Showing papers in "Journal of Geometry in 1987"
TL;DR: In this paper, the authors derive the desarguesian plane of order q2 for every prime power q and construct a unital in this plane which can be embedded in both the Hall plane and the dual of q 2 which are non-isomorphic projective planes.
Abstract: By deriving the desarguesian plane of order q2 for every prime power q a unital of order q is constructed which can be embedded in both the Hall plane and the dual of the Hall plane of order q2 which are non-isomorphic projective planes. The representation of translation planes in the fourdimensional projective space of J. Andre and F. Buekenhouts construction of unitals in these planes are used. It is shown that the full automorphism groups of these unitals are just the collineation groups inherited from the classical unitals.
39 citations
TL;DR: In this article, all possible sets of type (l, n) in a finite projective plane are characterized from the arithmetical point of view, and necessary existence conditions are proved for sets of types (m,n) which are stronger then the previously known ones.
Abstract: All possible sets of type (l,n) in a finite projective plane are completely characterized from the arithmetical point of view. Furthermore, necessary existence conditions are proved for sets of type (m,n) which are stronger then the previously known ones and provide the unique possible arithmetical solutions whenever they are satisfied.
36 citations
TL;DR: In this paper, a new proof is given using methods of classical projective geometry, based on the observation that by a converse of the theorem of IVORY each pair of incongruent octahedra with the same edge lengths is connected with a certain pair of confocal guadrics.
Abstract: In 1896 R. BRICARD gave a complete description of all flexible octahedra; a different approach is due to R. CONNELLY (1978). In this paper a new proof is given using methods of classical projective geometry. The proof is based on the observation that by a converse of the theorem of IVORY each pair of incongruent octahedra with the same edge lengths is connected with a certain pair of confocal guadrics.
29 citations
TL;DR: In this article, a subset of a set K of which the points of which are not in a collinear triplet of K are studied, and it is shown that K is a (q+1)-set of PG(2,q).
Abstract: In this paper we study subset N(K) of a set K the points of which are not in a collinear triplet of K and prove that ¦N(K)¦≤(q+1)/2 or N(K)=K if K is a (q+1)-set of PG(2,q). We describe all the k-arcs of AG(2,q) the secants of which meet the ideal line exactly in k points.
24 citations
22 citations
TL;DR: In this paper, it was shown that every convex polyhedron in Euclidean 3-space is isomorphic to one all vertices of which lie on a sphere.
Abstract: Jacob Steiner asked, more than 150 years ago, whether every convex polyhedron in Euclidean 3-space is isomorphic to one all vertices of which lie on a sphere. It is well known that the answer to this question is negative, but many related problems are still unsolved.
21 citations
16 citations
12 citations
TL;DR: In this paper, the method of differences was used to establish that every 2-regular multigraph on v− 1≡ 0,2 (mod 3) points occurs as the neighbourhood graph of an element in a two-fold triple system of orderv, with two exceptions: C2∪C3 and C3∫C3.
Abstract: The method of differences is used to establish that every 2-regular multigraph onv− 1≡0,2 (mod 3) points occurs as the neighbourhood graph of an element in a twofold triple system of orderv, with two exceptions: C2∪C3and C3∪C3.
10 citations
TL;DR: In this paper, the authors introduce d-injective planes, i.e., the shadow structures coming from the Buekenhout diagram, which provide a diagram representation for high rank D-Injective geometries, and give two new constructions using pointwise and setwise action of a class of mappings.
Abstract: F-squashed geometries, one of the many recent generalizations of matroids, include a wide range of combinatorial structures but still admit a direct extension of many matroidal axiomatizations and also provide a good framework for studying the performance of the greedy algorithm in any independence system. Here, after giving all necessary preliminaries in section 1, we consider in section 2F-squashed geometries which are exactly the shadow structures coming from the Buekenhout diagram:
, i.e. bouquets of matroids. We introduce d-injective planes:
(generalizing the case of dual net for d=1) which provide a diagram representation for high rank d-injective geometries. In section 3, after a brief survey of known constructions for d-injective geometries, we give two new constructions using pointwise and setwise action of a class of mappings. The first one, using some features of permutation geometries (i.e. 2-injection geometries), produces bouquets of pairwise isomorphic matroids. The last section 4 presents briefly some related problems for squashed geometries.
