Finite geometry — the series

The Fano plane, and the incidence table behind it. Seven points joined by six straight lines and one circle, beside the seven-by-seven table of which point lies on which line.

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.

part 1 · computation

A schedule on 9 points where every pair meets exactly once. Points around a circle with the triples of a Steiner system drawn between them, beside the list of triples.

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.

part 2 · computation

A design on 7 points cannot have fewer than 7 blocks. The incidence matrix of a design on 7 points and 7 blocks beside the product of it with its own transpose, which has a constant off the diagonal and a determinant computed exactly.

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.

part 3 · computation

A plane of 13 points from a list of 4 numbers. A ring of 13 points with one block of 4 of them drawn as a closed path, beside the table of the 13 blocks its shifts produce.

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.

part 4 · computation

The one line from which the nearfield plane looks Desarguesian. A grid of the 91 lines of the nearfield plane of order nine shaded by how many of 40 Desargues configurations with that line as axis failed; only the line at infinity has none, and every other line at least 20.

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.

part 5 · computation

The conic y = x² in the plane of order 7. A 7 by 7 grid of the affine plane over GF(7) with the points of the conic y = x² filled and its point at infinity marked: 8 points, no three collinear.

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.

part 6 · computation

The plane of order 3 as a table, and the table times its transpose. The 13 × 13 incidence table of the projective plane of order 3 and its product with its transpose, which has 4 on the diagonal and 1 in every other cell.

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.

part 7 · computation

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.

part 8 · computation