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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. He also founded intuitionism, under which the non-constructive proof he gave of his own theorem would not count as a proof at all.

Named by 7 essays across 2 fields — each of them below, with the objects they name alongside it.

A map of the interval must fix a point. a continuous map of the interval, drawn with the diagonal. Every continuous map of the interval into itself meets the diagonal somewhere; this one does so at x = 0.6944.

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 The field that cannot be combed. A tangent field on the sphere, flowing along the meridians. Every arrow is tangent to the surface, and at the two poles there is no direction for an arrow to take — the field is zero there, and no rearrangement removes both zeros.

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 One line, and both shapes halved. Two shapes and the single straight cut that divides each of them into two equal areas. The direction was found by sweeping every angle and watching the imbalance change sign.

One line that halves them both

Two shapes lying anywhere on a page, of any sizes and any shapes at all. There is always a single straight line that cuts both of them into two equal halves at once — and finding it needs no cleverness, only the observation that a quantity which reverses sign has to pass through zero.

topology · Borsuk ulam A three-coloured triangulation, and the walk that finds a rainbow triangle. A triangle cut into 36 smaller ones, its corners coloured under Sperner's rule. The 9 small triangles carrying all three colours are shaded, and a path enters through a door on one edge and ends inside one of them.

Three colours force a triangle

Cut a triangle into small ones and colour the corners under one restriction. However the cutting and the colouring are done, some small triangle ends up with all three colours — and the number of them is always odd.

discrete · Fixed points Three sets where a fixed point escapes, and one where it cannot. A ring turned about its centre, an open disc halved toward a point of its rim, the plane shifted sideways, and the closed disc turned and shrunk. Only the last has a point that its map leaves where it is.

Where the fixed point escapes

The theorem asks for a set that is closed, bounded and free of holes. Drop any one of the three and a map appears that moves every single point — and in each case the point that should have stayed still can be seen leaving.

topology · Fixed points Two opposite points on a globe with the same temperature and the same pressure. A world map in longitude and latitude with one curve where each point's temperature matches its opposite point's and another where the pressures match, crossing at a pair of opposite points that are marked.

Two opposite points that agree twice

At any moment there are two points on opposite sides of the Earth with the same temperature and the same pressure. On a seeded globe they sit at 11.9°N 44.6°E and 11.9°S 135.4°W. The reason is the circle argument that halved two shapes, run one dimension up: the differences between opposite readings, walked round the equator, wind round zero an odd number of times — and an odd number cannot be zero.

topology · Borsuk ulam A rule that meets the diagonal, and the same rule with a hole in it. Two panels, each the graph of a rule assigning a set of values to every point of the unit interval, drawn against the diagonal: the first meets the diagonal at a filled-in jump, the second has the jump left open and misses it.

A map that offers a choice

Brouwer's theorem needs a function, and the object it was most wanted for is not one — a best reply is a whole set whenever a chooser is indifferent. Allow a point to be sent to a set and the fixed point survives, provided the sets are convex, and the convexity is the entire hypothesis.

topology · Fixed points

Named alongside it

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

Fixed pointContinuityNonconstructiveExistence proofWinding numberAntipodal pairBoundaryDegreeFair divisionIntermediate value theoremTopological invariantBest reply

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