One that particularly caught my eye was the möbius cast on (which I have yet to attempt, and I suggest you don't hold your breath) where you create a knitted möbius strip. (For more about what that is, try the wikipedia article.)
By a strange coincidence, at about the same time as I bought the book (in the flush of enthusiasm that followed completing a very long project), my brother-in-law sent me a link to an article by Sarah-Marie Belcastro in the American Scientist, featuring some truly awesome mathematical knitting: including the möbius strip, and something called a klein bottle:
Neither my knitting nor my mathematical understanding are in this league, and I struggled with phrases like "a physical instantiation of an abstraction", but I am fascinated by the combination of an old-fashioned craft and difficult three-dimensional concepts. Here's the author's explanation of why it works so well:
You might wonder why one would want to knit mathematical objects. One reason is that the finished objects make good teaching aids; a knitted object is flexible and can be physically manipulated, unlike beautiful and mathematically perfect computer graphics. And the process itself offers insights: In creating an object anew, not following someone else’s pattern, there is deep understanding to be gained.
She then discusses knitting as geometry, the differences between knitting a surface or a 3-D object, and about the complexities of texture and colour in representing shapes and structure. Something for the true geek knitter to get their teeth into?
