The Möbius strip

The strip is built around a circle of radius R (3 here). For each angle u around the circle, a short line segment of half-width w is placed across it, tilted by u/2. As u goes once around, the segment turns through half a revolution, which is the half twist. That gives x = (R + v cos(u/2)) cos u, y = (R + v cos(u/2)) sin u, z = v sin(u/2), with −w ≤ v ≤ w.

Start painting the strip anywhere and keep going without crossing the edge: you will paint the whole thing, both “sides”, and come back to where you started. Run a finger along the edge and the same thing happens: there is only one edge, a single loop. Surfaces like this, where you cannot consistently choose an inside and an outside, are called non-orientable.

Cut a paper Möbius strip along its centre line and it does not fall into two pieces: you get one longer loop with two full twists, which is two-sided. Cut a strip a third of the way in from the edge instead and you get two interlinked loops, one a Möbius strip and one a longer two-sided band. These are good experiments to try before looking at the explanation.

• Conveyor belts and old typewriter ribbons made as Möbius strips wear evenly on both faces, because there is only one face.
• The international recycling symbol is drawn as a Möbius strip.
• M. C. Escher’s 1963 woodcut Möbius Strip II shows ants walking along one.