Möbius band
Named by 6 essays across one field — each of them below, with the objects they name alongside it.
The surface with one side, and what happens when it is cut
A strip of paper, half a twist, and a join. The result has one side and one edge, and cutting it down the middle does not produce two of anything.
Every surface is a sphere with handles
Take a square and say which edges are to be glued to which, and which way round. Four such rules give four different surfaces — and two numbers computed from the rule, without ever building the surface, say which one.
The third number a surface needs
Leave an edge unpaired in a gluing word and the surface acquires an edge of its own. Two numbers no longer name it — a count of boundary circles is needed as well — and with that third number the list is complete again, every triple occurring exactly once.
Every section of the band must vanish
Read the Möbius band as a line standing over each point of a circle, and choose a point on each line continuously: a section. On a cylinder a section can stay away from zero all the way round. On the band it cannot — every section crosses zero an odd number of times — and that single fact is what orientability means for a bundle. There are exactly two line bundles over a circle, told apart by one sign, and two Möbius bands added together make the plain cylinder's twin.
The room a projective space needs, read off Pascal's triangle
The projective plane cannot sit in three-dimensional space without crossing itself, and the reason can be written as arithmetic: a polynomial that records how a shape twists, which a room must cancel. For the n-dimensional projective space that polynomial is a row of Pascal's triangle read mod 2, its inverse is another row, and the inverse's last term says how many extra dimensions the room must have — exactly enough, at every power of two.
![Five squares on one octagon. Octagon [[1.0756,0.5516],[0.6082,1.2964],[-0.466,1.2294],[-0.8758,0.3153],[-0.797,-0.3348],[-0.3706,-0.7683],[0.2718,-0.6558],[1.1693,-0.4453]]; squares with sides 1.4975, 1.4588, 1.4407, 1.3929, 1.3921.](/f/five-squares-on-one-octagon-d1904a63.svg)
An odd number of squares on every polygon
Does every closed curve that does not cross itself pass through the four corners of some square? Toeplitz asked in 1911, and for polygons and smooth curves the answer is yes, because the squares can be counted and the count is odd. For rectangles of a fixed shape the count is even, which is why they needed a different proof — and for an arbitrary continuous curve the square is still not proved to exist.
Named alongside it
The objects these essays reach for when they reach for this one.
OrientationEuler characteristicGluing diagramNon-orientableBoundaryGenusKlein bottleTopological invariantVector bundleBinomial coefficientConjectureConnectedness