Concept

Non-orientable

A surface on which a consistent sense of clockwise cannot be chosen, because some route reverses it. A shape carried round the offending route comes back mirrored, and no consistent choice of normal exists.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

A Möbius band. A strip joined end to end after a half twist, so it has one side and one edge.

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.

topology · Orientability

A point, a ray, and 9 crossings. A closed curve wound into a spiral corridor, with a marked point, a ray from it and every crossing marked; an odd count means the point is inside.

Which side of the line is inside

A closed curve with no self-crossings divides the plane into an inside and an outside. Nobody doubts it, almost nobody can prove it, and on a curve wound tightly enough nobody can see which side a given point is on either.

topology · Jordan curve

The four ways to glue a square's edges in pairs. Squares with their edges arrowed to show which is glued to which and which way round, each with the vertices, edges and faces the gluing leaves, and the surface those numbers name.

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.

topology · Surface classification

Named alongside it

The objects these essays reach for when they reach for this one.

Boundary Connectedness Euler characteristic Gluing diagram Klein bottle Möbius band Orientation Topological invariant Closed curve Continuity Counterexample Genus

All concepts