Intermediate value theorem
Named by 8 essays across 4 fields — each of them below, with the objects they name alongside it.
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.
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.
A matrix that counts the returns
Draw a map straight through the cycle 0 → 1/3 → 1 and its two pieces carry each other in a fixed pattern: the left piece only across the right, the right across both. The orbits' words are then walks on a two-node graph, and the number of points that come back after n steps is the trace of that graph's matrix to the nth power — 1, 3, 4, 7, 11, 18 — each one checked by solving for the points exactly.
As many cuts as colours
Two thieves steal a necklace and want half of every colour of bead each. However the beads are strung, they never need more cuts than there are colours — three cuts for three colours, four for four — and sometimes they need every one. The guarantee is the Borsuk–Ulam theorem again, with a point on a sphere read as a way of cutting the necklace, and every necklace of several small kinds has been checked against it.
A rent nobody envies
Three housemates, three rooms that are not alike, one rent. Every way of splitting the rent is a point of a triangle; ask, at each point of a fine grid, which room one housemate would take at those prices, taking turns so that each small triangle has one corner for each of them. Sperner's lemma then promises a small triangle where all three would choose different rooms — and as the grid is refined, the envy at that triangle shrinks to nothing.
![The slope of x² sin(1/x): discontinuous at nought, and never jumping. The derivative of x^2 sin(1/x) on [−0.12, 0.12], with the value at nought marked and the level 0.5 crossed 77 times.](/f/the-slope-of-x2-sin-1-x-discontinuous-at-nought-and-never-60857308.svg)
A slope can swing but never jump
A function can have a slope at every point without that slope changing continuously: x² sin(1/x) has slope nought at the origin and a slope that swings between −1 and 1 however close to the origin it is taken. What a slope cannot do is jump. Darboux proved in 1875 that a derivative takes every value between any two of its values, so a step is never a derivative — and the only way a slope can be discontinuous is by oscillating.
Every section of the band must vanish
Read the Möbius band as a line standing over each point of a circle, and choose a point on each line continuously: a section. On a cylinder a section can stay away from zero all the way round. On the band it cannot — every section crosses zero an odd number of times — and that single fact is what orientability means for a bundle. There are exactly two line bundles over a circle, told apart by one sign, and two Möbius bands added together make the plain cylinder's twin.
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.
Named alongside it
The objects these essays reach for when they reach for this one.
Fair divisionContinuityFixed pointNonconstructiveAntipodal pairBrouwerExistence proofBisectionBorsuk ulam theoremBoundaryCharacteristic polynomialComplexity