Институт Золотого Сечения — Математика Гармонии

Аlexey Stakhov
The generalized golden proportions, a new theory of real numbers, and ternary mirror-symmetrical arithmetic
Abstract

We consider two important generalizations of the golden proportion: golden p-proportions [Stakhov A.P. Introduction into algorithmic measurement theory. Soviet Radio, Moscow, 1977 [in Russian]] and «metallic means» [Spinadel V.W. La familia de numeros metalicos en Diseflo. Primer Seminario Nacional de Grafica Digital, Sesion de Morfologia у Matematica, FADU, UBA, 11-13 Junio de 1997, vol. II, ISBN 950-25-0424-9 [in Spanish]; Spinadel VW. New sma-randache sequences. In: Proceedings of the first international conference on smarandache type notions in number theory, 21-24 August 1997. Lupton: American Research Press; 1997, p. 81-116. ISBN 1-879585-58-8]. We develop a constructive approach to the theory of real numbers that is based on the number systems with irrational radices (Bergman's number system and Stakhov's codes of the golden p-proportions). It follows from this approach ternary mirror-symmetrical arithmetic that is the basis of new computer projects.

1. Introduction

A theory of Fibonacci numbers and the golden section is an important branch of modern mathematics [1—3]. Let us consider the basic notions of this theory. It is known from the ancient times that a problem of the «division of a line segment AB with a point С in the extreme and middle ratio»
 (1)

This problem came to us from the «Euclidean Elements». In modern science the problem is known as the golden section problem. Its solution reduces to the following algebraic equation:
 (2)

Its positive root
 (3)

is called the golden proportion, the golden mean, or the golden ratio.

It follows from (2) the following remarkable identity for the golden ratio:
 (4)

where n takes its values from the set:

The Fibonacci numbers
 (5)

are a numerical sequence given by the following recurrence relation:
 (6)

with the initial terms
 (7)

The Lucas numbers
 (8)

are a numerical sequence given by the following recurrence relation:
 (9)

with the initial terms
 (10)

The Fibonacci and Lucas numbers can be extended to the negative values of the index n. The «extended» Fibonacci and Lucas numbers are presented in Table 1.

As can be seen from Table 1, the terms of the «extended» series Fn and Ln have a number of wonderful mathematical properties. For example, for the odd n = 2k + 1 the terms of the sequences Fn and F-n coincide, that is, F2k+1 = F-2k-1, and for the even n = 2k they are opposite by the sign, that is, F2k = —F-2k. As to the Lucas numbers Ln, here all are opposite, that is, L2k = L-2k; L2k+1 = -L-2k-1.

In the last decades this theory and its applications were developing intensively. In this connection the works on the generalization of Fibonacci numbers and golden proportion are of the greatest interest. There are two of the most important generalizations of the golden proportion. The first of them, which was called the golden p-proportions {p = 0,1,2,3,...}, was made by the present author in a book [4] published in 1977. The author's works [4-21] in this field develops and generalizes the theory of Fibonacci numbers and the golden proportion and gives many interesting applications of the golden p-proportions to the different fields of mathematics and computer science.

Another generalization of the golden proportion called the metallic means or metallic proportions was developed by the Argentinean mathematician Vera Spinadel in the series of the papers and books [22-32]. The first Spinadel's paper in this area [22] was published in 1997.