Desargues Configurations Research Articles

Note on Discovering Doily in PG(2,5)

W. L. Edge proved that the internal points of a conic in PG(2,5), together with the collinear triples on the non-secant lines, form the Desargues configuration. M. Saniga showed an intimate connection between Desargues configurations and the generalized quadrangles of order 2, GQ(2,2), whose representation was dubbed “the doily” by Stan Payne in 1973. In this note, we prove that the external points of a conic in PG(2,5), together with the collinear and non-collinear triples on the non-tangent lines, form the generalized quadrangle of order 2.

Inherited conics in Hall planes

The existence of ovals and hyperovals is an old question in the theory of non-Desarguesian planes. The aim of this paper is to describe when a conic of PG(2,q) remains an arc in the Hall plane obtained by derivation. Some combinatorial properties of the inherited conics are obtained also in those cases when it is not an arc. The key ingredient of the proof is an old lemma by Segre–Korchmáros on Desargues configurations with perspective triangles inscribed in a conic.

Desargues configurations with four self-conjugate points

In a projective plane over a field F, the diagonal points of a quadrangle are collinear if and only if F has characteristic 2. Such a quadrangle together with its diagonal points and the lines connecting these points form the subplane of order 2, called a Fano plane. Using Desargues configurations and polarities, we provide a similar type of synthetic criterion and construction for characteristic 3 fields. Let F be a field with characteristic not equal to 2. From any quadrangle and one of its diagonal points V, we construct a pair of triangles $$\Delta _1,\Delta _2$$ in perspective from V, and the resulting Desargues configuration D such that the vertices of $$\Delta _1$$ are self-conjugate under a particular polarity. For this Desargues configuration D, the vertex of perspectivity V of the pair $$\Delta _1,\Delta _2$$ is a fourth self-conjugate point if and only if F has characteristic 3. If F has characteristic 3, then the 10 points and 10 lines of D together with three additional points and three additional lines yield a projective subplane of order 3 of $$\pi $$ .

A Combinatorial Grassmannian Representation of the Magic Three-Qubit Veldkamp Line

It is demonstrated that the magic three-qubit Veldkamp line occurs naturally within the Veldkamp space of a combinatorial Grassmannian of type G 2 ( 7 ) , V ( G 2 ( 7 ) ) . The lines of the ambient symplectic polar space are those lines of V ( G 2 ( 7 ) ) whose cores feature an odd number of points of G 2 ( 7 ) . After introducing the basic properties of three different types of points and seven distinct types of lines of V ( G 2 ( 7 ) ) , we explicitly show the combinatorial Grassmannian composition of the magic Veldkamp line; we first give representatives of points and lines of its core generalized quadrangle GQ ( 2 , 2 ) , and then additional points and lines of a specific elliptic quadric Q - (5, 2), a hyperbolic quadric Q + (5, 2), and a quadratic cone Q ^ (4, 2) that are centered on the GQ ( 2 , 2 ) . In particular, each point of Q + (5, 2) is represented by a Pasch configuration and its complementary line, the (Schläfli) double-six of points in Q - (5, 2) comprise six Cayley–Salmon configurations and six Desargues configurations with their complementary points, and the remaining Cayley–Salmon configuration stands for the vertex of Q ^ (4, 2).

Quantum angular momentum, projective geometry and the networks of seven and ten spins: Fano, Desargues and alternative incidence configurations

The basic ingredients of the quantum theory of orbital and spin angular momentum (vector coefficients, 3nj symbols) encounter continuing relevance in wide areas beyond the traditional ones (molecular, atomic and nuclear spectroscopies and dynamics). This paper offers insight on the connection at the most elementary of levels with the diagrammatic approaches to projective geometry. In particular here we exhibit how the Fano, Desargues and related incidence configurations emerge in the Racah and in the Biedenharn-Elliott identities, corresponding respectively to the hexagonal and pentagonal relationships that provide the basis for the construction of 3nj symbols and of spin networks. It is shown that the treatment, although mostly confined to the quadrangulation of the real projective plane, permits however the introduction of networks involving seven and ten spins, and preludes to developments towards computational and asymptotic approaches for quantum and semi-classical applications to spectroscopy and dynamics.

One-point extensions in n_3 configurations

Given an n 3 configuration, a 1-point extension is a technique that constructs an ( n + 1) 3 configuration from it. It is proved that all ( n + 1) 3 configurations can be constructed from an n 3 configuration using a 1-point extension, except for the Fano, Pappus, and Desargues configurations, and a family of Fano-type configurations. A 3-point extension is also described. A 3-point extension of the Fano configuration produces the Desargues and anti-Pappian configurations. The significance of the 1-point extension is that it can frequently be used to construct real and/or rational coordinatizations in the plane of an ( n + 1) 3 configuration, whenever it is geometric, and the corresponding n 3 configuration is also geometric.

