441,420 research outputs found
Cayley-Dickson Algebras and Finite Geometry
Given a -dimensional Cayley-Dickson algebra, where , we
first observe that the multiplication table of its imaginary units , , is encoded in the properties of the projective space
PG if one regards these imaginary units as points and distinguished
triads of them , and , as lines. This projective space is seen to feature two distinct kinds
of lines according as or . 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, 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 -configuration ; in particular,
(octonions) is isomorphic to the Pasch -configuration,
(sedenions) is the famous Desargues -configuration,
(32-nions) coincides with the Cayley-Salmon -configuration found
in the well-known Pascal mystic hexagram and (64-nions) is
identical with a particular -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 occurs as a geometric hyperplane of . Finally, a brief
examination of the structure of generic leads to a conjecture that
is isomorphic to a combinatorial Grassmannian of type .Comment: 26 pages, 20 figures; V2 - the basis made explicit, a footnote and a
couple of references adde
Singer quadrangles
[no abstract available
- …