Knitted Hyperbolic Surfaces

Large pieces of high-curvature hyperbolic space are better crocheted than knitted, I think, unless you are willing to use double-pointed needles (which I am not). Still, I made a silly hyperbolic toddler tutu, visible on a public ravelry page...

It's fun to toss these at students and ask (a) is the curvature positive or negative on these objects? and (b) which one has greater curvature?

I had knitted the above bits from the inside out, so to speak, by beginning with a small number of stitches and putting an increase on every stitch or every-other stitch. (That's exactly what one does when crocheting.) In summer of 2004, Ari Turner asked me what would happen if I knit them outside in, i.e. by starting with a needleful of stitches and uniformly decreasing. Here are the results:

These were done at 2:1, 3:2, and 4:3 stitch ratios, respectively. And it's much, much easier to knit outside->in than inside->out for these things...The recipe is as follows:

Cast on as many stitches as your needle will hold. (I use circular needles, always.)
2:1 stitch ratio? *K2 togtbl* each row until you feel done. Cast off.
3:2 stitich ratio? *K1, K2togtbl* each row until you feel done. Cast off.
4:3 stitch ratio? *K2, K2togtbl* each row until you feel done. Cast off.

Taken together, Ari and I agreed they look much like Xmas ornaments.

A nice, loose pseudosphere (photo on public ravelry page here) may be made in the following way:
Cast on as many stitches as your circular needle will hold. Join without twisting to work in the round.
*K10, K2togtbl* until one stitch remains. Yes, really, that works, and yes, really, that's all there is to it.

One cool thing that can be done with a hyperbolic octagon is folding it into pants. Knitting instructions for baby pants can be found in Making Mathematics with Needlework and a customizable version of the pattern is at the Wolfram Demonstrations Project. An eight-colored pair of hyperbolic pants may be seen here (public ravelry page); really, they are a dual map to K8 embedded on a 2-holed torus.

I did once make a hyperbolic toddler tutu (public ravelry page)...

Miscellaneous Mathematical Knitting

The color orders in the stripes of this scarf form the group S3; think of S3 as acting on the triple (green, grey, black).

Here are a few braids.

More braids and some knots appear on my public ravelry pages:
triquetra (flattened trefoil knot)
triple braid
(5,3) torus knot embedded on a torus
(3,2) torus knot embedded on a torus with one chirality
(3,2) torus knot embedded on a torus with the other chirality