Concept

Pascals triangle

The triangle of binomial coefficients, each entry the sum of the two above it. Row n lists the ways to choose 0, 1, … n things from n, and the triangle's patterns — symmetry, row sums of 2ⁿ, divisibility — are identities about those choices.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

Elementary cellular automaton, rule 90. A row of cells evolving downward, each cell decided by the three above 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.

dynamics · Cellular automata

12 points, a cup of 5 and a cap of 5. 12 points in general position with the longest convex-upward chain (5 points) and the longest convex-downward chain (5 points) marked.

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.

discrete · Ramsey theory

Named alongside it

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

Binary Binomial coefficient Cellular automaton Convex position Determinism Exhaustive search Extremal combinatorics General position Iteration Locality Pigeonhole principle Ramsey theory

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