Concept

Intermediate value theorem

A continuous quantity that takes two values also takes every value between them. It is what makes bisection work, and it is the one-dimensional ancestor of every fixed-point theorem in this collection.

Named by 2 essays across one field — 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

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

Named alongside it

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

Brouwer Continuity Fixed point Nonconstructive Antipodal pair Bisection Boundary Degree Existence proof Fair division Orbit Retraction

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