Simplex
Named by 6 essays across 4 fields — each of them below, with the objects they name alongside it.
When the whole histogram deviates
A rare average has a price, an exponent that grows with the number of trials. Ask instead for the chance that the whole tally of outcomes comes out wrong, and the exponent is no longer a function of one number — it is a distance between two distributions, and every rare-average rate is a shadow of it.
A split nobody can walk away from
Every way of dividing what a group earns is a point of a triangle, and every coalition's threat to leave cuts a straight line across it. What survives all the cuts is the set of stable divisions — and for one three-player game there is nothing left.
Six in four dimensions, and three forever after
The count of regular solids goes five in three dimensions, six in four, and then three in every dimension above — for good. Four dimensions is the last place anything unusual happens, and it happens twice.
Zero in four dimensions
Corners minus edges plus faces is two for every solid. One dimension up, corners minus edges plus faces minus cells is zero for every one of the six regular four-dimensional solids, from the five-cell to the six-hundred-cell, and for every other convex solid in four dimensions. The alternating sum does not break when the dimension rises: it alternates, two in odd dimensions and zero in even ones, because it is measuring a sphere and not a solid.
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.
Two-thirds of a step past the line
Add random numbers between nought and one until the total passes one, and it takes e of them on average. Ask the same about passing any t and the answer is an exact finite sum that settles, within a few units, onto the line 2t + 2/3. The slope is one over the average step. The two-thirds is the interesting part: it is what the one step that crosses the line charges for being chosen by its length, and it is set entirely by how much the steps vary.
Named alongside it
The objects these essays reach for when they reach for this one.
PolytopeDimensionDualityAllocationCentral limit theoremClassificationCoalitionConstraintConvergence rateCooperative gameCoreCounting argument