10 citations
TL;DR: In this paper, the group of all automorphisms of the geometric structure which preserve both parallelisms is shown to consist exactly of those automorphism of the algebraic structure that preserve the fibration.
Abstract: In any kinematic space (with non trivial fibration) the group of all automorphisms of the geometric structure which preserve both parallelisms is shown to consist exactly of those automorphisms of the algebraic structure which preserve the fibration. Moreover a characterization of such group is given for a particular class of kinematic spaces.
TL;DR: In this article, it was shown that every finite Desarguesian affine plane is Pappian without using the representation theorem and a theorem of J.H.M.WEDDERBURN.
Abstract: Without using the representation theorem and a theorem of J.H.M. WEDDERBURN we show that every finite Desarguesian affine plane is Pappian.
TL;DR: In this article, it was shown that every point in the plane which can be constructed by a compass and a ruler, given a set S of points, can also be constructed using a compass alone, in such a way that the centres of all the circles used are on one particular segment OK, where O and K are two arbitrarily chosen distinct points of S.
Abstract: We show that every point in the plane which can be constructed by a compass and a ruler, given a set S of points, can be constructed using a compass alone, in such a way that the centres of all the circles used are on one particular segment OK, where O and K are two arbitrarily chosen distinct points of S. This strengthens (and at the same time also gives an alternative simple proof to) a famous theorem of Mascheroni and Mohr.
TL;DR: The question of whether the Archimedean ordering of only one of the ternary rings of a projective plane implies that it is an Archimediate plane is answered in the negative by the construction of local-Archimedean orderings of free planes as mentioned in this paper.
Abstract: The question, whether the Archimedean ordering of only one of the ternary rings of a projective plane Π implies that Π is Archimedean, i.e. that every ternary ring of Π is Archimedean, is answered in the negative by the construction of local-Archimedean orderings of free planes. There exists even Archimedean affine planes with non-Archimedean associated projective planes.
TL;DR: In this article, the authors introduced the concept of (0, 2)-set in generalized quadrangles, in order to obtain characterizations for P(S,(∞)) and T 2 * O (O).
Abstract: In [3], [ 4 ] we introduced the concept of (0,2)-set in generalized quadrangles, in order to obtain characterizations for P(S,(∞)) and T
2
*
(O). Using these sets we are now able to formulate a characterization for Q(4,q), q odd, by assuming local conditions in an antiregular point x of a generalized quadrangle of order s.
TL;DR: In this paper, a series of incidence propositions for projective planes, which in the Desarguesian case is equivalent to Pappos' theorem, are studied. But they do not consider the case where the plane is projective.
Abstract: We present and study a series of incidence propositions: for projective planes, which in the Desarguesian case is equivalent to Pappos' theorem.
TL;DR: In this paper, general properties of motions in the flag space are investigated, and some special motions can be constructed by one-parameter groups in an easy way, such as motions with W-curve paths.
Abstract: In the first part (1. -4.) of this paper general properties of motions in the flag space I
3
(1)
are investigated. Orbits of points and remarkable points on orbits are regarded. In the second part (5.–6.) some special motions are regarded, which can be constructed by one-parameter groups in an easy way. At last (7.) special classes of motions such as motions with W-curve paths are characterized.
TL;DR: In this paper, the possibility of finite simple groups acting as collineation groups on finite translation planes of odd order with special attention paid to the sporadic simple groups was discussed, and it was shown that if the characteristic of the plane does not divide the order of the group, then the group cannot be one of eleven simple groups and if one of the Mathieu groups acts irreducibly on a finite translation plane then it is either M11 or M23.
Abstract: We discuss the possibility of finite simple groups acting as collineation groups on finite translation planes of odd order with special attention paid to the sporadic simple groups. We assume such a group acts irreducibly (in the vector space sense) on the plane. It is shown that if the characteristic of the plane does not divide the order of the group, then the group cannot be one of eleven sporadic simple groups. Also, if one of the Mathieu groups acts irreducibly on a finite translation plane then it is either M11 or M23.
TL;DR: In this paper, it was shown that Π is a translation plane of odd order p2R for p a prime and admits at least two p-groups, Bi, i, i = 1, 2, B1≠B2 of orders > pR/2 which fix Baer subplanes.
Abstract: Let Π be a translation plane of odd order p2R for p a prime. Let Π admit at least two p-groups, Bi, i=1, 2, B1≠B2 of orders > pR/2 which fix Baer subplanes Πi, i = 1,2, Π1≠Π2 pointwise. It is shown that under these assumptions Π must be a Hall plane.
