We consider the following `even hyperplane construction¿ of flats in the projective space which are external to the Grassmannian of lines of PG(4,2). Let the Grassmann image in of a partial spread in PG(4,2) be . Then is an r-cap on . Using the recent classification [N.A. Gordon, R. Shaw, L.H. Soicher, Classification of Partial Spreads in PG(4,2), pp. 63, available from: http://www.hull.ac.uk/maths/people/rs/staffdetails.html] of partial spreads in PG(4,2), we determine those partial spreads such that the projective space generated by the points mijmi+mj is an external flat. We show that, in this simple manner, we may construct seven out of the ten GL(5,2)-orbits of external flats. Keywords: Grassmannian G1,4,2; Partial spreads; External flats; GL(5,2)-orbits

Constructions are given of different kinds of flats in the projective space which are external to the Grassmannian of lines of PG(4,2). In particular it is shown that there exist precisely two GL(5,2)-orbits of external 4-flats, each with stabilizer group 31:5. (No 5-flat is external.) For each k=1,2,3, two distinct kinds of external k-flats are simply constructed out of certain partial spreads in PG(4,2) of size k+2. A third kind of external plane, with stabilizer 23:(7:3), is also shown to exist. With the aid of a certain key counting lemma, it is proved that the foregoing amounts to a complete classification of external flats.

The hidden universal Clifford algebra structure of M4(R) is ambiguous in the sense that this matrix algebra is the universal geometric algebra belonging to each of the real four-dimensional quadratic vector spaces R1,3 and R2,2. As a non-universal Clifford algebra, however, M4(R) is a geometric algebra of the unique five-dimensional quadratic vector space R2,3.

It is well-known that Clifford algebras are geometrical algebras in the sense that they afford the spin representation of the orthogonal groups. By virtue of modern results, however, it is clear that Clifford algebras make a structure which is suited also to other purposes. Such another purpose we want to pay attention to is the representation of conformai groups. We shall present a most rewarding Clifford algebra approach to conformai geometry. Our employment of the modulo (1,1) periodicity of Clifford algebras brings about the full generalization of the classical Möbius transformations. Moreover, the treatment of conformai groups which we advocate, gives rise to a generalization of twistors (this subject shall not be pursued here). In this paper all proofs are omitted. For a detailed treatment of the subject one is referred to [5].