Fundamental figures, in four and six dimensions, over GF(2)

Introduction. This paper falls into three parts.In §§ 1–5 it is explained how, when the base field of the geometry is GF(2), there are figures of n + 2 interlocking polygons in [n], every two polygons sharing a vertex. When n is even these ½(n + 1) (n + 2) vertices lie in an [n − 1], and two of them are conjugate in a certain null polarity when, and only when, they do not belong to the same polygon.

Pencils of Null Polarities

A theorem due to von Staudt states that a null polarity in complex projective space of three dimensions is determined by a selfpolar skew pentagon. By allowing an element of the self-polar pentagon to vary in a suitable manner we can arrive at a family of ∞1 null polarities, which we term a pencil of null polarities. Each polarity of the pencil distinguishes a linear complex as the class of self-polar lines. Thus, associated with the pencil is a family of ∞1 linear complexes, which we term a pencil of linear complexes.It is the purpose of this paper to continue an earlier investigation of pencils of polarities (2), by applying analogous techniques to the study of pencils of null polarities and pencils of linear complexes.Since it develops that the lines common to all linear complexes of a pencil are the lines of a linear congruence, the central question has been: How many of the different types of linear congruences can be achieved in this manner? Happily, it can be reported that the classification of pencils of null polarities yields all of the three types of linear congruences (4, pp. 140-141).