The pentagon relation and incidence geometry

We define a map \documentclass[12pt]{minimal}\begin{document}$S:{\mathbb {D}}^2\times {\mathbb {D}}^2 \dashrightarrow {\mathbb {D}}^2\times {\mathbb {D}}^2$\end{document}S:D2×D2⤏D2×D2, where \documentclass[12pt]{minimal}\begin{document}${\mathbb {D}}$\end{document}D is an arbitrary division ring (skew field), associated with the Veblen configuration, and we show that such a map provides solutions to the functional dynamical pentagon equation. We explain that fact in elementary geometric terms using the symmetry of the Veblen and Desargues configurations. We introduce also another map of a geometric origin with the pentagon property. We show equivalence of these maps with recently introduced Desargues maps which provide geometric interpretation to a non-commutative version of Hirota's discrete Kadomtsev–Petviashvili equation. Finally, we demonstrate that in an appropriate gauge the (commutative version of the) maps preserves a natural Poisson structure—the quasiclassical limit of the Weyl commutation relations. The corresponding quantum reduction is then studied. In particular, we discuss uniqueness of the Weyl relations for the ultra-local reduction of the map. We give then the corresponding solution of the quantum pentagon equation in terms of the non-compact quantum dilogarithm function.

Modelling of Ring Geometry from von Neumann’s Point of View

The idea to consider different unions of points, lines, planes, etc., is rather old. Many important configurations of such kinds are geometric (or matroidal) lattices. In this work, we study Desargues, Pappus, and Pasch configurations in D-semimodular lattices.

Fundamental group of Desargues configuration spaces

We compute the fundamental group of various spaces of Desargues configurations in complex projective spaces: planar and non-planar configurations, with a fixed center and also with an arbitrary center.

Desargues Configurations via Polar Conics of Plane Cubics

Using polar conics of plane cubics we define a rational map \(\Psi_m: \mathcal{M}_6^b \rightarrow \mathcal{M}_D\) from the moduli space of stable binary sextics into the moduli space of Desargues configurations. We show that this map is the inverse of a birational map \(\Phi_s: \mathcal{M}_D \rightarrow \mathcal{M}_6^b\) defined via the von Staudt conic. In particular Ψm is a birational map.

Curves of genus 2 and Desargues configurations

A Desargues configuration is the configuration of 10 points and 10 lines of the classical theorem of Desargues in the complex projective plane. For a precise definition see Section 2. The Greek mathematician C. Stephanos showed in 1883 (see [10]) that one can associate to every Desargues configuration a curve of genus 2 in a canonical way. Moreover he proved that the induced map from the moduli space of Desargues configurations to the moduli space of curves of genus 2 is birational. Our main motivation for writing this paper was to understand the last result. Stephanos needed about a hundred pages of classical invariant theory to prove it. We apply instead a simple argument of Schubert calculus to prove a slightly more precise version of his result. Let MD denote the (coarse) moduli space of Desargues configurations. It is a threedimensional quasiprojective variety. On the other hand, let M 6 denote the moduli space of stable binary sextics. There is a canonical isomorphism H2nD1 GM 6 (see [1]). Here H2 denotes the moduli space of stable curves of genus 2 in the sense of Deligne–Mumford and D1 the boundary divisor parametrizing two curves of genus 1 intersecting transversely in one point. So instead of curves of genus 2, we may speak of binary sextics. The main result of the paper consists of the following two statements:

A Configurational Characterization of the Moufang Generalized Polygons

We generalize Baer's theorem on Desargues configurations in projective planes to all generalized polygons, thus obtaining a common geometric characterization of all (finite and infinite) Moufang generalized polygons.

On the existence of little Desargues configurations in planes satisfying certain weak configurational conditions

On the existence of little Desargues configurations in planes satisfying certain weak configurational conditions

Desargues configurations and their collineation groups

When two triangles, ABC and A′B′C′, are perspective from a point 0, their pairs of corresponding sides meet on a line 0, the axis of perspective. The line OA passes through A′ and some point on 0. These four points have a certain cross ratio which is the same if B or C is used instead of A. The reciprocal cross ratio arises if A′B′C′ is regarded as the first triangle and ABC the second. The complete figure contains ten pairs of perspective triangles, yielding twenty cross ratios. In section 6 a technique, suggested by D. W. Babbage and John Rigby, is used to express these cross ratios in the form

Desargues Configurations Admitting. A Collineation Group*

Journal of the London Mathematical SocietyVolume s1-8, Issue 1 p. 34-39 Notes and Papers Desargues Configurations Admitting. A Collineation Group† B. Ramamurti, B. RamamurtiSearch for more papers by this author B. Ramamurti, B. RamamurtiSearch for more papers by this author First published: January 1933 https://doi.org/10.1112/jlms/s1-8.1.34Citations: 1 † * I am indebted to Dr. R. Vaidyanathaswamy and to the Referee for suggestions and criticism in the preparation of this paper. AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Citing Literature Volumes1-8, Issue1January 1933Pages 34-39 RelatedInformation