Concept

Borsuk ulam theorem

The theorem that any continuous map from a sphere to space of one dimension fewer sends some pair of opposite points to the same place. Its relatives, built on other symmetries, prove that fair divisions, colourings and intersections must exist without constructing them.

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

Four equal quarters of an L-shaped plate. An L-shaped plate cut by two perpendicular halving lines at 48.8 degrees into four pieces of equal area.

Four equal quarters with two lines

Any flat shape, however lopsided, can be cut into four pieces of equal area by two perpendicular straight lines. The proof turns the pair of lines like the hands of a clock: the area in one quadrant, minus a quarter, reverses its sign every quarter-turn, so somewhere it is zero. The same kind of argument cuts a solid into eight equal pieces with three planes. It stops working in five dimensions, where some masses cannot be cut into thirty-two equal pieces by five hyperplanes — and in four, nobody knows.

topology · Borsuk ulam

Seven points split three ways

Any seven points in the plane can be divided into three groups whose convex hulls share a point, and six points in general position never can. The theorem is Helge Tverberg's; its topological version, in which straight lines may bend, is true when the number of groups is a power of a prime and false otherwise — and the proof that decides which is the same antipodal argument that halves a sandwich.

topology · Borsuk ulam

Named alongside it

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

Convex hull Fair division General position Ham sandwich theorem Intermediate value theorem Mass partition Polygon clipping Prime power Radon theorem Tverberg theorem

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