Concept

Affine plane

A projective plane with one line and its points removed, so that lines split into classes of parallels that never meet. Over a field of q elements it is the grid of q² points with coordinates, and its q² + q lines are the solutions of linear equations.

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

The affine plane of order 3, one parallel class at a time. The n² cells of a complete set of orthogonal Latin squares of order 3, with the rows, the columns and each square's symbol classes drawn as lines of a plane.

The plane hiding in the squares

A complete family of orthogonal squares is not a collection of squares that happen to agree nowhere. It is a geometry — a plane with n² points in which every two points lie on exactly one line — and reading it that way is how the impossible orders were found.

computation · Latin squares

Eighty-one cards and twenty with no SET among them. A three-by-three arrangement of three-by-three grids covering the eighty-one points of four-dimensional space modulo three, with twenty cells marked that contain no three on a line.

Twenty cards with no set among them

The card game SET is a four-dimensional space over the integers mod 3, and a set is a line in it. Twenty cards can avoid every line and twenty-one cannot — a fact that took a proof in 1970 — while laying cards down at random and stopping when nothing more fits reaches twenty about once in two thousand tries.

computation · Finite fields

Named alongside it

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

Counting argument Exhaustive search Finite field Arithmetic progression Cap set Fano plane Greedy algorithm Impossibility Incidence Latin square Orthogonal latin squares Pigeonhole principle

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