Nonclassical properties of a discrete geometries space
- PDF / 153,938 Bytes
- 9 Pages / 595.276 x 793.701 pts Page_size
- 0 Downloads / 170 Views
NONCLASSICAL PROPERTIES OF A DISCRETE GEOMETRIES SPACE
UDC 514.01
Yu. G. Grigoryan
Non-classical issues arising in analyzing the Discrete Geometries Space N( D ) are considered. An arithmetic model of the Discrete Euclidean Geometry is constructed on the Cartesian plane. This geometry is shown to reject the ordering axiom of the Classical Euclidean Geometry. A formula is constructed that allows one to determine the asymmetry ratio of the space N( D ) and is based on the asymmetry of arithmetic graphs with a polyhedral structure. It is noted that N( D ) is a space satisfying the Riemann hypotheses. Keywords: nonclassical space, discrete geometry, model, asymmetry, symmetry.
In [1–4], the discrete metric space N( D ) is constructed in which, based on a unified system of transformations, three geometries that are discrete analogues of Euclidean, non-Euclidean, and projective geometries are formulated. A minimal axiomatic base, the system relatedness, and the uniqueness of these geometries testify to the perfection, universality, and fundamentality of the space N( D ) . This article is a continuation of [1–5] connected with the investigation of new nonclassical properties of space that are based on a special trigonometry constructed earlier [4]. This allows one to determine the asymmetry ratio of a triangle and each subspace (layer) of the space N( D ) . It is shown that, in the space N( D ) , two classical axioms, namely, the axioms of ordering and parallelism are negated and a comparative analysis of properties of the constructed geometries with Riemann hypotheses [6] is performed. For convenience, we introduce some definitions and theorems from [4] in a simpler form. Definition 1. A set of real numbers R (| R | ³ 3) without zero is called a scalar set if it satisfies the following two conditions: (a) for any pair of different A , B Î R, we have A + B > 0 ; (b) for any triple of different A , B , C Î R, we have AB + AC + BC ³ 0 . For example, the triple ( -2, 3, 6) Î R and the triple ( -2, 3, 4 ) Ï R. It may be noted that, in [1–4], the set R is called an arithmetic set. In this article, it is called a scalar set, which more corresponds to its content. THEOREM 1. One can associate with each triple of real numbers A , B , C Î R some DA1 B1 C1 whose side lengths are equal to a =| B1 C1 |, b =| A1 C1 |, c =| A1 B1 | , where A=
b2 + c2 - a2 , a2 + c2 - b2 , a2 + b2 - c2 . B= C= 2 2 2
(1)
(2)
Proof. We solve system (2) with respect to a, b, and c. It has the unique solution a = B + C , b = A + C , c = A + B.
(3)
Russian-Armenian Modern Humanitarian Institute, Yerevan, Armenia, [email protected] and [email protected]. Translated from Kibernetika i Sistemnyi Analiz, No. 5, pp. 51–59, September–October 2009. Original article submitted February 6, 2009. 714
1060-0396/09/4505-0714
©
2009 Springer Science+Business Media, Inc.
Since A , B , C Î R, all the expressions under the radicals in solution (3) are positive, as well as the numbers a, b, and c. Let us show that these numbers satisfy the triangle axiom. We take
Data Loading...
