Möbius strip

The picture is a strip glued into a loop after 0, 1, 2 or 3 half-twists. With one half-twist it is a Möbius strip. The inset above the panel is the same paper cut open and laid flat; when you paint, it shows the front and the back in two rows. Here you can paint the strip, walk an arrow round it and cut it, and see which results depend only on whether the number of half-twists is odd and which depend on the number itself.

Whether the strip has one side depends on how its faces meet at the glued ends. With an even number of half-twists, front is glued to front and back to back, so each face closes up on itself. The roller goes once round on its own face and cannot reach the other without crossing an edge: two sides. With an odd number, the front’s right end is glued to the back’s left end, upside down, and the two faces join into one band twice as long. After one lap the roller is back at its start but on the other face, and after a second lap both faces are painted. It never crossed an edge, so the two sides are one.

The roller looks from outside. The arrow is printed in the paper and seen through it, as a creature living in the paper would see it. Walked once along the middle line of a Möbius strip, it comes back to where it set out, turning anticlockwise against its clockwise ghost. Off the middle line it ends at the matching point across it instead. A surface on which some loop turns clockwise into anticlockwise is called non-orientable. For a surface in ordinary space this is the same as being one-sided, but orientability belongs to the surface alone: a creature that cannot leave the paper can still find it by walking.

With Off the middle at 1 the arrow walks along an edge. On a Möbius strip one lap leads to what looks like the other edge, and only two laps lead back: there is one edge. With three half-twists that single edge is tied in a trefoil knot, the simplest knot. With an even number $k$ there are two edges, linked $k/2$ times.

Cut a Möbius strip down the middle and it does not fall in two. The paper on either side of the cut is joined at the seam, upper side to lower, into one long band that goes twice round, with two sides and two edges. Its two edges are linked twice. With its middle line laid out as a flat circle, each link is one full twist, so the long band has two full twists.

Cut a Möbius strip a third of its width in from an edge and the scissors need two laps to come back: the two lines on either side of the middle line in the inset join at the seam into one cut. The cut leaves a thinner Möbius strip in the middle, and the two margins join into a long band; the two pieces are linked once. On an even strip every cut closes after one lap and simply adds a piece.

So the number of half-twists does two jobs. Whether it is odd or even decides the surface: as surfaces, without the space round them, all the odd strips are the same, one-sided with one edge, and so are all the even ones. The exact number decides how the surface sits in space: how its edges are linked or knotted, which no movement in space can undo without the paper passing through itself.

August Ferdinand Möbius and Johann Benedict Listing found the strip independently in 1858.