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Concept

Ordinary line

A line through exactly two points of a finite set in the plane. The Sylvester–Gallai theorem says that any finite set of points not all on one line has at least one, and later work bounds how few there can be.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

Also named here as point set — the same set of essays touches all of them, so they are one junction rather than several.

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.

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 · Ordinary lines A triangle, its midpoints and its centroid, turned into lines. The dual arrangement of 7 points: one line per point, crossing where points were collinear. 3 crossings are of exactly two lines, the dual of the ordinary lines; the others are where three or more meet.

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.

geometry · Ordinary lines Böröczky's 12 points and their 6 ordinary lines. A disc standing for the projective plane: the 6 corners of a regular polygon inside, and 6 points at infinity marked in pairs on the rim. All 22 connecting lines are drawn, the 6 ordinary ones solid.

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 · Ordinary lines

Named alongside it

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

Extremal configurationIncidencePoint setProjective planeCollinearConfigurationConjectureCounting two waysDualityEuler characteristicExhaustive searchMinimal counterexample

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