-Bewegungen mit den -Automorphismen
The simple incidence structure formed by points and unordered pairs of distinct parallel lines of a finite affine plane of order is a design. If , is the complementary design of . If , is isomorphic to the geometric design (see [2; Theorem 1.2]). In this paper we give necessary and sufficient conditions for a design to be of the form for some finite affine plane of order . As a consequence we obtain a characterization of small designs .
We study 2-frieze patterns generalizing that of the classical Coxeter-Conway frieze patterns. The geometric realization of this space is the space of -gons (in the projective plane and in 3-dimensional vector space) which is a close relative of the moduli space of genus curves with marked points. We show that the space of 2-frieze patterns is a cluster manifold and study its algebraic and arithmetic properties.