Pascals triangle
Named by 5 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.
How often a number can appear in Pascal's triangle
Every number bigger than one appears in Pascal's triangle at least twice, next to the 1 at each end of its own row. A few appear more often: 120 appears six times, and 3003 eight — as C(3003, 1), C(78, 2), C(15, 5) and C(14, 6), each twice. Counting every entry up to a million million finds seven numbers that appear more than four times and none that appear five, seven or more than eight. Whether any number appears ten times, or whether there is any limit at all, is unknown.
Differences round a ring
Write four numbers at the corners of a square and put on each side the difference of its two corners. Repeat with the new square. Whatever the four whole numbers are, the process reaches nought within a few steps. Write five numbers and it almost never does. Rings whose size is a power of two always finish, and every other ring settles into a cycle whose length depends on how a polynomial factors over two elements: 7 at seven, 341 at eleven, 31 at thirty-one.
Named alongside it
The objects these essays reach for when they reach for this one.
Binomial coefficientExhaustive searchIterationBinaryCellular automatonConjectureConvex positionDeterminismDiophantine equationEmbeddingExtremal combinatoricsFibonacci numbers