Three Vertex and Parallelograms in the Affine Plane: Similarity and Addition Abelian Groups of Similarly <i>n</i>-Vertexes in the Desargues Affine Plane

Abstract

In this article will do a' concept generalization n-gon. By renouncing the metrics in much axiomatic geometry, the need arises for a new label to this concept. In this paper will use the meaning of n-vertexes. As you know in affine and projective plane simply set of points, blocks and incidence relation, which is argued in [1], [2], [3]. In this paper will focus on affine plane. Will describe the meaning of the similarity n-vertexes. Will determine the addition of similar three-vertexes in Desargues affine plane, which is argued in [1], [2], [3], and show that this set of three-vertexes forms an commutative group associated with additions of three-vertexes. At the end of this paperare making a generalization of the meeting of similarity nvertexes in Desargues affine plane, also here it turns out to have a commutative group, associated with additions of similarity n-vertexes.

Ordered Line and Skew-Fields in the Desargues Affine Plane

“…From the translation properties described at [20], we have the following parallelisms, Clearly, we see that the translation ϕ can be seen as parallel projection P p , from line ℓ to line ℓ ′ , with direction the line ℓ Aϕ (A) .…”

Skew-Field of Trace-Preserving Endomorphisms, of Translation Group in Affine Plane

“…In an affine plane related to dilation δ id P all traces Pδ (P) for all P ∈ P , or cross the by a single point, or are parallel between themselves (see [12], [13]).…”

“…The Algebra (Tr A ) Tr A , +, • , is called the algebra of maps of Tr A , on himself. A map α : Tr A −→ Tr A , is an endomorphism of the group (Tr A , •), on himself (see [16], [12]), namely such that, ( 6)…”

“…Also great contributions in this direction have been made by, E.Artin in [1], H. S. M. Coxeter, in [3], Marcel Berger in [2], Robin Hartshrone in [5], etc. Even earlier, in my works ' [15], [18], [16], [4], [13], [14], [17], [20], [12], [19]' I have brought up quite a few interesting facts about the association of algebraic structures with affine planes and with 'Desargues affine planes', and vice versa.…”

Isomorphic-Dilations of the skew-fields constructed over parallel lines in the Desargues affine plane

“…The foundations for the study of the connections between axiomatic geometry and algebraic structures were set forth by Hilbert [7], recently elaborated and extended in terms of the algebra of affine planes in, for example, [8], [3, §IX.3, p. 574](affine transformation of the plane transforms a lattice into a lattice) and in the Desargues affine plane (see, e.g., [20,21,6,18,19,4,17]). Properties of geometric structures such as adjacency and proximity [2, §III.…”