Generator

The nine-point grid, and the lines they force

A generator in the geometry library, called 28 times across 5 essays. Below: what it draws with nothing chosen and at each mode an essay asks for, what it checks while drawing, and everywhere it is used.

ordinary 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.

With nothing chosen

The nine-point grid, and the lines they force. 9 points with all 20 of their connecting lines drawn. The 12 carrying exactly two points are drawn solid and the rest faintly; the count is computed from the coordinates rather than read off the drawing.

Removing one end of an ordinary line from a triangle, its midpoints and its centroid

Lines against points, every subset of a 4 by 4 grid

Lines against points, four kinds of set

6 trees on a cubic, 4 rows of three

The cubic y = x³ − 12x at whole numbers from −4 to 4

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 even construction on 6 points has n/2 ×8
Melchior's inequality holds for 8 random points of a 4 × 4 grid (1) ×3
a line meets a cubic in at most three points ×1
a line with three points on it yields a strictly smaller distance ×1
a polygon of 3 to 10 sides ×1
a regular m-gon's chords point in exactly m directions ×1
and Csima and Sawyer's ×1
between four and twelve points on the curve ×1
Böröczky's set has exactly n/2 ordinary lines ×1
de Bruijn and Erdős: at least as many lines as points ×1
each pair of points lies on one connecting line only ×1
every construction respects Kelly and Moser's bound ×1
every family has at least as many lines as points ×1
every line through P either vanishes (if ordinary) or survives without P ×1
every point on a connecting line gives a dual line through its crossing ×1
every set with exactly as many lines as points is a near-pencil ×1
every triple is either collinear or clearly not — no decision rests on rounding ×1
every zero-sum triple of whole numbers is a row ×1
Melchior's inequality holds for a triangle, its midpoints and its centroid ×1
Melchior's inequality holds for five in a line and one off it ×1
Melchior's inequality holds for four points in general position ×1
Melchior's inequality holds for the nine-point grid ×1
Melchior's inequality holds for three points ×1
Melchior's inequality holds on every configuration ×1
no configuration beats one row per three pairs ×1
no configuration searched avoids ordinary lines altogether ×1
no set searched has fewer lines than points ×1
no two dual lines are parallel after the turn ×1
removing an end of an ordinary line removes that line entirely ×1
some connecting line holds exactly two of the points ×1
some point is off some connecting line ×1
some set of this size is not all in one line ×1
the census runs to between four and eight points ×1
the closest point-and-line pair uses a line with exactly two points ×1
the connecting lines between them account for every pair of points ×1
the cyclic group on the cubic meets Sylvester's count at every n ×1
the distance the argument produces is strictly the smaller one ×1
the drawn witch has the counted rows ×1
the five-by-five grid has 140 connecting lines ×1
the grid searched is three or four across ×1
the grid searched is three or four points across ×1
the highlighted triple adds to zero modulo n ×1
the highlighted triple is three point indices ×1
the ordinary line chosen exists ×1
the point set is one this family knows ×1
the points are drawn between 12 and 60 units apart ×1
the search runs to between four and seven points ×1
the set has an ordinary line to remove a point from ×1
the set is not all in one line ×1
the smaller set, not in one line, already has as many lines as points ×1
the three-point lines are the zero-sum triples of the cyclic group ×1
the view is one the family draws ×1
the window runs from −k to k, k between 2 and 5 ×1
the witch never beats the known maximum ×1
three points of the witch are in line exactly when their indices add to zero modulo n ×1
three points of y = x³ + ax are in line exactly when their x add to zero ×1

Where it is called

Every figure on this list is drawn by the same rule, so a change to the rule changes all of them at once. That is why the list is published.

Geometry

At least as many lines as points

Sylvester's theorem says some line through two of the points misses all the rest. Remove one end of that line and the line itself disappears, taking at least one line away with one point. Run that backwards and it proves that n points not all in a line determine at least n lines — and the only sets that manage exactly n are a line of n − 1 points with one point off it.

Geometry

Rows of three, planted on a cubic

Nine trees can be planted in ten rows of three, and the arrangement that does it is not a grid or a star but nine points on a cubic curve. On the curve three points are in line exactly when their angles add up to a right angle, so choosing the points as a cyclic group turns collinearity into addition — and the count of rows it produces is the number Green and Tao proved is the most any planting can reach.

Geometry

The fewest ordinary lines a polygon allows

Take the corners of a regular polygon and add the points at infinity where its parallel chords meet. Every chord then carries three points, the line at infinity carries all the new ones, and the only lines left with exactly two points are the tangents at the corners — half as many as there are points. Dirac guessed in 1951 that nothing does better, and Green and Tao proved it in 2013.

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.

Geometry

Three ordinary lines from a count

Kelly's proof finds one line through exactly two of the points by minimising a distance. Melchior, seven years earlier, had found three — by turning every point into a line and counting the corners, edges and regions of the picture that results. Euler's formula for the projective plane does the rest, and it says exactly which configurations have no more than three.

The whole library · What the figures prove