TL;DR: In this paper, the notion of locally affine Minkowski and Laguerre planes with neighbor relation was introduced and two other classes of circle geometries were introduced.
Abstract: In [7] we have introduced the notion of a Mobius plane with neighbor relation as a generalization of ordinary Mobius planes. In this paper we present two other classes of circle geometries which are locally affine Klingenberg planes: Laguerre and Minkowski planes with neighbor relation.
TL;DR: In this paper, the authors investigate {1, 2, 3}-semiaffine planes and show that they are embeddable into projective planes of the same order n in the most natural way.
Abstract: In this paper, we investigate {1,2,3}-semiaffine planes All such planes of order n >51 shall be classified It turns out that they are embeddable into projective planes of the same order n in the most natural way
TL;DR: An Ordered Incidence Geometry as discussed by the authors is a geometry with certain axioms of incidence and order, which is a minimal setting for the fundamental convexity theorems, which usually appear in the context of a linear vector space.
Abstract: An Ordered Incidence Geometry, that is a geometry with certain axioms of incidence and order, is proposed as a minimal setting for the fundamental convexity theorems, which usually appear in the context of a linear vector space, but require only incidence, order (and for separation, completeness), and none of the linear structure of a vector space.
TL;DR: In this article, a Baer subspace V* of a skew subspace R and V of P =PG(d,q) is defined and the set C(R,V*) is defined, where the union is taken over all PeV*, aBaer cone of type (r,v).
Abstract: Let R and V be two skew subspaces with dimensions r and v of P=PG(d,q). If q is a square, then there is a Baer subspace V* of V, i.e. a subspace of dimension v and order √q. We call the set C(R,V*)=\(\mathop \cup \limits_p \), where the union is taken over all PeV*, aBaer cone oftype (r,v).
TL;DR: In this paper, the analogous motions with point paths all being plane or spherical curves (DARBOUX-motions or BRICARDmotions) are shown to be at least two-parametric.
Abstract: In Euclidean kinematics the motions with point paths all being plane or spherical curves (DARBOUX-motions or BRICARD-motions) are well known. Here we determine the analogous motions in the three-dimensional flag space and show that these motions are at least two-parametric.
TL;DR: In this paper, it was shown that every bijective map which preserves blockingsets of a fixed level 1 is a collineation, and that every affine space of infinite order has a blockingset.
Abstract: In every projective or affine space of infinite order there exist blockingsets. Moreover, we prove that every bijective map which preserves blockingsets of a fixed level 1 is a collineation.
TL;DR: In this article, it was shown that a finite Cn geometry (n ≥ 4) in which all lines are thick and all C3 residues are either buildings or flat has to be a building.
Abstract: In this paper we extend a result of [2] to cover a more general situation. In [2] it was shown that a finite Cn geometry (n ≥4) in which all lines are thick and all C3 residues are either buildings or flat has to be a building. Here we observe that the finiteness assumption of [2] was unnecessary in order to achieve the major part of the result. If we drop the finiteness assumption we can still prove that such a geometry is either a quotient of a building or flat. Flat Cn geometries for n ≥4 are seen to be degenerate in a certain sense. In the finite case with thick lines such degenerate geometries are easily shown not to exist, while finite buildings of type Cn with thick lines do not admit non-trivial quotients (Brouwer and Cohen, [1]). Thus the result of [2] follows as an immediate corollary of this more general case. The result does not hold when we drop the assumption that all lines are thick. In Section 3 we produce some examples of geometries of this type.
TL;DR: In this article, the authors consider the projective plane of order five from a combinatorial point of view and show that the complete graph on six vertices is a complete graph.
Abstract: In this paper, we consider the projective plane of order five from a combinatorial point of view. We shall see many of its properties (such as its uniqueness and existence, the order of the full collineation group and Segre's theorem) by looking at a structure as simple as the complete graph on six vertices.
TL;DR: In this article, an arbitrary set S is expressed as an increasing union of star-like sets and every three points of S are clearly visible via S from a common point of cl S if and only if each bounded subset of S may be extended to a starlike set in S.
Abstract: Let S be an arbitrary set in R2. Every three points of cl S are clearly visible via S from a common point of cl S if and only if each bounded subset of S may be extended to a starlike set in S. When this occurs, set S is expressible as an increasing union of starlike sets.