Seven points, seven lines

Euclid’s parallel postulate says that through a point off a line there is exactly one line missing it. There is a geometry in which there is none, it has seven points in it, and it fits on a postage stamp.

The axioms, and how few there are #

A projective plane is a set of points, a set of lines, and a relation saying which points lie on which lines, satisfying three conditions:

1. any two distinct points lie on exactly one common line;
2. any two distinct lines meet in exactly one common point;
3. there exist four points with no three of them on a line.

The first is Euclid’s. The second is Euclid’s with the parallel case deleted — in this geometry there are no parallels at all. The third rules out degenerate arrangements, such as all the points sitting on a single line with one extra point off it, which would satisfy the first two vacuously.

Nothing about size is assumed and nothing about straightness is mentioned. A “line” is a set of points, and the axioms are conditions on how those sets intersect.

The counting that fixes the size #

From those three conditions alone, the shape of the whole object follows.

Suppose some line has $n+1$ points on it; call $n$ the order. Then:

• Every line has $n+1$ points. Given two lines, pick a point off both — the third axiom guarantees one — and project one line onto the other through it. The projection is a bijection, since every line through the chosen point meets both lines exactly once.
• Every point lies on $n+1$ lines. Same argument dualised.
• There are $n^2+n+1$ points. Fix a point $P$. Every other point lies on exactly one of the $n+1$ lines through $P$, each of which carries $n$ points besides $P$. So the total is $1 + (n+1)n = n^2+n+1$.
• There are $n^2+n+1$ lines, by the same count with the roles swapped.

For $n = 2$ that gives seven points and seven lines, which is the figure above. For $n = 3$, thirteen of each; for $n = 4$, twenty-one; for $n=5$, thirty-one.

The generator does not take any of this on trust. It counts the points, the lines, the points on each line and the lines through each point, then checks every pair of points for exactly one common line and every pair of lines for exactly one common point — which at order five is nine hundred and thirty pairs of each kind.

Where the planes come from #

The construction needs a finite field, and it is the most economical use of one in this collection.

Take three-dimensional space over $GF(q)$: triples of field elements, with the usual addition and scaling. Now:

• a point of the plane is a line through the origin of that space — a set $\{\lambda v\}$ for a non-zero $v$;
• a line of the plane is a plane through the origin;
• a point lies on a line when the one-dimensional space is contained in the two-dimensional one.

Counting: there are $q^3 - 1$ non-zero triples, and each line through the origin contains $q-1$ of them, so there are $(q^3-1)/(q-1) = q^2+q+1$ points. The same count applies to the planes through the origin, by pairing each with the direction perpendicular to it.

The axioms come free from linear algebra. Two distinct lines through the origin span exactly one plane; two distinct planes through the origin meet in exactly one line. Those are facts about dimensions, and dimension arithmetic over a finite field works exactly as it does over the reals.

So there is a projective plane of order $q$ for every prime power $q$. Whether there is one of any other order is a question this essay ends on.

The dependence is worth stating precisely, because it is the reason the two anchors of this field are the same subject. Nothing in the construction uses geometry; it uses the fact that scaling by a non-zero element is invertible, which is what makes “up to scale” an equivalence with classes all of the same size. Take that away — work over the integers modulo six, say — and the classes come out with different sizes, the count $q^2+q+1$ collapses, and the axioms fail. A field is required, and there is no field of order six.

The plane at infinity, which is where the parallels went #

There is a way to arrive at a projective plane without any finiteness at all, and it explains the missing parallels.

Take the ordinary plane and add one new point for each direction — one for horizontal, one for vertical, one for each slope — and declare that all the new points lie on a single new line. Two parallel lines now meet, at the new point belonging to their shared direction. The exceptional case in Euclid’s second axiom disappears, and every two lines meet exactly once.

The finite planes are that construction done over a finite field instead of the reals, and the coordinates make it exact. A point of the plane is a triple $(x:y:z)$ up to scaling; the ones with $z \ne 0$ can be scaled to $(x:y:1)$ and are the ordinary plane; the ones with $z = 0$ are the added directions, and they form one line.

So the Fano plane’s seven points are four “ordinary” ones and three at infinity, and which three depends entirely on which coordinate is called $z$ — a choice with no meaning inside the object. That is the sharpest statement of what projective geometry is for: the line at infinity is a convention, and every theorem is a statement that survives changing it.

Is the drawing the plane? #

The picture at the top of this page is a triangle with its midpoints, its centre, and a circle. That is a drawing; whether it is the Fano plane is a separate question, and the sort of question a figure on this site is required to settle rather than assert.

The drawing offers seven point-positions and seven three-element sets: the three sides, the three medians, and the circle through the midpoints. The algebra offers seven one-dimensional subspaces of $GF(2)^3$ and seven two-dimensional ones.

The two are the same object if there is a relabelling of the seven points carrying the algebra’s line-sets onto the drawing’s. There are $7! = 5{,}040$ relabellings, which is a small enough number to try all of them, so the generator does: it walks every permutation and asks whether the image of the algebraic line system is the drawn one.

One is found. That is a proof of isomorphism rather than a claim of it, and the point labels in the figure are the algebra’s coordinates carried across the map it found.

Why one line has to bend #

The circle in the drawing is not an artistic choice, and it cannot be removed.

