Matpic
mathematics, in pictures
Concept

Planarity

The property of a graph that it can be drawn in the plane with no two edges crossing.

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

the mapwho touches whom

Four colours, and a proof nobody can read

Every map on a plane can be coloured with four colours so that no two neighbours match. The statement is understandable by a child, it resisted a century of attempts, and the proof that settled it cannot be checked by a human being.

discrete · graph colouring K4no crossingsK51 crossing, and no drawing has noneK3,31 crossing, and no drawing has noneK4: 4 points, 6 edges, and at most 6 allowed; K5: 5 points, 10 edges, and at most 9 allowed; K3,3: 6 points, 9 edges,and at most 8 allowed6,000 layouts were tried for each; the best found is drawn, and the bound is what rules out anything better

Two graphs that will not lie flat

Five points, every pair joined: no matter how the points are placed or how the lines are drawn, two of the lines cross. The proof is not about drawing at all — it counts edges against faces and finds one edge too many.

discrete · planarity

Named alongside it

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

GraphGraph colouringPlanar graphChromatic numberComplete graphComputer assisted proofCounting argumentCrossing numberEuler characteristicEuler formulaSubdivisionTopological invariant

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