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Decided by exhaustion — page 9

Questions with finitely many cases, settled by going through all of them — and what changes when a claim about every argument becomes a count.

The 4 reduced forms of discriminant −84, each in its place. The reduced binary quadratic forms of discriminant −84 (x² + 21y²; 2x² + 2xy + 11y²; 3x² + 7y²; 5x² + 4xy + 5y²) plotted at their roots in the upper half-plane, all inside the modular fundamental region.

Number

Counting the classes that break factorisation

The class number measures how badly unique factorisation fails in a field, and defined through ideals it looks impossible to compute. Gauss computed it by hand, for every field he wanted, by counting quadratic forms — and every form can be squeezed, by changes of variable that keep its values, into exactly one small standard shape.

The narrowest door in two 6-cliques joined by one edge, and the gap it pins down. two 6-cliques joined by one edge, with the vertex set of smallest conductance coloured and the 1 edges leaving it thickened. Beside it a logarithmic ruler marks half the conductance squared, the spectral gap and twice the conductance, in that order from the bottom.

Probability

The narrowest door sets the pace

How fast a chain forgets is an eigenvalue, and nobody can compute the eigenvalues of a chain worth studying. Cheeger's inequality trades the eigenvalue for a picture — the narrowest door in the state space — and pins the one between the square of the other and twice it. Both ends of that range are reached, on graphs small enough to search completely.

Every way to cut a genus-2 surface into pairs of pants. 2 thickened graphs, one for each type of pants decomposition of the closed surface of genus 2, each with 2 coloured junctions and 3 cutting circles.

Topology

Every surface is sewn from pants

A sphere with three holes — a pair of pants — is the smallest piece a surface with two or more handles can be cut into. Every such surface falls apart into them, and the Euler characteristic alone says how many pieces and how many cuts, however the cutting is done. What it does not say is the pattern, and the patterns are counted by drawing each one as a graph: two for two handles, five for three, seventeen for four.

The seven-vertex torus, unrolled onto a lattice. A triangular lattice with every point labelled a + 3b mod 7 and fourteen triangles shaded as one copy of the torus.

Topology

The fewest corners a surface needs

Build a closed surface from triangles, any two meeting along a whole edge, at a single corner or not at all, and ask for the fewest corners. Two lines of counting give a floor for every surface, in terms of its Euler characteristic alone. The torus meets it with seven, the projective plane with six — and the Klein bottle, which the counting says could be built from seven, cannot, as a search of every possible arrangement shows.

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