## The golden ratio and Fibonacci numbers by Richard A Dunlap

By Richard A Dunlap

During this worthy publication, the fundamental mathematical homes of the golden ratio and its prevalence within the dimensions of 2- and 3-dimensional figures with fivefold symmetry are mentioned. additionally, the iteration of the Fibonacci sequence and generalized Fibonacci sequence and their dating to the golden ratio are provided. those techniques are utilized to algorithms for looking out and serve as minimization.Read more...

summary: during this priceless publication, the fundamental mathematical homes of the golden ratio and its incidence within the dimensions of 2- and three-d figures with fivefold symmetry are mentioned. furthermore, the new release of the Fibonacci sequence and generalized Fibonacci sequence and their courting to the golden ratio are offered. those innovations are utilized to algorithms for looking out and serve as minimization. The Fibonacci series is seen as a one-dimensional aperiodic, lattice and those rules are prolonged to 2- and 3-dimensional Penrose tilings and the concept that of incommensurate p

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**Example text**

The large rabbit icon represents a pair of adult rabbits and the small icon a pair of baby rabbits. From Dunlap (1990). This sequence of A's and b's is d e t e ~ ~ ~ s tthat i c ; is, it may be extended indefinitely in a unique way because the rules for generating the next character in the sequence are well defined, In this sense it is distinct from a random sequence of A's and b's. However, it is also different from what is referred to as a periodic sequence. A simple example of a periodic sequence might be Fibonacci Numbers AbAbAbAbAbAbAbAbAb...

Will approach Tin the limit of large n. Certain choices of seed values will, however, yield additive sequences which are distinctly different from ~ j studied ~ extensively in the late the Fibonacci sequence. One such ~ s s i b iwas nineteenth century by the French mathematician Edouard Lucas who published the results of these investigations in 1877. He considered the next smallest seed values LO= 2 and LI = 1. These values will generate the additive sequence 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, ...

This is climbed by taking either one step or two steps at a time and the number of different ways of climbing the stairs, S,,, is to be determined. E n is 1 then the solution is simple, S,, = 1. e. 1 + 1, or 2. For n = 3 there are three different ways; 1 + 2, 2 + 1 or 1 + 1 + 1. Thus the number of possibilities for n stairs is equal to the mun of Sn-i and S+2. That is; - - which is equivalent to Eq. 3). This shows that the values of S,, follow the Fibonacci sequence with s, = Fn+, and the values of S,, for small values of n as given above confirm this.