However you spell it, a Mobius strip is a one-sided strip, conceived by August Mbius, a mathematician best know for his work in topology. Physlink.com says a Mobius strip is a
non-orientable surface: you can build one with a strip of paper (twist the strip
and glue end together to form a ring) and verify that it has only one side: it is
not possible to paint it with two colours. Keep that "non-orientable" in mind.
I began my Moebius-banded hat today, using Cat Bordhi's way of casting on from "A Treasury of Magical Knitting". Cat's description, with pictures, is as clear as can be. In order to make it work you have to Stop Thinking and just follow her instructions. You start with a slip knot, and by the time you have cast on and knit a few rows that slip knot is in the middle of the piece - not on an edge as you would expect. You actually cast on in such a way that, as you work around the coiled up needle, you are knitting from the middle out - in two directions at once. Remember that "non-orientable" from the first paragraph.
Brilliant! I recommend you buy the book just to read the first 16 pages. So here's the chemo hat I'm making, and a page from her book.
I like the purple heather much better than the rust color in the book, but that's just my personal taste.
My hat is being worked on a 40" circular needle that is coiled up to create both edges of the strip.
It just blows my mind.