Showing papers in "Journal of Geometry in 1978"
TL;DR: In this paper, the transposition of translation planes in the topological context has been studied and it has been shown that every hyperplane of ℝ2n contains a component of a topological congruence.
Abstract: This note deals with the transposition of translation planes in the topological context. We show that a topological congruenceC of the real vector space ℝ2n has the property that every hyperplane of ℝ2n contains a component ofC. This makes it possible to define the transposePτ of the topological translation planeP associated withC; it is proved that the translation planePτ is topological also. The relationship between collineation groups and the relationship between coordinatizing quasifields ofP andPτ are also discussed.
18 citations
TL;DR: In this article, it was shown that the boundary-complex of a convex polytope can be "shelled" using the theory of stellar operations, and several results on special shelling procedures and on non-shellable complexes were obtained.
Abstract: Brugesser and Mani proved that the boundary-complex of a convex polytope can be “shelled”. This result lead to McMullen's proof of the “Upper-bound-conjecture”. We show that the “shellability” of complexes has a close connection to the theory of stellar operations. Several results on special shelling procedures and on non-shellable complexes are obtained.
9 citations
TL;DR: In this article, it was shown that a surface of order three with a peak contains one, two or three lines and there are four types of such surfaces based upon the configuration of these lines.
Abstract: In [1], we presented a theory of surfaces of order three in real projective three-spaceP3. In the present paper, we prove that a surfaceF of order three with a peak contains one, two or three lines and there are four types of suchF based upon the configuration of these lines. We describe eachF by determining the existence and the distribution of elliptic, parabolic and hyperbolic points; that is, points ofF not lying on any line contained inF.
2 citations
TL;DR: In this paper, the affine Klingenberg structures (AK-structures) were defined as a generalization of affine AK-planes and the Desarguesian AK-structure was defined.
Abstract: In this paper we define the affine Klingenberg structures (AK-structures) as a generalization of the affine Klingenberg planes (AK-planes) [2]. By means of moduls over the local rings (which need not be free as with the coordinate AK-planes is the case) there is constructed a class of the coordinate AK-structures. In II and III we define the Desarguesian AK-structures. Any coordinate AK-structure is a Desarguesian AK-structure. With the methods established by E.Artin and applied by W.Klingenberg in [4] we show that any Desarguesian AK-structure is isomorphic with the coordinate AK-structure. Some of the results obtained have been applied to the theory of the AK-planes. Thus, for example, it has been shown that it is possible to assume less with the definition of the Desarguesian AK-planes than with [4] or even with [5].
2 citations
TL;DR: In this article, the authors presented a characterization of the Hall planes in terms of the order of the affine central collineations in a derived semifield plane, which essentially allows the extension of the theorems of Kirkpatrick and Rahilly on generalized Hall planes to arbitrary derived semidefinite planes.
Abstract: LetG denote the collineation group generated by the set of all affine central collineations in a derived semifield plane. We present a characterization of the Hall planes in terms of the order ofG. This essentially allows the extension of the theorems of Kirkpatrick and Rahilly on generalized Hall planes to arbitrary derived semifield planes. That is, a derived semifield plane of order q2 is a Hall plane precisely when it admits q+1 involutory central collineations.
2 citations
TL;DR: In this paper, the aperture distance of a curve orthogonal to the generating lines of a ruled closed surface is considered, and the aperture angle of an orthogonally circumscribed tangent surface.
Abstract: On a ruled closed surface Φ in the elliptic 3-space two integral invariants are considered: the aperture distance of a curve orthogonal to the generating lines of Φ, and the aperture angle of an orthogonally circumscribed tangent surfaces. By means of these integral invariants and by considering certain ruled surfaces associated to Φ one finds the geometric meaning of further integral invariants. If Φ is generated by the binormals of a curve one obtains some properties of closed curves in the elliptic 3-space.
2 citations
TL;DR: In this paper, it was shown that under certain conditions on the coordinating nearfield, each metrizable finite dimensional desarguesian topological incidence group has a unique completion, and the analogous problem for topological projective planes has been studied by Breitsprecher.
Abstract: In this paper it is shown that under certain conditions on the coordinating nearfield, each metrizable finite dimensional desarguesian topological incidence group has a unique completion The analogous problem for topological projective planes has been studied by Breitsprecher [2]
1 citations
TL;DR: In this article, the authors used the operation of extension to construct the linear hull of a set A in a convexity space and showed that 〈A〉=jA if j=[log2(n−1)]+3, [x] meaning the greatest integer contained in x.
Abstract: We use the operation of extension to construct the linear hull 〈A〉 of a set A in a convexity space: if A is n-dimensional (n>1) and1A=A/A,jA=(j−1)A/(j−1)A, we show that 〈A〉=jA if j=[log2(n−1)]+3, [x] meaning the greatest integer contained in x.
TL;DR: In this article, it was shown that chains of perpendiculars in Hjelmslev-groups are closely related to the group generated by the points of a set of lines and a symmetric orthogonality relation.
Abstract: Let be given a geometric structure with a set of lines and a symmetric orthogonality relation. F. Bachmann defined in [3] chains of perpendiculars in such a structure, and studied those chains in Hjelmslev-groups. All AGS-groups (= groups treated in [1]) are Hjelmslev-groups, and the theory of Hjelmslev-groups includes also Hjelmslev's concept of “Allgemeine Kongruenzlehre”. In Hjelmslev-groups, chains of perpendiculars are closely related to the group generated by the points. We establish some results about chains of perpendiculars in special classes of Hjelmslev-groups.