![]() ![]() The Golden Ratio is also a way of dividing a segment into two pieces that reproduce the statement at the beginning of this article: the small is to the large as the large is to the whole. Where F(n) is the n th element of the Fibonacci sequence starting as F(0) = F(1) = 1.īy the way, it is inmediately apparent from Table 3 that φ itself establishes a "Fibonacci" sequence, which at the same time is a geometric progression, both above and below unity: We are preparing an article on this subject that will be available soon in the corresponding section of our site.īefore going on, we should point out some mathematical properties that follow from the fact that φ satisfies the second order equation φ 2 = φ + 1:ġ/φ n = 1/φ n-2-1/φ n-1 = (-1) n The Fibonacci sequence has many more interesting properties and is more involved in the reality that we perceive than we suspect. Why is the Fibonacci sequence so special? Well, at least from the fact that, as Drunvalo Melchizedek points out in his first book "The Ancient Secret of the Flower of Life", Nature uses this property to construct sequences of lengths that converge to the Golden Ratio, such as in the distances between successive branches of a tree, or successive leaves in a branch, or the dimensions of our own body:įigure 1: Two examples of biological constructs that obey the Fibonacci recurrence. Table 2: The quotient of consecutive elements in the Fibonacci sequence converge to the Golden Ratio. Table 1: The quotient of consecutive elements in the Lucas sequence converge to the Golden Ratio.īut there is one very special sequence that is closely related to the Golden Ratio, and this is the Fibonacci sequence ( a 0=1, a 1=1):įibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, … For example when a 0=2 and a 1=1 you get Lucas sequence: You could come to an infinite number of such sequences depending on the initial values a 0 and a 1. Notice that the Golden Ratio is connected with how the series is constructed, but not with any particular example of that construction. What this result states is that the ratio between consecutive values of any sequence of this type always aproaches φ. This leads to the well known second order equation whose positive solution is φ: ![]() ![]() ![]() If the consecutive ratios q n tend to a limiting value Q, this must satisfy the equation: Given two positive initial elements a 0 and a 1, compute a general element a n by the addition of the two preceding ones:Īs a consequence, the ratio of two consecutive elements in the series q n = a n / a n-1 also satisfies a recurrence:Ī n+1 a n = 1 + a n-1 a n ⇒ q n+1 = 1 + 1 q n This ratio can be obtained as a limiting form of the following general type of succession (we noticed this property after reading R.W. It is important to stress that, although the Greeks gave to the Golden Ration its name Phi (φ), as its name indicates it is actually a ratio. The "applications" are left for the separate pages that you can find at the "And much more." section, although I encourge you to read this article first. We will be talking about the definition of the Golden Ratio, its geometrical construction, some mathematical properties, and some important geometrical objects where you can find it. As I don't like to take things for granted, I don't expect you to do so, so I will try to provide short mathematical or geometrical demostrations of my statements (don't be afraid, you won't face any partial differential equation). In this article I will present the basics about the Golden Ratio. Such is the case, for example, of Atomic Physics or DNA codon populations of the whole human genome. In this site we also provide some examples of disciplines where the presence of the Golden Ratio was unsuspected until recently. We can find it in art, music composition, even in the proportions of our own body, and elsewhere in Nature "hidden" behind the Fibonacci sequence. This ratio has been venerated by every culture in the planet. This is usually applied to proportions between segments. Simply stated, the Golden Ratio establishes that the small is to the large as the large is to the whole. ![]()
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |