, Mike, Infinity In Your Pocket (2005) Elwin Street Ltd p.25
Hemmons Hiatt, June, The Principles of Knitting (2012) Touchstone pp. 368-370
Patmore, Frederica & Haffenden Vikki, The Knitting Book (2011) Dorling Kindersley Ltd p.197
Wikipedia, Möbius strip, Retrieved 05.12.12. from http://en.wikipedia.org/wiki/M%C3%B6bius_strip
Bordhi, C - Intro to Moebius Knitting, Retrieved 05.12.12 from http://www.youtube.com/watch?v=LVnTda7F2V4