Concept
Genus
2 essays name this object, across one field. What follows is each of them, and the objects they name alongside it.
Every corner pays for itself
Count the corners of any solid, subtract the edges, add the faces. The answer is two. It is two for a cube, for a pyramid, for a football, for anything squashed or stretched — and the number is measuring the shape it is wrapped around rather than the shape itself.
Nothing on a sphere can be combed flat
Point an arrow along the surface at every place on a sphere, continuously, and somewhere an arrow has to vanish. On a doughnut it can be done. The difference between the two is a number that was already known from counting corners.
Named alongside it
The objects these essays reach for when they reach for this one.
Euler characteristicTopological invariantContinuityFixed pointGraphIndexNonconstructiveOrientationPlanar graphPolyhedronVector fieldWinding number