Concept

Closed surface

A surface with no edge and no boundary, finite in extent. Every one of them is a sphere with some number of handles or with some number of cross-caps, and which of the two lists it is on is decided by whether a sense of turning survives being carried round every loop.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Also named here as immersion — the same set of essays touches all of them, so they are one junction rather than several.

The Klein bottle, drawn where it does not fit. A closed one-sided surface in three dimensions, drawn as a tube with a figure-eight cross-section that turns over once on the way round, with the circle where the drawing passes through itself marked.

The bottle that needs a fourth dimension

Take the Möbius band's rectangle and glue the second pair of edges too. The result is closed, one-sided, and cannot be built in three dimensions without passing through itself — which is a fact about the room rather than about the surface.

topology · orientability

The gluing abab makes a projective plane. A polygon whose edges carry the word abab, with arrows for the direction each edge is glued and the corners coloured by which vertex they become.

A disc sewn to a Möbius band

The band has one boundary curve, and a disc has one boundary curve. Sew them together and the result is the smallest closed surface with one side — the one every other one-sided surface is built out of.

topology · orientability

Named alongside it

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

Euler characteristic Gluing diagram Immersion Orientability Antipodal map Boundary Classification Codimension Connected sum Cross cap Embedding Klein bottle

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