Let's start with something simple. Just about every kid learns about these early on. You take a strip of paper and join the ends in a loop after giving one end a half-twist so it joins up with the other one upside-down. If you've never tried this before, you should immediately (a) feel ashamed, and (b) try it out. The big deal about this is that it has only one side and one edge. The cool trick is that if you cut it in half lengthwise, it doesn't fall into two pieces. Instead, it makes one loop with two half-twists in it. Comes in handy in industry too: a Möbius fan-belt, with a surface that's twice as long as an ordinary one, takes twice as long to wear smooth, and you don't have to take it off to reverse it when one side wears out.
OK, so making a Möbius strip is easy, why bother mentioning it? Well, because I had fun making it in unusual ways. A page on the web mentioned knitting one from the edge. Obviously it's simple enough to knit one: you can knit a strip and graft the edges together. But knitting it from the edge is another matter: You wind up with a strip with no seam! It also rather freaked me out when I realized it was working. I didn't actually understand the description used in the web page, so I did my best to interpret it and basically worked it out myself. The result is pretty cool.
Here's a knitted, seamless Möbius strip. Click for a bigger image. It's made of a funky variegated acrylic yarn in seed stitch.
Here's what I did: start with a nice big circular needle (29-inch or so). I used one size 9, but it shouldn't matter much. Cast on a mess of stitches, at least enough to cover half the needle. Better to make it close to the whole thing, but make sure to cast on loosely (the success of the whole thing depends on a lot of give everywhere). Also, don't cast on too many stitches. You need them well spaced out, since the needle is actually going to hold twice as many. Also, you can't even get started if the last stitches are on the metal needle at the end of the flexible line. Once you have them cast on, carefully line up the two ends, preparing to close the circle. Make very sure that the stitches are not twisted... except for the twist you put in! Carefully make a half-twist, so that you are knitting into the bottoms of the cast-on stitches. This is bizarre, so pay attention. You are not knitting into the loops as usual, you are knitting into the stuff underneath the loops that holds them together. There are two strings running parallel there; I usually knit into the farther one. I suppose you could knit through both and maybe get better results. Note also that as you knit, you are not removing anything from the needle (or it would unravel). So you knit a stitch but leave it on the needle, and you move on and knit the next one. And so you wind up with your needle looping around itself another time: when you get back to the start and finally start knitting into the tops of the stitches, the needle has looped around twice. It's hard to describe, and a little mind-blowing the first time you see it, but it isn't all that complicated actually. For one-sided shapes like this I often find it good to use a stitch that looks the same from both sides, like "seed stitch" (I think that's what it's called: K 1, P 1 across, then K over all P stitches and P over all K stitches from then on). It also has the advantage over stockinette that it doesn't have the same tendency to curl. Still, the first one I did was stockinette, and you should try one: it's a little brain-bending to see how it works out, with the knit and purl sides and how they line up seamlessly. I suppose ideally one should work out the parity of the number of stitches so things work out right with the alternating knits and purls, but I've never bothered (so what if there's a small discontinuity?) It's a weird feeling: you knit around twice to truly do a row; the stitches go around, pass under your needles without being knitted, and then come back around again to be knitted. And of course at the end there's only a cast-off edge. With only one edge, obviously you can't have a cast-on and a cast-off edge! The yarn you started with sticks out of the center of the band.
Try it out; it's kind of fun!
Top view of a crocheted Möbius strip with three half-twists, folded into a flat triangle. Note the variegated strip around the edge. This is not the one I wear as a kipah, but was used as a proof-of-concept for it.
There are other possibilities too. Crocheting a Möbius strip is somewhat easier than knitting one. Just make a chain and close it off into a loop, but make a half-twist, so you first crochet into the "wrong" side and then into the "right" side as you go around. Then just single-crochet 'round and 'round, building up the strip. Crocheting is assymmetrical much as knits and purls are, but there's no purl for crocheting, so you wind up with a band which has different stitching on each half. Which is not really a bad thing. Crocheting is easy and quick, so this is an easy way to throw a quick Möbius strip together, in case you need one for a last-minute topologist party or something. It also isn't limited in size like working with a circular needle is: you can make it as big or small as you like. My wife made me a nice Möbius scarf several feet in diameter. You can also have fun with the colors; remember that when you make a full circuit around the work, it adds a stripe to both "edges." In that vein, I now have a bit of wearable topological art. Being of the Jewish persuasion, I wear a kipah (or yarmulka: a skullcap) all the time. And now I wear one that is a three half-twist Möbius strip, folded into a flat triangle. Remember, it's a one-sided strip so long as it has an odd number of half-twists, and all odd-half-twisted strips are topologically equivalent. You can change them into each other--if you're allowed to work in four dimensions. If you picture that the surface could pass through itself, you can see how it would go. I used three half-twists instead of one because it has better symmetry that way when folded up. My wife crochets faster and better than I do, so I asked her to make it for me (and all through it, she kept saying, "Mark, you are weird.") The one pictured here is not the one I wear; the one I wear is made of much thinner thread and finer stitches, and is colored differently. It's black with one light blue stripe and one white/silver stripe (though of course it looks like two of each because of how they loop around each other). Here's a picture of it.
I like pentagons, so thought maybe a five half-twist kipah would be cool... but it doesn't work. I think I've worked out the math and it can't close up the middle of the pentagon. Still, you can look at an example of the one I was trying it out with.