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Ordinary lines — the series

5 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · geometry
  2. 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.

    part 2 · geometry
  3. 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.

    part 3 · geometry
  4. Removing one end of an ordinary line from a triangle, its midpoints and its centroid. Two panels. Left: 7 points with all 9 connecting lines, one ordinary line solid and one of its ends ringed. Right: the same points with that end removed, 7 connecting lines left.

    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.

    part 4 · geometry
  5. 9 trees on a cubic, 10 rows of three. The curve y = 1/(1 + x²) with 8 marked points on it and a 9th at infinity. Every line through three of them is drawn: 10 in all.

    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.

    part 5 · geometry

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