The ‘‘golden’’ hyperbolic models of Universe Alexey Stakhov, Boris Rozin * International Club of the Golden Section, 6 McCreary Trail, Bolton, Ont., Canada L7E 2C8 Accepted 31 March 2006 Abstract This article presents a review of new mathematical models of the hyperbolic space. These models are based on the golden section. In this article, the authors discuss the hyperbolic Fibonacci and Lucas functions and the surface of the golden shofar, which are the most important of these models. The authors also introduce, within this article, the golden hyperbolic approach for modeling the universe. Ó 2006 Elsevier Ltd. All rights reserved. I want to know how God created this world. I am not interested in this or that phenomenon, in the spectrum of this or that element. I want to know His thoughts; the rest are details. Albert Einstein 1. Introduction Discovering that the world around us is hyperbolic is probably one of the major achievements of science. The pri- ority of creation of the non-Euclidean geometry belongs to the Russian mathematician Nikolay Lobachevsky. In 1827, Lobachevsky offered a new geometric system based on hyperbolic functions. The need for new geometrical ideas became apparent in physics at the beginning of the 20th century and proceeded from Einstein’s special relativity theory, pre- sented in 1905. In 1908, three years after publication of special relativity theory, the German mathematician Herman Minkovsky gave the original geometrical interpretation of special relativity theory based on hyperbolic ideas. In 1993, the Ukrainian mathematicians Stakhov and Tkachenko developed a new approach to the hyperbolic geom- etry [4]. Using Binet formulas, they developed a new class of hyperbolic functions called the Hyperbolic Fibonacci and Lucas functions [4,5]. The authors of the present article further developed the ideas of the hyperbolic Fibonacci and Lucas functions and introduced the symmetric hyperbolic Fibonacci and Lucas functions in paper [6]. In [7], the authors developed a new surface of the second degree, called the golden shofar. The hyperbolic Fibonacci and Lucas functions and the surface of the golden shofar are the representatives of the golden mathematical models used for mod- eling hyperbolic space. Further development of the models of the universe requires the concepts of the generalized Fibonacci numbers and the generalized golden proportions [8], the golden algebraic equations [9], the generalized Binet formulas and the 0960-0779/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.04.046 * Corresponding author. E-mail addresses: goldenmuseum@rogers.com (A. Stakhov), rozinb@yahoo.com (B. Rozin). URLs: www.goldenmuseum.com (A. Stakhov), www.goldensection.net (B. Rozin). Chaos, Solitons and Fractals 34 (2007) 159–171 www.elsevier.com/locate/chaos