Incidence
Named by 20 essays across 3 fields — each of them below, with the objects they name alongside it.
What two points can build
A compass and a straightedge are not a craft. They are two operations on a set of points, applied over and over, and writing them that way turns "can this be drawn?" into a question with an answer.
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.
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.
Nine points on one circle
Three midpoints, three feet of altitudes and three more midpoints. Nine points defined in three unrelated ways, on an arbitrary triangle, and all nine sit on one circle — checked here on two hundred and forty triangles as well as on the drawn one.
The straightedge buys nothing
Every point a compass and a straightedge can construct together can be constructed by the compass alone. The straightedge draws lines nobody needs; the compass does the work, and the proof that it does is an inversion performed with arcs.
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 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.
One circle touching four
The nine-point circle touches the inscribed circle and each of the three escribed ones. Nothing in its construction mentions them, the two families of centres are built from different kinds of number, and the tangency is four exact equalities between distances and radii.
A centre is three weights
Write each classical centre as a weighted average of the corners and a coincidence becomes a determinant. The Euler line is then one number rather than a construction, the whole catalogue becomes mechanical, and the reason one centre is missing from it is visible in the weights.
More blocks than points
A schedule in which every pair meets once cannot use fewer groups than it has people. Nothing about the counting conditions says so, and the proof is not combinatorial at all — it is a determinant, computed over a field the schedules have nothing to do with.
A plane in a list of numbers
A projective plane of order three has thirteen points and thirteen lines and fifty-two incidences. All of it is in the four numbers 0, 1, 3, 9 — because their pairwise differences hit every non-zero residue modulo thirteen exactly once, and the plane is that list's thirteen shifts.
A plane no field built
Every finite field builds a projective plane, and for a long time every known plane was built that way. The plane over Dickson's nearfield of order nine has ninety-one points, ninety-one lines and every incidence right — and Desargues' theorem fails in it on most configurations tried, except for one line, from which it never fails at all.
The curve that no three points in line define
In a finite plane, take as many points as possible with no three on a line. In odd order the largest such sets have one more point than the order — and every one of them, searched exhaustively in the small planes and proved by Segre for all odd orders, is a conic. In even order every tangent meets at one point, which can be added, and the curves stop being forced.
The densest graph without a square
Forbid four points joined in a cycle and a graph can keep only about ½n^(3/2) of its edges — far fewer than the quarter of all pairs a triangle-free graph keeps. Counting pairs of neighbours proves the ceiling in two lines. What reaches it is not a random graph but a finite geometry: the points of a projective plane, joined when they are orthogonal.
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 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.
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.
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.
The orders a plane cannot have
Every counting condition allows a projective plane of order six, and there is none. The proof that rules it out looks at one matrix identity — each point on seven lines, each two points on one — and turns it, by way of Lagrange's four squares, into the statement that six would have to be a sum of two squares. Run on the planes that do exist, the same argument hands back their orders as sums of two squares; run on six, it asks for something no arithmetic can supply.
Every power of x that draws a hyperoval
In a plane of order 2^h, the graph of x^k plus two points at infinity is sometimes a hyperoval — as many points as a plane allows with no three in line. Searching every exponent in every plane from order 4 to 4096 finds hundreds that work, and once six symmetries of the problem are applied they fall into exactly the families already known: the conic, the translation curves, Segre's x⁶ and Glynn's two. Whether that list is complete in every order is open.
Named alongside it
The objects these essays reach for when they reach for this one.
Projective planeCounting argumentExhaustive searchFinite fieldExtremal configurationOrdinary linePoint setSymmetryCircleConjectureCounterexampleExistence proof