A Note on Fibonacci Numbers and the Golden Ratio of Order k.

We define and study the notion of the golden ratio of order k = 0, denoted ϕk, as a generalized form of the golden ratio ϕ for any real number k = 0. We show that similar to the special case of ϕ and its conjugate ϕ, ϕk and ϕk are the two distinct roots of a quadratic polynomial for any fixed real k = 0. We express some numerical and algebraic properties of ϕk and ϕk and write their relations to ϕ and ϕ, respectively, with some examples for some special values of k. In particular, it is shown that ϕk = ϕ and ϕk = ϕ if and only if k = 0. We show that Z[(k + 1)ϕk] is a subring of the ring Z[ϕ] for any nonnegative integer k. We will define the golden rectangle of order k (or k-golden rectangle for short) with a class of examples for all k = 0. We also discuss some cases of two Fibonacci numbers in connection to the golden ratio. We will show that the ratio of height to width of the pages of the Gutenberg Bible is the golden ratio of order k ϕ= 0. Actually, some erroneous ideas and examples of disputed observations related to the golden ratio are good reasons to apply ϕk to improve the measurements regarding ?ϕ for some k ϕ= 0. Finally, we end the paper by posing a question related to the Penrose tiling and quasicrystals in connection to the golden ratio of order k > 0. [ABSTRACT FROM AUTHOR]