Pascals triangle
Named by 3 essays across 3 fields — each of them below, with the objects they name alongside it.
Eight rules and a triangle
A row of cells, each one deciding its next state from the three above it. Eight cases, one bit of output each — a rule that fits in a byte, and 256 of them in total. One of those bytes draws Pascal's triangle.
Every crowd holds a bowl or a dome
Among enough points in the plane, some k of them always bend upward like a bowl or some l bend downward like a dome. The number that forces it is a binomial coefficient, it is exactly right, and it proves that every large enough crowd contains a convex polygon — with a bound nobody could lower to the true answer for eighty years.
The room a projective space needs, read off Pascal's triangle
The projective plane cannot sit in three-dimensional space without crossing itself, and the reason can be written as arithmetic: a polynomial that records how a shape twists, which a room must cancel. For the n-dimensional projective space that polynomial is a row of Pascal's triangle read mod 2, its inverse is another row, and the inverse's last term says how many extra dimensions the room must have — exactly enough, at every power of two.
Named alongside it
The objects these essays reach for when they reach for this one.
Binomial coefficientBinaryCellular automatonConvex positionDeterminismEmbeddingExhaustive searchExtremal combinatoricsGeneral positionImmersionIterationLocality