Paradoxically it is easier to make an infinity scarf than it is to describe one, but in the service of enlightenment, here goes.
An infinity scarf is the fashionable version of a Möbius Strip, which, in simple terms, is a loop of material that incorporates a half-twist along its length, which lends it the fascinating properties of having only one surface, and only one edge. Our mind may argue but a fingertip run along the surface does not lie (nor do M.C. Escher's ants, shown below in ceaseless transit around a Möbius grid).
Möbius strips are mathematical oddities that refuse to stay put in their natural realm--even music has been subjected to their warped yet inviolate topography. Below a fascinating version of J.S. Bach's Crab Canon, given the Möbius treatment.
Fashionistas need not look at the mathematicians and artists and musicians with envy, for we too can make a Möbius marvel out of an ordinary strip, in our case a knit scarf. It will, in turn, perform the small miracle of eliminating the problem of how to deal artfully with the ends: this scarf has no ends. Simply loop it several times around your neck and drape to your satisfaction. Something no doubt mathematical about its twisted nature causes it to settle around the cylinder of your neck in a way far more pleasing than would a simple ring of the same fabric.
Tips on making an Infinity Scarf:
1) Find a long and narrow knit scarf in your drawer or at a charity shop/thrift store--it should be at least long enough to loop a couple times around your neck before you do anything to it. If it has fringes, cut them off.
2) A stretchy, looser knit will work far better than a tightly woven one. Also, a looser knit lends a less obvious seam.
3) Find some sewing thread to match the scarf. It's always best to match thread type to fabric, so if you can get hold of some wooly thread, all the better, although this isn't critical.
4) Now comes the (slightly) tricky part. Lay the scarf out flat. Then, pick up one of the ends, one corner with each hand. Cross your wrists so the underside of the fabric flips around up top. Keeping fabric oriented like this, swap your hands on the corners. Bring the edge up to meet its opposite, pin together, and sew them together as neatly as you can.
Presto! At top is one I made last year, from an originally beautifully knitted by my sister-in-law . . .
To the highbrows who claim fashion is all surface--in this particular case, yes indeed.
(Mobius Strip II (Red Ants) © Cordon Art BV - Baarn - the Netherlands, from Math and the Art of MC Escher, many thanks)