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mathematics, in pictures
Ladder

Surface classification — the ladder

One essay so far against this idea. A ladder is the distinct arguments that stand against one idea, and this one has room to grow.
  1. 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.

    rung 1 · topology

All ladders