Matpic
mathematics, in pictures
Concept

Non orientable

A surface on which a consistent sense of clockwise cannot be chosen, because some route reverses it.

Named by 3 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.

topology · orientability insidethe ray crosses the curve 9 times, so the point is insidefired in any of 360 directions from the same point, the count changes and its parity does not

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 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.

BoundaryConnectednessEuler characteristicGluing diagramKlein bottleMöbius bandOrientationTopological invariantClosed curveContinuityCounterexampleGenus

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