Matpic
mathematics, in pictures
Generator

incidence

A generator in the computation library, called 13 times across 3 essays. Below: what it draws at its defaults and at each mode an essay asks for, what it checks while drawing, and everywhere it is used.

incidence is one function. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page — so a figure here is the same figure a reader meets in an essay, and if the generator changes, this page changes with it.

At its defaults

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

show: "steiner"

A schedule on 9 points where every pair meets exactly oncePoints around a circle with the triples of a Steiner system drawn between them, beside the list of triples.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

show: "sizes"

Which sizes admit a system of triples, up to 25A grid of sizes with the ones admitting a Steiner triple system filled in.n ≡ 1 or 3 (mod 6)3456789101112131415161718192021222324253 · 7 · 9 · 13 · 15 · 19 · 21 · 25 — the sizes at which a schedule of triplescovering every pair once can existthey are exactly 1 and 3 mod 6, and the reason is two divisions: (n−1)/2 peopleto meet in pairs, n(n−1)/6 triples to hold them

show: "matrix"

The incidence table of the projective plane of order 3A square grid with a mark wherever a point lies on a line, for the projective plane over a small field.13 × 13, 52 markslines ↓ points →the plane over GF(3): 13 points, 13 lines, 4 points on each line and 4 lines througheach pointevery one of the 78 pairs of points was checked to lie on exactly one line, andevery pair of lines to meet exactly once

What it checks while it draws

Collected by running the family and recording what it asserted, not written here. The count is how many separate times the claim was put to the test while these drawings were made.

Where it is called

Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.

Computation

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

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

The thirty-six officers

Six regiments send six officers each, one of every rank. Arrange all thirty-six in a square so that each row and each column holds every rank once and every regiment once. Euler could not, guessed why, and was wrong about the reason.

The whole library · What the figures prove