Concept

Brouwer

The Dutch topologist whose fixed-point theorem says that every continuous map of a disc to itself leaves some point exactly where it was.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Also named here as fixed point — the same set of essays touches all of them, so they are one junction rather than several.

Something always stays put

Stir a cup of coffee however violently and let it settle. Some molecule is exactly where it started. Crumple a map and drop it on the region it depicts, and one point lies over the place it names.

topology · fixed points

Nothing on a sphere can be combed flat

Point an arrow along the surface at every place on a sphere, continuously, and somewhere an arrow has to vanish. On a doughnut it can be done. The difference between the two is a number that was already known from counting corners.

topology · fixed points

Named alongside it

The objects these essays reach for when they reach for this one.

Continuity Fixed point Nonconstructive Boundary Euler characteristic Genus Index Intermediate value theorem Orbit Orientation Retraction Topological invariant

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