Generator

The Fano plane, and the incidence table behind it

A generator in the computation library, called 53 times across 12 essays. Below: what it draws with nothing chosen and at each mode an essay asks for, what it checks while drawing, and everywhere it is used.

incidence 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.

With nothing chosen

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.

A plane of 13 points from a list of 4 numbers

A schedule on 13 points where every pair meets exactly once

A design on 13 points cannot have fewer than 13 blocks

Desargues' theorem in the ordinary plane

Multiplication in the nearfield of order nine

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.

blocks 0 and 1 meet in exactly one point ×210
points 0 and 1 lie on exactly one block ×210
points 0 and 1 share 1 blocks ×156
the pair 0,1 appears once ×105
GF(9): 1 has exactly one inverse ×31
point 0 is in 4 blocks ×20
the difference 1 occurs exactly once ×20
the diagonal of the Gram matrix is the block count at point 0 ×13
a prime power 2 is never excluded by Bruck–Ryser ×11
order 2 has a recorded status ×11
the generator of GF(4) has order 3 ×11
the powers of the generator of GF(4) do not repeat early ×11
PG(2, 5) has n² + n + 1 lines ×7
PG(2, 5) has n² + n + 1 points ×7
PG(2, 5): every two points lie on a line ×7
PG(2, 5): two points lie on only one line ×7
the parabola y = x² closes to an arc in order 3 ×6
in odd order 3 no other monomial graph is an oval ×4
the plane of order 3 has n² + n + 1 lines ×4
the plane of order 3 has n² + n + 1 points ×4
the plane of order 3: every two points lie on a line ×4
the plane of order 3: two points lie on only one line ×4
in PG(2, 3) the ovals number q⁵ − q², the count of non-degenerate conics ×2
no set of q + 2 points of PG(2, 3) avoids three on a line ×2
PG(2, 2) has arcs of q + 2 points ×2
the power of two is a whole number between 3 and 11 ×2
a construction is known here for this many points ×1
a design on this many points was actually found by search ×1
a failing configuration was found ×1
a hyperoval has no tangent line ×1
a number is a sum of two squares exactly when no prime 3 mod 4 divides it an odd number of times ×1
a point lies on n + 1 lines and two points share exactly one ×1
a prime power is never excluded ×1
a projective plane over GF(q) has q² + q + 1 points ×1
a sum of two squares is divisible by 3 only when both are ×1
after the elimination only the last variable is left ×1
and distributes from the right ×1
and each is a conic with its nucleus ×1
and fails in the nearfield plane ×1
and no pair differs by nothing ×1
and none larger ×1
and the remaining lines miss it ×1
and the two squares it hands back add to n ×1
and therefore there are at least as many blocks as points ×1
any two lines meet in exactly one point ×1
Bruck–Ryser rules out 6 but not 10 or 12 ×1
but not from the left, and it is not commutative ×1
Desargues never fails in the plane over GF(9) ×1
each point is in more blocks than it shares with any other ×1
each point lies in (v−1)/2 triples ×1
each vertical line holds one affine point of the curve ×1
enough configurations were drawn ×1
every block's size times the block count is the point count times r ×1
every class found belongs to one of the known families ×1
every exponent equivalent to one that works also works ×1
every line carries q + 1 points ×1
every member of the base block is a residue modulo n ×1
every other point off the oval lies on exactly one ×1
every oval of PG(2, 5) lies on a conic ×1
every pair of its q + 2 points spans its own line ×1
every pair of points appears ×1
every point lies on q + 1 lines ×1
every tangent passes through the horizontal direction ×1
every whole number is a sum of four squares ×1
exactly one line — the line at infinity — passes every test ×1
exactly one line passes through any two points ×1
in every other position it fails on a large share ×1
in order eight the exponents 2, 4 and 6 give hyperovals ×1
no fraction has a zero denominator ×1
no line meets the curve in three points ×1
no point off the oval lies on exactly one tangent ×1
no three of the points are collinear ×1
one point lies on every tangent: the nucleus ×1
q(q+1)/2 points lie on two tangents ×1
q(q−1)/2 points lie on none ×1
so the Gram matrix is invertible and the incidence matrix has full column rank ×1
the base block is between three and six distinct whole numbers ×1
the chosen sign leaves the variable solvable ×1
the design drawn is one that exists ×1
the design has v(v−1)/6 triples ×1
the design is one the family knows ×1
