Gluing diagram
Named by 10 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.
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.
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.
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.
The third number a surface needs
Leave an edge unpaired in a gluing word and the surface acquires an edge of its own. Two numbers no longer name it — a count of boundary circles is needed as well — and with that third number the list is complete again, every triple occurring exactly once.
Every way to pair a polygon's edges
A hexagon's six edges can be paired in fifteen ways. Glue each pair head to tail and five of the fifteen give a sphere and ten give a torus; an octagon's 105 pairings give 14 spheres, 70 tori and 21 surfaces with two handles. The spheres are exactly the pairings whose chords never cross, and the whole table obeys one recurrence found in 1986.
The surface a random gluing makes
Pair the edges of a large polygon at random and glue each pair head to tail. The surface almost always has nearly as many handles as the polygon allows: a thousand edges leave about seven and a half vertices, and the genus is within four of its ceiling of 250. The vertices behave like the cycles of a random permutation, and their average is a harmonic number.
Every surface is sewn from pants
A sphere with three holes — a pair of pants — is the smallest piece a surface with two or more handles can be cut into. Every such surface falls apart into them, and the Euler characteristic alone says how many pieces and how many cuts, however the cutting is done. What it does not say is the pattern, and the patterns are counted by drawing each one as a graph: two for two handles, five for three, seventeen for four.
Two copies of the polygon, cross-matched
The two-sided surface that lies over a one-sided one can be built from the gluing word alone: take two copies of the polygon, one read each way round, glue a letter within the copies when it is used once each way, and across them when it is used twice the same way. The recipe works for every surface at once, doubles every count, and shows that a sphere with k cross-caps is covered by the surface with k − 1 handles — and that among all its connected double covers, exactly one is two-sided.
Named alongside it
The objects these essays reach for when they reach for this one.
Euler characteristicGenusNon-orientableOrientationTopological invariantBoundaryKlein bottleMöbius bandOrientabilityCatalan numbersClassificationClosed surface