Computer-assisted proof
Named by 3 essays across 3 fields — each of them below, with the objects they name alongside it.
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.
A region the orbit cannot leave
That the Hénon map and the Lorenz flow have attractors takes two lines of arithmetic each — a region mapped strictly inside itself, a quantity that falls outside an ellipsoid. That the attractors are strange took a computer and twenty years for Lorenz, and for Hénon's classical parameters it has never been done. The difference between the two questions is the difference between a region and what lives in it.
The machines that need a reason never to stop
Run every three-state machine on a blank tape and the ones that halt announce themselves: the longest stops after 21 steps. The work is in the others. Each needs a reason it will never stop, and four simple kinds of reason settle all but 27 of them — machines that sweep back and forth over a growing tape and defeat every check that looks for a repeat.
Named alongside it
The objects these essays reach for when they reach for this one.
Busy beaverChromatic numberDecidabilityEuler characteristicExhaustive searchGraphGraph colouringHalting problemHenon mapInterval arithmeticLorenz systemLyapunov exponent