The Sylvester–Gallai theorem says that in the ordinary plane, any finite set of points not all on one line has some line through exactly two of them. The Fano plane’s seven points have no such line: every line of the configuration carries three, and the theorem forbids that arrangement of straight lines in the real plane.

So the Fano plane is not drawable with seven straight lines, whatever the arrangement. Every published picture of it bends one line, and every published picture is honest about the same thing: the object exists, and the plane it is being drawn in cannot hold it.

That is worth pausing on, because it is a clean example of a limit on figures rather than on mathematics. The incidence table in the previous figure carries the whole object with nothing bent, and is much harder to read. The bent drawing is easier to read and is a picture of something whose straightness is a lie. Both are shown here for that reason.

The lesson generalises to every finite geometry. Their points are not positions and their lines are not straight; both are labels for sets, and any drawing imposes a geometry the object does not have.

It is a different situation from a Voronoi diagram, where the picture is the definition and the regions really are regions. Here the drawing is a representation chosen by the author, and the site’s rule for such cases — established one field earlier, where logic’s objects had no natural positions either — is that the choice must be checkable. The relabelling search is that check, and it is the reason the bent circle can be shown without apology: the picture has been proved to be a picture of the right thing, whatever its curvature suggests.

Duality, which is free #

The axioms are symmetric under swapping the words point and line: “two points lie on one line” becomes “two lines meet in one point”, and the third axiom becomes its own mirror.

So every theorem about projective planes has a dual theorem, obtained by swapping the two words throughout, and it needs no separate proof.

That symmetry is visible in the incidence tables. Transposing the matrix — swapping rows for columns — turns the plane into another plane, its dual. For the planes built from fields the dual is isomorphic to the original, so the tables are symmetric up to relabelling; for some exotic planes it is not, and a plane and its dual are genuinely different objects of the same size.

Duality is the reason all the counts in this essay came in pairs. Nothing had to be proved twice.

Seven objects, seven objects, and a symmetry group #

One more count is worth doing, because it says how rigid the object is.

A symmetry of the Fano plane is a relabelling of the seven points carrying lines to lines. The search that established the drawing’s identity found one such map; running it to the end finds all of them, and there are $168$.

That number can be arrived at directly. A symmetry is determined by where it sends a chosen basis: the first point may go to any of $7$, the second to any of the remaining $6$, and the third to any of the $4$ not on the line through the first two — since a triple on a common line does not determine the rest. Multiplying gives $7 \times 6 \times 4 = 168$.

One hundred and sixty-eight symmetries on seven points is a great many. For comparison, the complete graph on seven vertices has $5{,}040$, and a generic configuration of seven points would have one. The Fano plane is close to being as symmetric as a seven-point object can be, and its symmetry group is a famous one — the second-smallest non-abelian simple group, appearing elsewhere as the symmetries of a particular curve and as a group of matrices over $GF(2)$.

Symmetry of that order is what makes finite geometries useful for building other things. A design with many symmetries can be permuted into itself, so a construction using it can be assumed to start anywhere, and the schedule of the next essay is built by exactly that kind of rotation.

Which orders exist #

The construction gives a plane at every prime power: $2, 3, 4, 5, 7, 8, 9, 11, 13, 16, \ldots$

Whether any others exist is one of the oldest open problems in combinatorics, and the state of knowledge is unusual.

Order 6 does not exist. The Bruck–Ryser theorem says that if $n \equiv 1$ or $2$ modulo $4$, a plane of order $n$ exists only if $n$ is a sum of two squares. Six is $2$ mod $4$ and is not a sum of two squares, so there is no plane of order six. That result also settles $14$, $21$, $22$ and infinitely many others.

Order 10 does not exist, and Bruck–Ryser says nothing about it, since $10 = 1^2+3^2$. It was settled in 1989 by Clement Lam and colleagues after several thousand hours of computer search over the possible weight distributions of an associated code. The proof is an exhaustion no person can check, and its status was debated for exactly that reason — the same debate the four-colour theorem had provoked fifteen years earlier.

Order 12 is open. So is every non-prime-power order the two results above do not reach. Nobody has found a plane of non-prime-power order and nobody has proved that none exists, and the gap between those two states has not moved in a hundred years.

That is the situation this whole field keeps producing: a construction that works for a well-understood set of sizes, a necessary condition that rules out some others, and a stubborn silence in between.

That last figure is a preview of a result worth stating here even though it belongs to another essay: a projective plane of order $n$ exists exactly when there are $n-1$ mutually orthogonal Latin squares of order $n$. The two objects are interchangeable, and the non-existence at order six was first proved in the Latin-square language, by Gaston Tarry, in 1900 — long before Bruck and Ryser, and by pure exhaustion.

Where this anchor goes #

A projective plane is one kind of structure defined by an incidence condition. There are others, and the smallest interesting family drops the requirement that the lines all have the same relationship to each other and keeps only the requirement about pairs.

That family is the Steiner triple system: a set of points and a set of triples, such that every pair of points lies in exactly one triple. The Fano plane is one — its seven lines are triples, and every pair of its seven points is on exactly one.

Comparing that figure with the one at the top of the page is instructive. Both are the Fano plane. One arranges the points to make the lines look like lines and pays for it with a bent circle; the other makes no spatial claim at all and is unreadable as a geometry. Neither drawing is the object, and the incidence table is closer to being it than either. Which sizes admit one turns out to have a complete and much friendlier answer than the projective plane question, and the answer comes from two divisions.