Four Colour Theorem
A constructive colouring of planar graphs
Every planar graph G satisfies χ(G) ≤ 4. This site gathers the working material for a constructive proof: interactive exhibits on the classical theorem and the structures met along the way, the writing that grew around the work, and a live map of the proof tree.
Interactive
Explore
Maps and dual graphs, Euler’s formula, Kempe chains, discharging, and a colouring playground for the classical theorem.
02Wrapping the Sphere
A 4-coloured triangulation shown as a panelled ball and wrapped onto a tetrahedron, with the folding degree and the degree-5 hole.
03The Tidy Trap
The colouring that defeats every cheap descent score, and three ways around it, including avoidance.
04The Dead-End Region
A symmetric hub where descent meets a ring of ten dead ends, and breadcrumb runs that find a way out.
05The Six-Ring Trap
A hole ringed by degree-six vertices: 74 ring patterns, a trap no local rule escapes, and a state four swaps deep.
06Cages and Diamonds
Fullerene-dual cages with no Birkhoff diamond, the diamond's ring colourings that extend or need a swap, and the one cycle of states at a (5,5,6,5,6) hole.
07The Spin-Glass View
Kempe chains as the cluster moves of a four-state antiferromagnet: what is proved on a torus and a sphere, and what the energy landscape looks like.
08The 9-8 Alternation
Near-rigid π-runs.
09Counting Kempe Chains
The chain count N around a degree-five vertex, Kempe classes, and the chain-parity law.
10Diamonds and Holes
The Birkhoff diamond, 2.122, the frame class, and the link words of a hole.
Writing
The Long Table
A play in one long night. Three colleagues trade rumours of the structural proof, find the last gate, and brainstorm until dawn.
PlayThe Afternoon Call
The sequel: a day of work, two leaked documents, and a video call that rewrites the plan.
PlayThe Vacancy
The third play, after Beckett. On a park bench, the same three colleagues sort the night’s loose pages and find that the hole now moves.
PoemThe Fifth Degree
The other team’s rhyming account of the proof, the creative team’s reply, and a navigator reading of both.
StoryA Mathematical Fever Dream
The inspiration story. Kempe swaps as charge flow on an anti-ferromagnetic Potts model, lifted to an SO(3) rotation of a tetrahedral magic gem.