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mathematics, in pictures
Ladder

Symmetry groups — the ladder

2 distinct arguments against one idea, from the one that introduces it to the one that assumes the rest.
  1. 123412431324134214231432213421432314234124132431312431423214324134123421412341324213423143124321all 24 ways of relabelling the 4 corners, and the 8 that move no distancethe other 16 change at least one distance between corners, so no motion of the plane performs them

    Eight ways to leave a square alone

    A square can be picked up and put back so that nothing looks different. There are exactly eight ways to do it, and the number is not asserted here — it is what a search through all twenty-four relabellings of the corners comes back with.

    rung 1 · algebra
  2. 1way4ways4ways2ways4ways1way16 colourings of 4 corners in 2 colours, pooled into 6 classestwo colourings share a row exactly when some motion carries one to the other

    Colourings nobody can tell apart

    Sixteen ways to colour four corners in two colours, and only six of them are genuinely different. The count can be got by pooling the sixteen — or by never forming a single class and instead averaging how many colourings each motion leaves untouched.

    rung 2 · algebra

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