Matpic
mathematics, in pictures
Concept

Induction

A method of proof in which a statement about every whole number follows from the first case and from each case implying the next. Its strength is that a single argument covers infinitely many claims, and its cost is that the conclusion is only as sharp as the step.

Named by 4 essays across 3 fields — each of them below, with the objects they name alongside it.

One swap frees a colour. A vertex of degree five whose neighbours carry five different colours, before and after a Kempe chain is recoloured. The swap frees one colour for the middle vertex.

Five colours, and a chain that can be followed

The four-colour theorem cannot be checked by a person. The five-colour theorem can, in a page, and the argument that does it is the one Kempe thought had settled four — with the exact step where it fails visible in the picture.

discrete · Graph colouring 17 points coloured by whether their difference is a square. 17 points on a circle with every pair joined, coloured by whether the difference of their labels is a square modulo 17; the largest set of points all joined by one colour has 3 members.

Eighteen people, and the seventeen that escape

Among any eighteen people, four are mutual acquaintances or four are mutual strangers. Seventeen can be arranged so that neither happens, and the arrangement is not a lucky find — it is a rule about squares.

discrete · Ramsey theory The order type of a nonstandard model of arithmetic. The ordinary numbers as a run of dots, followed by 9 galaxies — copies of the integers — at the positions of c/2 up to 2c, ordered densely like the rationals, with c² beyond. Infinitely many more galaxies lie between those drawn.

A number larger than every number

Ask for a number bigger than 0, bigger than 1, bigger than 2, and so on for ever. Every finite piece of that request is granted by an ordinary number, so compactness grants all of it at once — in a structure that satisfies every sentence true of the whole numbers and still contains something beyond all of them. Nothing in first-order logic can say 'and nothing else'.

logic · Models Removing one end of an ordinary line from a triangle, its midpoints and its centroid. Two panels. Left: 7 points with all 9 connecting lines, one ordinary line solid and one of its ends ringed. Right: the same points with that end removed, 7 connecting lines left.

At least as many lines as points

Sylvester's theorem says some line through two of the points misses all the rest. Remove one end of that line and the line itself disappears, taking at least one line away with one point. Run that backwards and it proves that n points not all in a line determine at least n lines — and the only sets that manage exactly n are a line of n − 1 points with one point off it.

geometry · Ordinary lines

Named alongside it

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

AxiomChromatic numberCompactnessComplete graphConsistencyCounterexampleCounting argumentEuler formulaExhaustive searchExistence proofExpressive powerExtremal configuration

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