Matpic
mathematics, in pictures
Concept

Fano plane

The smallest projective plane, with seven points and seven lines, three points on each line and three lines through each point.

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

001010100110101011111point on line?0123456•••••••••••••••••••••L0L1L2L3L4L5L6seven points, seven lines, three points on every line and three lines through every pointthe drawing was checked against the algebra by searching all 5,040 relabellings — one of them carries GF(2)³ onto thispicture

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 01234567812 triples0 1 20 3 40 5 60 7 81 3 51 4 71 6 82 3 82 4 62 5 73 6 74 5 89 points, 12 triples, each point in 4 of them — and every one of the 36 pairs appears exactly oncefound by backtracking over the pairs, which decides existence rather than assuming it

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

Named alongside it

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

Counting argumentIncidenceProjective planeBasisBlock designDivisibilityDualityExistence proofFinite fieldPrime powerSteiner triple system

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