arXiv:1405.6888v2 [math.CO] 2 Jun 2014 Cayley-Dickson Algebras and Finite Geometry Metod Saniga, 1,2 Fr´ ed´ eric Holweck 3 and Petr Pracna 4 1 Institute for Discrete Mathematics and Geometry, Vienna University of Technology Wiedner Hauptstraße 8–10, A-1040 Vienna, Austria (metod.saniga@tuwien.ac.at) 2 Astronomical Institute, Slovak Academy of Sciences SK-05960 Tatransk´ a Lomnica, Slovak Republic (msaniga@astro.sk) 3 Laboratoire IRTES/M3M, Universit´ e de Technologie de Belfort-Montb´ eliard F-90010 Belfort, France (frederic.holweck@utbm.fr) 4 J. Heyrovsk´ y Institute of Physical Chemistry, v. v. i. Academy of Sciences of the Czech Republic Dolejˇ skova 3, CZ-18223 Prague, Czech Republic (pracna@jh-inst.cas.cz) Abstract 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 one regards these imaginary units 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 = 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 binomial  ( N+1 2 ) N−1 , ( N+1 3 ) 3  -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. We also draw attention to a remarkable nesting pattern formed by these configurations, where C N−1 occurs as a geometric hyperplane of C N . Finally, a brief examination of the structure of generic C N leads to a conjecture that C N is isomorphic to a combinatorial Grassmannian of type G 2 (N + 1). Keywords: Cayley-Dickson Algebras – Veldkamp Spaces – Finite Geometries 1 Introduction As it is well known (see, e. g., [1, 2]), the Cayley-Dickson algebras represent a nested sequence A 0 ,A 1 ,A 2 ,...,A N ,... of 2 N -dimensional (in general non-associative) R-algebras with A N ⊂ A N+1 , where A 0 = R and where for any N ≥ 0 A N+1 comprises all ordered pairs of elements from A N with conjugation defined by (x, y) ∗ =(x ∗ , −y) (1) and multiplication usually by (x, y)(X, Y )=(xX − Yy ∗ ,x ∗ Y + Xy). (2) 1