Related papers: The bridge between Desargues' and Pappus' theorems

In geometry, Monge's theorem states that for any three nonoverlapping circles of distinct radii in the two dimensional analytical plane equipped with the Euclidean metric, none of which is completely inside one of the others, the…

In complete analogy with Seiberg-Witten map defined in noncommutative geometry we introduce a new map between a q-deformed gauge theory and an ordinary gauge theory. The construction of this map is elaborated in order to fit the Hopf…

By the methods of the synthetic geometry we investigate properties of objects generated from a complete quadrangle and a line, which lies in its plane. We start with a problem from the book of Sharygin "Problems in Plane Geometry". We…

We consider how the problem of determining normal forms for a specific class of nonholonomic systems leads to various interesting and concrete bridges between two apparently unrelated themes. Various ideas that traditionally pertain to the…

In classical geometric algebra, there have been several treatments of affine and projective planes based on fields. In this thesis we approach affine and projective planes from a constructive point of view and we base our geometry on local…

Using the theory of extensors developed in a previous paper we present a theory of the parallelism structure on arbitrary smooth manifold. Two kinds of Cartan connection operators are introduced and both appear in intrinsic versions (i.e.,…

A classical theory of Desarguesian geometry, originating with D. Hilbert in his 1899 treatise, Grundlagen der Geometrie, leads from axioms to the construction of a division ring from which coordinates may be assigned to points, and…

We propose two new proofs of the Pythagorean theorem via area rearrangement arguments starting from very simple geometric configurations. The constructions depend on an angular parameter, each choice of which yields a proof. For specific…

In this article we use the Desargues' theorem and its reciprocal to solve two problems.

It is shown that the structure of the generalized quadrangle of order two is fully encoded in the properties of the Desargues configuration. A point of the quadrangle is represented by a geometric hyperplane of the Desargues configuration…

We study central configurations when the set of positions is symmetric. We use a theorem from representation theory of finite groups to explore the symmetry properties of equations for central configurations. This approach simplifies…

Canonical duality-triality is a breakthrough methodological theory, which can be used not only for modeling complex systems within a unified framework, but also for solving a wide class of challenging problems from real-world applications.…

Recently the duality map between electric-like asymptotic charges of $p$-form gauge theories is studied. The outcome is an existence and uniqueness theorem and the topological nature of the duality map. The goal of this work is to extend…

The results here presented are a continuation of the algebraic research line which attempts to find properties of multiple-valued systems based on a poset of two agents. The aim of this paper is to exhibit two relationships between some…

Using techniques of projective geometry, we give elementary proofs of two theorems concerning Hagge configurations.

We try to understand complete types over a somewhat saturated model of a complete first order theory which is dependent (previously called NIP), by "decomposition theorems for such types". Our thesis is that the picture of dependent theory…

The classical theorems relating integral binary quadratic forms and ideal classes of quadratic orders have been of tremendous importance in mathematics, and many authors have given extensions of these theorems to rings other than the…

We describe a mathematical link between aspects of information theory, called pairwise comparisons, and discretized gauge theories. The link is made by the notion of holonomy along the edges of a simplex. This correspondance leads to open…

We start by introducing the basics of configurations of points and lines, and then move into discussing symmetry groups of these configurations. Specifically, we explore how we might classify the symmetries of $(9_3)$ and $(10_3)$ geometric…

The notion of geometric construction is introduced. This notion allows to compare incidence configurations in the algebraic and tropical plane. We provide an algorithm such that, given a tropical instance of a geometric construction, it…