From Cayley-Dickson Algebras to Combinatorial Grassmannians

Given a 2(N) -dimensional Cayley-Dickson algebra, where 3 <= N <= 6, we first observe that the multiplication table of its imaginary units e(a), 1 <= a <= 2(N) -1, is encoded in the properties of the projective space PG (N -1, 2) if these imaginary units are regarded as points and distinguished triads of them {e(a), e(b), e(c)}, 1 <= a < b < c <= 2(N) - 1 and e(a)e(b) = +/- e(c), as lines. This projective space is seen to feature two distinct kinds of lines according as a + b = c or a + b not equal c. Consequently, it also exhibits (at least two) different types of points in dependence on how many lines of either kind pass through each of them. In order to account for such partition of the PG (N -1, 2), the concept of Veldkamp space of a finite point-line incidence structure is employed. The corresponding point-line incidence structure is found to be a specific binomial configuration C-N; in particular, C-3 (octonions) is isomorphic to the Pasch (6(2), 4(3)) -configuration, C-4 (sedenions) is the famous Desargues (10(3)) -configuration, C-5 (32-nions) coincides with the Cayley-Salmon (15(4), 20(3)) -configuration found in the well-known Pascal mystic hexagram and C-6 (64-nions) is identical with a particular (21(5), 35(3)) -configuration that can be viewed as four triangles in perspective from a line where the points of perspectivity of six pairs of them form a Pasch configuration. Finally, a brief examination of the structure of generic CN leads to a conjecture that C-N is isomorphic to a combinatorial Grassmannian of type G(2) (N + 1).