Vector bundle
Named by 3 essays across one field — each of them below, with the objects they name alongside it.
Orientation is a sign
Carry a pair of arrows once round a loop and compare them with the pair they started as. The comparison is a determinant, its sign is the whole of the answer, and no room, no side and no normal vector appears anywhere in the statement.
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.
Named alongside it
The objects these essays reach for when they reach for this one.
DeterminantFrameMöbius bandOrientabilityOrientationTransportBinomial coefficientEmbeddingHandednessImmersionIntermediate value theoremIntrinsic property