Fano plane
Named by 5 essays across 3 fields — each of them below, with the objects they name alongside it.
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.
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.
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.
The symmetric graphs no tour can close
Take the words of five places with two ones and join two when they share no one. The ten words look exactly alike to the graph, and no closed tour runs through them — because every way of leaving out one edge at each word leaves two pentagons. Only four connected graphs this symmetric are known to fail like that, and nobody knows whether a fifth exists.
Two fields that every prime confuses
Reduce x⁸ − 3 and x⁸ − 48 modulo any prime other than 2 and 3 and factor them: the two lists of factor degrees are the same, at all 9,590 such primes below 100,000. The fields the two equations define are still different. The reason is a pair of ways for one group to permute eight roots, with the same cycle lengths element by element and different fixed points — the same coincidence, in degree seven, as the points and lines of the Fano plane.
Named alongside it
The objects these essays reach for when they reach for this one.
Counting argumentFinite fieldIncidenceProjective planeAffine planeBasisBlock designConjugacy classDivisibilityDualityExhaustive searchExistence proof