Area functions on affine planes

Oriented area functions are functions defined on the set of ordered triangles of an affine plane which are antisymmetric under odd permutations of the vertices and which behave additively when triangles are cut into two. We compare several elementary properties which such an area function may have (roughly speaking shear invariance, equality of area of the two triangles obtained by cutting a parallelogram along a diagonal, and equality of area of the two triangles obtained by cutting a triangle along a median). It turns out purely by arguments of elementary affine geometry (if cleverly arranged) that these properties are grosso modo equivalent, although one has to be careful about “pathological” situations. Furthermore, all oriented area functions satisfying these properties are explicitly determined. Finally they are compared with so-called geometric valuations.     

A characterization of special laguerre planes and extended dual affine planes

A special Laguerre plane is a nondegenerate transversal 3-design such that the residue of each point is a dual affine plane. A special Laguerre plane is equivalent to an optimal code with three information digits and maximal length. An extended dual affine plane is an incidence structure (whose objects will be called points and blocks) such that the residue of each point is a dual affine plane, and each pair of points is in at least one block. Finite extended dual affine planes exist only of order 2, 4, and (dubiously) 10. We show that any finite incidence structure having the residue of each point a dual affine plane either is a transversal 3-design or has a block through each pair of points. Hence theorem: If a finite nondegenerate connected incidence structure has the residue of each point a dual affine plane, then is either an extended dual affine plane or a special Laguerre plane. This research was partially supported by NSF Grant MCS-8102361.     

Two families of flag-transitive affine planes

Two families of flag-transitive nondesarguesian affine planes of odd order are defined, and isomorphisms among the various planes are studied.Dedicated to Otto Wagner on the occasion of his 60th birthdayResearch supported in part by NSF grant DMS 87-01794 and NSA grant MDA 904-88-H-2040.     

Octagonality conditions in projective and affine planes

A projective confined configuration with axis and centre will be introduced in terms of a non-degenerate octagon satisfying some hypotheses on the position of its diagonal points (i.e. intersections of edges having distance 8 in the flag graph ( )) and its first minor diagonal lines (i.e. diagonal lines joining vertices of distance 6 in ( )). That confined configuration gives rise to a certain configurational condition whose affine specialization (i.e. the axis coincides with the line at infinity) is equivalent to the affine Pappos condition, whereas its little specialization (i.e. the centre lies on the axis) turns out to be equivalent to the little Desargues condition. In Pappian projective planes can be completed to a configuration of type (12₄, 16₃).     

Sharp homogeneity in affine planes,and in some affine generalized polygons

Let G be a collineation group of a generalized (2n + 1 )-gon Γ and let L be a line such that every symmetry σ of any ordinary (2n + 1 )-gon in Γ containing L with σ(L) = L extends uniquely to a collineation in G. We show that Γ is then a Desarguesian projective plane. We also describe the groups G that arise. As a corollary, we treat the analogous problem without the restriction σ(L) = L.     

On flag-transitive affine planes of orderq 3

Let be a translation plane of orderq ³,q an odd prime power, whose kern GF(q). Letl be the line at infinity of . LetG be a solvable collineation group of in the linear translation complement, which acts transitively onl , and letH be a maximal normal cyclic subgroup ofG. Then the restriction ofH onl acts semiregularly onl and {1, 2, 3, 6}, where is the restriction ofG onl (ifq –1(mod 3), then {1, 2}). Ifq {3, 5} and {1, 2}, then is determined completely, using a computer.     

Elementary constructions of locally compact affine planes

In this paper we describe several elementary constructions of 4-, 8- and 16-dimensional locally compact affine planes. The new planes share many properties with the classical ones and are very easy to handle. Among the new planes we find translation planes, planes that are constructed by gluing together two halves of different translation planes, 4-dimensional shift planes, etc. We discuss various applications of our constructions, e.g. the construction of 8- and 16-dimensional affine planes with a point-transitive collineation group which are neither translation planes nor dual translation planes, the proof that a 2-dimensional affine plane that can be coordinatized by a linear ternary field with continuous ternary operation can be embedded in 4-, 8- and 16-dimensional planes, the construction of 4-dimensional non-classical planes that admit at the same time orthogonal and non-orthogonal polarities. We also consider which of our planes have tangent translation planes in all their points. In a final section we generalize the Knarr-Weigand criterion for topological ternary fields.This research was supported by a Feodor Lynen fellowship.