Concept
Möbius band
A strip joined end to end after a half twist, leaving one side and one edge. It is the standard example of non-orientability, and cutting it down the middle gives one longer two-sided band rather than two.
Named by 2 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.
Named alongside it
The objects these essays reach for when they reach for this one.
Euler characteristicGluing diagramKlein bottleNon-orientableOrientationBoundaryConnectednessGenusPolyhedronTopological invariant