On projective and affine hyperplanes

Affine and projective planes

Affine and Projective Universal Geometry

By recasting metrical geometry in a purely algebraic setting, both Euclidean and non-Euclidean geometries can be studied over a general field with an arbitrary quadratic form. Both an affine and a projective version of this new theory are introduced here, and the main formulas extend those of rational trigonometry in the plane. This gives a unified, computational model of both spherical and hyp...

Affine Spaces within Projective Spaces

We endow the set of complements of a fixed subspace of a projective space with the structure of an affine space, and show that certain lines of such an affine space are affine reguli or cones over affine reguli. Moreover, we apply our concepts to the problem of describing dual spreads. We do not assume that the projective space is finitedimensional or pappian. Mathematics Subject Classification...

On Incidence Matrices of Finite Projective and Affine Spaces

It is welt-known that the rank of each incidence matrix of all points vs. all e-spaces of a finite d-dimensional projective or affine space is the number of points of the geometry, where 1 <_e<-d-1 (see [1], p. 20). In this note we shall generalize this fact: Theorem. Let 0 <-e < f <= d-e-1, and let Me, l be an incidence matrix of all e-spaces vs. all f-spaces of PG(d, q) or AG(d, q). Then the ...

Berlekamp's Switching Game on Finite Projective and Affine Planes

We adapt Berlekamp’s light bulb switching game to finite projective plans and finite affine planes, then find the worst arrangement of lit bulbs for planes of even and odd orders. The results are then extended from the planes to spaces of higher dimension. 1 Berlekamp’s Switching Game The original game is simple enough. 100 light bulbs arranged in 10 rows and 10 columns. Each column and row has...