Two remarks on generalized hexagons

In this paper, the point-distance-2-regular hexagons are characterized as the only hexagons for which the intersection sets have size one and containing on ovoidal subspace all the points of which are 3-regular. We also give a characterization of the finite split Cayley hexagon of even order.     

An inequality for generalized hexagons

We show that a generalized hexagon withs + 1 points on a line andt + 1 lines through a point satisfies s=1 orts ³.     

Primitive flag-transitive generalized hexagons and octagons

Suppose that an automorphism group G acts flag-transitively on a finite generalized hexagon or octagon S, and suppose that the action on both the point and line set is primitive. We show that G is an almost simple group of Lie type, that is, the socle of G is a simple Chevalley group.     

One-point extensions of generalized hexagons and octagons

In this note, we prove the uniqueness of the one-point extension S of a generalized hexagon of order 2 and prove the non-existence of such an extension S of any other finite generalized hexagon of classical order, different from the one of order 2, and of the known finite generalized octagons provided the following property holds: for any three points x, y and z of S, the graph-theoretic distance from y to z in the derived generalized hexagon S_x is the same as the distance from x to z in S_y.     

Exceptional presentations of three generalized hexagons of order 2

The generalized hexagons associated with L₃(2), U₃(3), ³D₄(2), respectively, are presented as subconfigurations of the projective plane over the complex, quaternionic, octonionic field, respectively. This leads to an embedding of Aut ³D₄(2) in the Lie group of type F₄(R).     

A note on the 3D4(q) generalized hexagons

The ³D₄(q) generalized hexagons are characterized by means of the regulus condition.     

Some combinatorial and geometric characterizations of the finite dual classical generalized hexagons

We characterize the dual of the generalized hexagons naturally associated to the groupsG ₂(q) and³ D ₄(q) by looking at certain configurations, and also by considering intersections of traces. For instance, the dual of a generalized hexagon of finite order (s, t) is associated to the Chevalley groups mentioned above if and only if the intersection of any two tracesx ^y andx ^z, with some additional condition, contains at mostt/s + 1 elements.     

Almost empty hexagons

In this work, new nontrivial bounds are obtained for the minimum number of points in general position on the plane, among which one certainly finds the set of vertices of a convex hexagon with not more than one point of the initial set inside.     

Almost regular integer hexagons

We establish the sequence of all Equiangular Integer Hexagons that have alternating sides whose lengths differ by one.The author is indebted to T.G. Ostrom, Richard Guy, and a referee for their assistance and suggestions in the development of this paper     

Generalized hexagons and Singer geometries

In this paper, we consider a set of lines of with the properties that (1) every plane contains 0, 1 or q + 1 elements of , (2) every solid contains no more than q ² + q + 1 and no less than q + 1 elements of , and (3) every point of is on q + 1 members of , and we show that, whenever (4) q ≠ 2 (respectively, q = 2) and the lines of through some point are contained in a solid (respectively, a plane), then is necessarily the set of lines of a regularly embedded split Cayley generalized hexagon in , with q even. We present examples of such sets not satisfying (4) based on a Singer cycle in , for all q.      

An interesting property of hexagons

These notes discuss several related problems in geometry that can be explored in a dynamic geometry environment. The problems involve an interesting property of hexagons.     

Valuations of glued near hexagons

Valuations of near polygons were introduced in [ 12 ] as an important tool for classifying dense near polygons. In the present article, we will introduce the class of the semi‐diagonal valuations. These valuations live in glued near hexagons. A glued near hexagon S can be coordinatized by a pair of admissible triples; such triples consist of a Steiner system , a group G, and a certain nice map . We will give a necessary and sufficient condition for the existence of semi‐diagonal valuations in in terms of these two admissible triples. For two classes of glued near hexagons, we will use this condition to determine all semi‐diagonal valuations. Each semi‐diagonal valuation will also give rise to a hyperplane of the glued near hexagon. © 2006 Wiley Periodicals, Inc. J Combin Designs 15: 35–48, 2007     

An extremal property of convex hexagons

The following conjecture is discussed: if K is a plane convex figure and T is a triangle of maximal area contained in K, then K is contained in ?5 \sqrt {5} T. It is shown that it suffices to check the conjecture in the case where K is a convex hexagon, but the conjecture is proved only in the case where K is a pentagon. Bibliography: 2 titles.