Topology — page 5
How short a paper Möbius band can be
A strip of paper ten times as long as it is wide makes a Möbius band easily. A square does not: paper bends but does not stretch, and a half-twist has to be paid for in length. Fold a strip of length √3 three times at sixty degrees and its ends meet turned over — a Möbius band, flat, in the shape of a triangle. Five folds manage it with 1.6246, and more folds approach π/2, but only by letting the paper pass through itself. Richard Schwartz proved in 2023 that √3 is the true limit for a band that does not.
Many crossings and a small linking number
Two random loops of a thousand steps, tangled round a common centre, cross each other about a thousand times in any picture of them. The linking number is half the signed count of those crossings — and it is typically about three. The signs cancel in nearby pairs, wherever one loop dips under the other and comes back, and what survives grows roughly as the fourth root of the length. Shut the same loops in a box and the cancellation stops: the crossings grow as n², and the linking number as n.