Matpic
mathematics, in pictures
Concept

Group action

A group applied to a collection, each element permuting it, so that composing the elements composes the permutations.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

1 string1 string5 strings5 strings5 strings5 strings5 strings5 strings32 strings fall into 8 necklaces32 strings in all: 2 constant ones, and 6 rings of 5so 32 − 2 = 5 × 6, and p divides a^p − a with nothing left over

Necklaces that prove a theorem

Thread five beads in two colours, thirty-two ways. Two of them are all one colour; the other thirty fall into rings of five. That count, and nothing else, is Fermat's little theorem.

number · fermats little theorem 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.

algebra · symmetry groups 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.

algebra · symmetry groups

Named alongside it

The objects these essays reach for when they reach for this one.

Cyclic groupCounting argumentCounting two waysDihedral groupLagrange theoremOrbitPermutationSymmetryClosureCompositeEquivalenceFermats little theorem

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