the determinant computed by elimination is the closed form ×1
the divisibility test and the residue rule agree ×1
the drawing is the projective plane over GF(2), up to relabelling ×1
the elimination ends at n x² = 1 + w², exactly ×1
the elimination is for orders that leave one or two over when divided by four ×1
the field order is one of 2, 3, 4, 5, 7, 8, 9 ×1
the field plane never fails (axis at infinity, centre off it) ×1
the field plane never fails (axis at infinity, centre on it) ×1
the field plane never fails (finite axis, centre off it) ×1
the field plane never fails (finite axis, centre on it) ×1
the first orders Bruck–Ryser excludes are 6, 14, 21, 22 and 30 ×1
the forced corners are new points off the axis ×1
the largest number of points in the table is a whole number between 9 and 25 ×1
the largest number of points tested is a whole number between 9 and 40 ×1
the largest order shown is a whole number between 20 and 120 ×1
the largest power of two searched is a whole number between 5 and 12 ×1
the last point-variable is not nought, so n is a ratio of two squares' sum ×1
the line at infinity is the last line built ×1
the nearfield plane has n² + n + 1 lines ×1
the nearfield plane has n² + n + 1 points ×1
the nearfield plane: every two points lie on a line ×1
the nearfield plane: two points lie on only one line ×1
the nearfield's multiplication is associative ×1
the number of points in the design drawn is a whole number between 7 and 15 ×1
the number of points in the design is a whole number between 3 and 21 ×1
the number written as four squares is a whole number between 1 and 99 ×1
the order is 3, 4, 5 or 7 ×1
the order is 3, 4, 5, 7 or 8 ×1
the orders run are among 2, 5 and 9 ×1
the plane drawn is of order 2 or 3 ×1
the plane is built over a prime between 2 and 5 ×1
the plane is built over a prime field between 2 and 5 ×1
the plane over GF(9) has n² + n + 1 lines ×1
the plane over GF(9) has n² + n + 1 points ×1
the plane over GF(9): every two points lie on a line ×1
the plane over GF(9): two points lie on only one line ×1
the point-variables and one more fill blocks of four exactly ×1
the points are labelled or not ×1
the prime is one that leaves three over when divided by four ×1
the residue rule and the divisibility rule agree at every size ×1
the rows are perpendicular and each has squared length n ×1
the search agrees with the arithmetic ×1
the shifts give n distinct blocks ×1
the size of the permuted set is a whole number between 1 and 8 ×1
the third meeting point is off the line through the other two ×1
the three meets of corresponding sides are collinear ×1
the three midpoints are the same distance from the centre, so one circle holds them ×1
the two divisibility conditions are exactly v ≡ 1 or 3 mod 6 ×1
the view is one of matrix, steiner, sizes, fisher, difference, desargues, nearfield, planecheck, desfail, affineconic, ovals, tangents, monomials, orders, axes, translation, gram, ordersmap, foursquare, brrun, twosq, monocensus, monoclasses, hypergrid ×1
there are exactly q + 1 tangent lines, one at each point of the oval ×1
with the axis at infinity and the centre on it, the nearfield plane never fails ×1

Where it is called

Every figure on this list is drawn by the same rule, so a change to the rule changes all of them at once. That is why the list is published.

Topology

A disc sewn to a Möbius band

The band has one boundary curve, and a disc has one boundary curve. Sew them together and the result is the smallest closed surface with one side — the one every other one-sided surface is built out of.

Computation

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.

Computation

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.

Computation

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.

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.

Computation

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.

Computation

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.

Computation

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.

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.

Computation

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.

Computation

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.

Computation

The thirty-six officers

Six regiments send six officers each, one of every rank. Arrange all thirty-six in a square so that each row and each column holds every rank once and every regiment once. Euler could not, guessed why, and was wrong about the reason.

The whole library · What the figures prove