The plane is not five-colourable (Hadwiger–Nelson)

About this app

How many colours does the plane need so that points at distance 1 never share a colour? Isbell's hexagons show that 7 suffice at the right size. A unit circle marks every same-coloured pair at distance 1; stripes, squares and 5-colour attempts visibly fail, and the Moser spindle and Golomb graph sit on the plane as unit-distance graphs. Result from the AI preprint catalogue openai/math (family 158, Lean-formalised): 5 colours never suffice, so the answer is 6 or 7.

Subject: Open problems, solved by AI

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