In an article published after his death, German mathematician and astronomer Augustus Ferdinand Möbius (1790-1868) described a remarkable paper surface as a strip which has no "other side".

This one-sided strip, hard to imagine but easy to construct, has some unexpected properties.

- A strip of paper is two-dimensional (well almost, assume it has no thickness) having length and breadth; it has two sides or surfaces and one edge.

- When the strip is joined to form a loop the paper itself is still two-dimensional, but occupies three-dimensional space. It has two sides and two edges.

- When the strip is given a half-twist before it is joined, the paper is still 2-d, still occupying 3-d space, BUT it has only ONE surface and ONE edge.

"Curiouser and curiouser," said Alice.

The one-sided Möbius strip

A Möbius strip is easily made from an ordinary flat strip of paper: first the strip is given a half twist and then the two ends are connected to make a closed ring.

"Halving" a Möbius strip

When a cut is made around the middle of a Möbius strip, it might be expected to divide the strip in two. But when a line is drawn around the strip and the strip is cut along the line the result is not two strips but a two-sided strip.

The mathematicians explanation: a Möbius strip has but one edge: the cut adds a second edge and a second side.

A Möbius strip cut in thirds

A Möbius strip cut one third of the way in from its edge produces a fresh surprise; the scissors make two complete trips around the strip but only a single continuous cut.

The end result is two strips intertwined. One of the strips is a two-sided hoop and the other is a new Möbius strip, with its one continuous side bounded by a single edge.