“… And so each venture Is a new beginning,
a raid on the inarticulate….”
— T. S. Eliot, “East Coker V” in Four Quartets
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arXiv:1409.5691v1 [math.CO] 17 Sep 2014
The Complement of Binary Klein Quadric as
a Combinatorial Grassmannian
Metod Saniga,
Institute for Discrete Mathematics and Geometry,
Vienna University of Technology,
Wiedner Hauptstraße 8–10, A-1040 Vienna, Austria
(metod.saniga@tuwien.ac.at) and
Astronomical Institute, Slovak Academy of Sciences,
SK-05960 Tatransk ́a Lomnica, Slovak Republic
(msaniga@astro.sk)
Abstract
Given a hyperbolic quadric of PG(5, 2), there are 28 points off this quadric and 56 lines skew to it. It is shown that the (286,563)-configuration formed by these points and lines is isomorphic to the combinatorial Grassmannian of type G2(8). It is also pointed out that a set of seven points of G2(8) whose labels share a mark corresponds to a Conwell heptad of PG(5, 2). Gradual removal of Conwell heptads from the (286,563)-configuration yields a nested sequence of binomial configurations identical with part of that found to be associated with Cayley-Dickson algebras (arXiv:1405.6888).
Keywords:
Combinatorial Grassmannian −
Binary Klein Quadric − Conwell Heptad
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See also this journal on the above date — 17 September 2014.