Roots : {Connections} ...between the Fibonacci Sequence and Phi, the Golden Proportion

The Fibonacci Sequence begins like this:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 600, 977, 1577, 2544, 4111, 6655…..

The observable pattern that each member is the sum of the two members immediately preceding it can be mathematically declared as a recurrence relation, and the Fibonacci sequence symbolically stated as:

F1=F2=1            Initial conditions
Fn=Fn-1+Fn-2        n>2    Recurrence relation

A related set of numbers is that of the ratios between two consecutive Fibonacci numbers:

1/1, 2/1, 3/2, 5/3, 8/5, 13/8, 21/13, 34/21, 55/34, 89/55, 144/89, 233/144, 377/233………

and in decimal form appears in approximation as:

1.000, 2.000, 1.500, 1.667, 1.600, 1.625, 1.615, 1.619, 1.618, 1.618, 1.618………..

As the sequence approaches infinity, the ratios converge to a limit of 1.618033989, a number with many names: Phi, the Golden Mean, the Golden Proportion, or the Divine Section.  Phi’s properties were known at least 16 centuries before Fibonacci, certainly to the ancient Greeks who contemplated it in geometry and applied it in art and architecture, and likely also familiar to the ancient Egyptians who built the Great Pyramid which embodies the proportion.  This proportion, Phi, is defined most easily on a line segment which is divided into two sections of unequal length, a and b.  A line segment thus divided will display the Golden Proportion if the ratio of the shorter section (b) to the longer section (a) is the same as the ratio of the longer section (a) to the length of the whole (a+b).  Another way to visualize the Golden Proportion is in a rectangle with sides of length in proportion 1 to 1.618.  If the interior of this rectangle is divided into a square with side length 1, then the remaining rectangle will have sides in an equal proportion to that of the original rectangle.  This process of dividing each consecutively smaller rectangle according to the Golden Proportion can go on forever, and the process can also be reversed, thus modeling infinite growth.  Both the Fibonacci numbers themselves, as well as the related Golden Proportion, are not only mathematical concepts invented and applied by humans in such fields as art, architecture, and music, but are actually observable in the natural forms around us—from the spiral of the galaxy to the spiral of a Nautilus seashell, from the growth patterns in plants to the generational pattern of animals, from the behavior of light to the pattern of a strand of DNA.

Downwards on tree : {Main Roots} {How the Fibonacci Sequence Got Its Name}


Sources: Summum website <summum.org>; Gruner <physics.ucla.edu>. See bibliography.