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Concept

Klein bottle

The one-sided closed surface made by gluing a square's edges with one pair joined straight and one flipped. It cannot be embedded in three dimensions without passing through itself, and cutting it in half gives two Mobius bands.

Named by 5 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 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 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 Reducing abcabc to a standard form. The gluing word abcabc rewritten step by step into one of the classification's standard forms, with the move used and the two invariants recomputed at each step.

Every word driven to a normal form

The classification is usually met as a statement: two numbers name the surface. The proof is a procedure — a short list of cut-and-reglue moves that drive any gluing word to one of the standard forms, with a measure that never rises to say why the procedure stops.

topology · Surface classification The seven-vertex torus, unrolled onto a lattice. A triangular lattice with every point labelled a + 3b mod 7 and fourteen triangles shaded as one copy of the torus.

The fewest corners a surface needs

Build a closed surface from triangles, any two meeting along a whole edge, at a single corner or not at all, and ask for the fewest corners. Two lines of counting give a floor for every surface, in terms of its Euler characteristic alone. The torus meets it with seven, the projective plane with six — and the Klein bottle, which the counting says could be built from seven, cannot, as a search of every possible arrangement shows.

topology · Surface classification

Named alongside it

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

Euler characteristicGluing diagramNon-orientableGenusOrientationTopological invariantBoundaryMöbius bandClosed surfaceCodimensionComplete graphConnectedness

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