This post is inspired by a nice article by Adrian Rice and Ezra Brown in the latest *BSHM Bulletin*, titled “Commutativity and collinearity: a historical case study of the interconnection of mathematical ideas, Part II”.

*Pappus’ Theorem* states that, if alternate vertices of a hexagon are collinear, then the three intersection points of pairs of opposite edges are also collinear. In other words, if we have six points *P*_{1}, …, *P*_{6} such that *P*_{1}, *P*_{3}, *P*_{5} are collinear, as are *P*_{2}, *P*_{4}, *P*_{6}, and *Q*_{1} is the intersection of *P*_{2}*P*_{3} and *P*_{5}*P*_{6}, with *Q*_{2} and *Q*_{3} similarly, then *Q*_{1}, *Q*_{2}, *Q*_{3} are collinear. This is a classical “configuration theorem” involving nine points and nine lines, asserting that if 8 of the 9 triples are collinear then so is the 9th.

Of course, this is a theorem of projective geometry. It is valid in the Euclidean plane, but remains valid if some of the points are at infinity (recognised in the affine plane by corresponding lines being parallel). There are quite a number of cases to consider. As in various similar situations such as the classification of conics, things are much simpler in the projective plane!

At a similar time, Diophantus discovered and studied the “two squares” identity

(*a*^{2}+*b*^{2})(*x*^{2}+*y*^{2}) = (*ax*−*by*)^{2}+(*ay*+*bx*)^{2}.

The analogous four squares inequality was discovered by Euler, and the eight squares inequality by Degen, Graves and Cayley (in that order, though the last of the three somehow got his name attached to it).

The authors’ thesis is that these two pieces of work were not seen to be related until the nineteenth century. In the spirit of the work of Cauchy and Weierstrass on calculus, mathematicians turned their attention to geometry, to the task of making Euclid’s axioms more rigorous. When studying just the incidence axioms, it was discovered that (together with the usual axioms for a projective plane) Pappus’ Theorem is equivalent to the property that the plane can be coordinatised by a commutative field. On the algebraic side, the two, four and eight squares identities express the facts that there are algebras of dimenions 2, 4 and 8 over the real numbers which have multiplicative norms (and are therefore division algebras), the complex numbers, quaternions and octonions; but the last two fail to be commutative. So Diophantus’ two squares identity is the related to the only finite extension of the real numbers which coordinatises a projective plane satisfying Pappus’ theorem.

At this point, let me note that one really has to consider various “degenerate” cases of Pappus’ theorem, when some pairs of points are identified. Though tedious, this is not a serious difficulty.

After the eight squares identity was discovered, mathematicians naturally wondered whether the sequence continues. The answer is no, as was proved by Hurwitz in 1898. (His result was slightly more general; he shows it not just for sums of squares but for arbitrary non-singular quadratic forms.)

Rice and Brown also discuss finite geometries, and in particular the Steiner triple systems on 7 and 9 points; these are the projective plane over the field of two elements, and the affine plane (which can be extended to a projective plane in a standard way) over the field of three elements.

Since these fields are commutative, the Fano plane (for example) should satisfy Pappus’ theorem. Because the Fano plane only has seven points, all instances of Pappus’ theorem in it are degenerate!

It is also the case that the octonions are conveniently described by the Fano plane with arrows on some of its lines. The seven basic units (apart from the identity) are matched with the seven points of the plane, and the triples whose product is plus or minus 1 are the lines of the plane; the arrows can be put on in such a way that each line has a cyclic orientation so that the product of two of its points is the third.

There is some interesting pre-history of Steiner triple systems. As is well known, Kirkman proved the existence theorem for Steiner triple systems in 1847 in answer to a question in the *Lady’s and Gentleman’s Diary*. Unaware of this, Jakob Steiner posed the same question in *Crelle’s Journal* in 1853, and it was answered by Reiss in 1859.

The problem that had attracted Kirkman’s attention was posed by Wesley Woolhouse, the editor of the *Lady’s and Gentleman’s Diary*, in 1846. Woolhouse and Steiner may both have got the problem from Julius Plücker. He had discovered the Steiner triple system of order 9 as the structure of the nine inflection points of a plane cubic curve, in 1835; in the same paper he had posed the existence question for “Steiner triple systems”. (He mistakenly thought that they could exist only for orders congruent to 3 (mod 6); he corrected his mistake and added 1 (mod 6) in 1839.)

Should the concept be named after the person who first proposed it (so they would be *Plücker triple systems*), or the person who first constructed it (so they would be *Kirkman triple systems*)? This problem arises in many other parts of mathematics; Lyons discovered evidence for a sporadic simple group which was constructed by Sims, for example. The matter is further complicated in the present case by the fact that the term *Kirkman triple system* is used with a different meaning, based on Kirkman’s schoolgirls problem.

But there is one further connection which Rice and Brown don’t seem to mention.

The Pappus configuration has nine points and nine lines. These lines cover 27 of the 36 pairs of distinct points. The nine pairs not covered correspond to a partition of the nine points into three sets of three points each containing no collinear pairs. If we add these three sets as new “lines”, we obtain the affine plane of order 3. The uniqueness of the Pappus configuration is thus related to the fact that the parallel classes of lines in the affine plane are equivalent under symmetries of the plane.