Euler formula
Named by 10 essays across 3 fields — each of them below, with the objects they name alongside it.
Four circles cannot do it
Three overlapping circles cut the plane into exactly the eight regions three sets need. Four circles cut it into fourteen, and sixteen are required — so the diagram everyone draws stops working at four, and the reason is a count.
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.
Two trees, and every edge in exactly one of them
Euler's formula is usually proved by deleting things until nothing is left. There is a better argument that deletes nothing — a tree through the corners and a tree through the faces, which between them use every edge once and can therefore be counted.
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.
The solid whose corners are triangulations
Take the triangulations of a hexagon as points and join two of them when a single diagonal can be swapped for another. The result is not merely a graph — it is the edge skeleton of a genuine convex polyhedron, with fourteen corners, three square faces and six pentagonal ones.
Five spokes squeezed into K5
The Petersen graph has no point with four neighbours, so no stretched copy of K5 can sit inside it. Contract its five spokes and K5 appears anyway. Kuratowski's theorem forbids stretched copies and Wagner's forbids squeezed ones, the two notions disagree on this graph — and they still name exactly the same planar graphs.
The few points that cut a flat graph
Any graph that can be drawn without crossings, however large, falls into pieces of at most two thirds once a few points are removed — about the square root of its size, never more than 2.83 times it. A grid shows the square root cannot be beaten, a ring of breadth-first neighbours comes close, and a cycle through a shallow tree finishes the job.
Twelve pentagons, whatever the hexagons
A football has twelve pentagons and twenty hexagons. A molecule of sixty carbon atoms has the same pattern, a molecule of seventy has twelve pentagons and twenty-five hexagons, and a geodesic dome of any size has twelve places where the pattern of six breaks. None of this is a coincidence of design: Euler's formula, rearranged, says that faces meeting three at a corner must fall short of hexagons by exactly twelve in total, and the hexagons are free.
Zero in four dimensions
Corners minus edges plus faces is two for every solid. One dimension up, corners minus edges plus faces minus cells is zero for every one of the six regular four-dimensional solids, from the five-cell to the six-hundred-cell, and for every other convex solid in four dimensions. The alternating sum does not break when the dimension rises: it alternates, two in odd dimensions and zero in even ones, because it is measuring a sphere and not a solid.
The crossings a graph cannot avoid
Five points all joined need one crossing, six need three, seven need nine. Euler's formula gives a lower bound that grows like the square of the number of points and is badly wrong; a sampling trick turns the same formula into a bound that grows like the fourth power and is right to within a constant. And for drawings with straight edges the count turns out to be something else entirely: the number of quadrilaterals the points make.
Named alongside it
The objects these essays reach for when they reach for this one.
Planar graphPlanarityComplete graphEuler characteristicGraphGraph colouringConvexityCrossing numberLower boundPolytopeSubdivisionAngle defect