Matpic
mathematics, in pictures
Concept

Gluing diagram

A polygon with its edges paired and arrowed, describing the surface those identifications produce.

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

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

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 abbaabb⁻¹a⁻¹V 3 − E 2 + F 1 = 2two-sideda sphereabababa⁻¹b⁻¹V 1 − E 2 + F 1 = 0two-sideda torusabababab⁻¹V 1 − E 2 + F 1 = 0one-sideda Klein bottleababababV 2 − E 2 + F 1 = 1one-sideda projective planethe corners are coloured by the vertex they become, so the count of colours is the V in V − E + Fthe characteristic and the orientability are computed from the word; the name underneath is looked up from that pair

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.

Euler characteristicKlein bottleMöbius bandNon orientableOrientationBoundaryConnectednessGenusPolyhedronTopological invariant

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