Generator

ordinary

A generator in the geometry library, called 6 times across 1 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.

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.

At its defaults

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.

`show: "kelly"`

The closest a point comes to a line it is not on. A point set with the point-and-line pair at least distance marked, and the strictly smaller distance the same argument produces from any line carrying three points.

`show: "search"`

The fewest ordinary lines on a 4 by 4 grid. A table of point-set sizes against how many configurations of that size a 4 by 4 grid holds and the fewest ordinary lines any of them achieves, every configuration enumerated.

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.

a line with three points on it yields a strictly smaller distance ×1
each pair of points lies on one connecting line only ×1
no configuration searched avoids ordinary lines altogether ×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 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 distance the argument produces is strictly the smaller one ×1
the grid searched is three or four points across ×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 view is one the family draws ×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.

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.

The whole library · What the figures prove