Generator

The Fano plane, and the incidence table behind it

A generator in the computation library, called 17 times across 6 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 it. Seven points joined by six straight lines and one circle, beside the seven-by-seven table of which point lies on which line.

`show: "steiner"`

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.

`show: "sizes"`

Which sizes admit a system of triples, up to 25. A grid of sizes with the ones admitting a Steiner triple system filled in.

`show: "matrix"`

The incidence table of the projective plane of order 3. A square grid with a mark wherever a point lies on a line, for the projective plane over a small field.

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.

the pair 0,1 appears once ×105
a construction is known here for this many points ×1
a design on this many points was actually found by search ×1
a projective plane over GF(q) has q² + q + 1 points ×1
any two lines meet in exactly one point ×1
each point lies in (v−1)/2 triples ×1
every line carries q + 1 points ×1
every pair of points appears ×1
every point lies on q + 1 lines ×1
exactly one line passes through any two points ×1
the design drawn is one that exists ×1
the design has v(v−1)/6 triples ×1
the divisibility test and the residue rule agree ×1
the drawing is the projective plane over GF(2), up to relabelling ×1
the largest number of points in the table is a whole number between 9 and 25 ×1
the largest number of points tested is a whole number between 9 and 40 ×1
the number of points in the design drawn is a whole number between 7 and 15 ×1
the number of points in the design is a whole number between 3 and 21 ×1
the plane is built over a prime between 2 and 5 ×1
the plane is built over a prime field between 2 and 5 ×1
the points are labelled or not ×1
the residue rule and the divisibility rule agree at every size ×1
the search agrees with the arithmetic ×1
the size of the permuted set is a whole number between 1 and 8 ×1
the three midpoints are the same distance from the centre, so one circle holds them ×1
the two divisibility conditions are exactly v ≡ 1 or 3 mod 6 ×1

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.

Topology

A disc sewn to a Möbius band

The band has one boundary curve, and a disc has one boundary curve. Sew them together and the result is the smallest closed surface with one side — the one every other one-sided surface is built out of.

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.

Geometry

The line with only two points on it

Scatter finitely many points on a page, not all in one line, and draw every line through two or more of them. However cunningly the points are placed, some line ends up carrying exactly two — and the proof is a minimisation with no algebra in it at all.

Computation

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

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