Rank three affine planes

Oriented Rank Three Matroids and Projective Planes

Quite recently, GOODMAN, POLLACK, WENGER and ZAMFIRESCU [7], [8] have proven two conjectures by GRÜNBAUM right, showing that any arrangement of pseudolines in the plane can be embedded into a flat projective plane and that there exists a universal topological projective plane in which every arrangement of pseudolines is stretchable. By FOLKMAN and LAWRENCE's theorem [6], this plane contains eve...

The Affine Permutation Groups of Rank Three

(iii) G is an ajfine group, that is, the socle of G is a vector space V, where V s= {ZpY for some prime p and n=p ; moreover, if Go is the stabilizer of the zero vector in V then G = VG0, Go is an irreducible subgroup of GLd(p), and GQ has exactly two orbits on the non-zero vectors of V. The rank 3 groups under (i) are given by the classification of the 2-transitive groups with simple socle (se...

Affine and projective planes

Mubs Inequivalence and Affine Planes

There are fairly large families of unitarily inequivalent complete sets of N+1 mutually unbiased bases (MUBs) in C for various prime powers N . The number of such sets is not bounded above by any polynomial as a function of N . While it is standard that there is a superficial similarity between complete sets of MUBs and finite affine planes, there is an intimate relationship between these large...

Convexity in Topological Affine Planes

We extend to topological affine planes the standard theorems of convexity, among them the separation theorem, the anti-exchange theorem, Radon’s, Helly’s, Carathéodory’s, and Kirchberger’s theorems, and the Minkowski theorem on extreme points. In a few cases the proofs are obtained by adapting proofs of the original results in the Euclidean plane; in others it is necessary to devise new proofs ...