A paper loop with a half-twist that has only one side and one edge (1858)
The construction itself — the most famous object in topology
Science · Mathematics
The Möbius strip
1858
Mathematics — topology
A surface with only one side and one edge, made by giving a strip of paper a half-twist and joining the ends. It was described independently in 1858 by two German mathematicians, August Ferdinand Möbius and Johann Benedict Listing — Listing published first, in 1861, but the name stuck to Möbius. It is the standard emblem of topology, the branch of mathematics concerned with the properties of shapes that survive stretching and bending but not cutting or gluing, and the classic example of a non-orientable surface.
How to spot them
- ▸One side and one edge, made from a half-twist in a loop of paper
- ▸Cut it along the middle and you get one long loop, not two
- ▸Non-orientable: a shape slid round it comes back mirror-reversed
- ▸The emblem of topology, alongside the coffee cup that is a doughnut
- ▸Listing described it first in print, but the name commemorates the other discoverer
Key facts
Cut it lengthways down the middle and you get a single longer loop
A two-sided loop with four half-twists, not two separate rings
The international recycling symbol, three arrows with a twist (1970)
Gary Anderson's winning design in a 1970 competition; the arrows form the surface
A bottle with no inside, which needs four dimensions to build without self-intersection (1882)
The Klein bottle, which can be cut into two copies of this surface
The conveyor belt patented to wear evenly on both faces — which are really one
A genuine industrial application, patented for belts and typewriter ribbons
The prefix meaning 'above' that labels the higher-dimensional analogues of cube and sphere
Hyper-: hypercube and hypersphere, as in the four-dimensional tesseract
M. C. Escher's 1963 woodcut of ants marching round a twisted band (1963)
Möbius Strip II (Red Ants)
The theorem that a doughnut and a coffee cup are the same shape
Topological equivalence: both have exactly one hole
Key ideas
- Topology studies properties preserved under continuous deformation. A coffee cup and a doughnut are equivalent because each has one hole; a sphere and a doughnut are not.
- Orientability is the key property this surface lacks. On an orientable surface you can define a consistent 'outward' normal everywhere; here you cannot.
- Higher-dimensional analogues take the prefix hyper-: a hypercube (tesseract) is the four-dimensional cube, a hypersphere the four-dimensional sphere.
Quick-fire facts
- •August Ferdinand Möbius (1790–1868) and Johann Benedict Listing (1808–1882) discovered it independently in 1858; Listing published in 1861
- •Listing also coined the word 'topology' (Topologie) in 1847
- •The surface has one side, one boundary curve and Euler characteristic 0, and is non-orientable
- •Cutting along the centre line gives one loop with four half-twists; cutting a third of the way in gives two interlinked loops of different lengths
- •The international recycling symbol, designed by Gary Anderson in 1970, is drawn as this surface
- •The Klein bottle, described by Felix Klein in 1882, can be cut into two such strips
- •Two such strips of opposite handedness joined along their edges also give a Klein bottle
- •M. C. Escher depicted it in Möbius Strip I (1961) and Möbius Strip II (Red Ants) (1963)
- •In August 2023 Richard Evan Schwartz proved the Halpern–Weaver conjecture of 1977: a smooth embedded paper band of this kind must have an aspect ratio greater than √3, about 1.73
- •Topology's other standard facts: the hairy ball theorem, the Brouwer fixed-point theorem, and the seven bridges of Königsberg problem solved by Euler in 1736
- •The Poincaré conjecture, on the topology of the 3-sphere, was proved by Grigori Perelman in 2002–03
Drill it — 8 cards
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Sources
- Plus Magazine, Millennium Mathematics Project, University of Cambridge — Möbius and his band
- MacTutor History of Mathematics Archive, University of St Andrews — August Ferdinand Möbius