Matpic
mathematics, in pictures
Concept

General position

Said of points none of which satisfy a coincidence they need not: no three on a line, no four on a circle, no three lines through a point. Statements proved for points in general position hold for almost every arrangement, and the exceptions have to be built on purpose.

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

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 Seven points split into three groups whose hulls share a point. Seven random points in a square split into three groups of sizes 2, 2, 3 whose convex hulls share a common point; 4 of the 301 splits work.

Seven points split three ways

Any seven points in the plane can be divided into three groups whose convex hulls share a point, and six points in general position never can. The theorem is Helge Tverberg's; its topological version, in which straight lines may bend, is true when the number of groups is a power of a prime and false otherwise — and the proof that decides which is the same antipodal argument that halves a sandwich.

topology · Borsuk ulam

Named alongside it

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

Binomial coefficientBorsuk ulam theoremConvex hullConvex positionExhaustive searchExtremal combinatoricsPascals trianglePigeonhole principlePrime powerRadon theoremRamsey theoryTverberg theorem

All concepts