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Concept

Fano plane

The smallest projective plane, with seven points and seven lines, three points on each line and three lines through each point. It is the smallest example of a finite geometry, and its incidence structure is also the multiplication table of the octonions.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

The Fano plane, and the incidence table behind it. Seven points joined by six straight lines and one circle, beside the seven-by-seven table of which point lies on which line.

Seven points, seven lines

A geometry with seven points, in which every two points lie on exactly one line and every two lines meet in exactly one point. There are no parallels, the whole thing is built out of the two-element field, and one of its lines has to be drawn as a circle.

computation · Finite geometry A schedule on 9 points where every pair meets exactly once. Points around a circle with the triples of a Steiner system drawn between them, beside the list of triples.

A schedule where every pair meets once

Sort n people into groups of three so that every two of them share a group exactly once. Two divisions have to come out whole, that rules out most sizes — and at every size the divisions permit, a schedule exists.

computation · Finite geometry 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

Named alongside it

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

Counting argumentIncidenceProjective planeFinite fieldAffine planeBasisBlock designDivisibilityDualityExhaustive searchExistence proofImpossibility

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