Appendix B

Fibonacci Numbers and the Golden Ratio

                         

        The golden ratio is an irrational mathematical constant, approximately 1.61803398875. This ratio is related to the Fibonacci numbers in which each number is the sum of the two preceding numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, .... The golden ratio can be obtained by dividing each of the Fibonacci numbers in the sequence by the previous number, to yield the series:

∞ (infinity), 1, 2, 1.5, 1.6667, 1.6000, 1.6250, 1.6154, 1.6190, 1.6176...

        Gradually each subsequent quotient approaches, ever closer, the number: 1.61803398875... which is the accurate number for the golden ratio. We also have to distinguish between two forms:

φ  =  1.61803398875...       and          Φ =  0.61803398875...

where φ is the result of dividing each Fibonacci number by the previous number and Φ is the result of dividing each Fibonacci number by the following number. Both of these numbers are considered by different authors as the Golden Ratio (although some select one, some the other). Fortunately, there is no real problem with this variation since one can quickly discover that:

1 / φ  =  1 / 1.61803398875...  =  0.61803398875...  =  Φ

        This relationship is accurate to however many decimal places one cares to carry it.
        It is quite amazing that the Fibonacci number patterns occur so frequently in nature ( flowers, shells, plants, leaves, to name a few) that this phenomenon appears to be one of the principal "laws of nature". Fibonacci sequences appear in biological settings, in two consecutive Fibonacci numbers, such as branching in trees, arrangement of leaves on a stem, the fruitlets of a pineapple, the flowering of artichoke, an uncurling fern and the arrangement of a pine cone. In addition, numerous claims of  the Fibonacci numbers or golden ratios in nature are found in popular sources, e.g. relating to the breeding of rabbits, the spirals of shells, and the curve of waves. The Fibonacci numbers or golden ratios are also found in the family tree of honeybees, etc. Humans have caught on to this fad-setting trend by composing the musical scale, where 1 piano has 1 keyboard with black keys arranged in groups of 2 and 3, consisting of 5 black keys (sharps and flats) and 8 white keys (whole tones) for a 13 note chromatic musical octave. Musically, we have thus accounted for seven consecutive Fibonacci numbers. And from the viewpoint of Nature, this form of music becomes harmonious with our physical, emotional, mental, and undoubtedly spiritual bodies. (There are countless articles on the Fibonacci numbers and golden ratio. If you are interested in the topic, click on the links listed at the bottom.) In brief, the phenomenon of the Fibonacci numbers can be summarized in the words of Stan Grist:

The Fibonacci numbers are Nature's numbering system. They appear everywhere in Nature, from the leaf arrangement in plants, to the pattern of the florets of a flower, the bracts of a pinecone, or the scales of a pineapple. The Fibonacci numbers are therefore applicable to the growth of every living thing, including a single cell, a grain of wheat, a hive of bees, and even all of mankind.

        The diagram below shows a golden rectangle (roughly). The large rectangle is divided into a square and a smaller rectangle. In a golden rectangle, the smaller rectangle is the same shape as the larger rectangle; in other words, their sides are proportional. In further words, the two rectangles are similar. This can be used as the definition of a golden rectangle. The proportions give us:  (a + b) / a = a / b = φ, which is the equation for the golden ratio. And then a golden rectangle becomes any rectangle that exhibits this golden ratio. When the longer side of the large rectangle is divided by the longer side of the smaller rectangle, we get φ, and in reverse we get Φ. Similarly, when the shorter side of the large rectangle is divided by the shorter side of the smaller rectangle, we get φ, and in reverse we get Φ.

        The smaller rectangle can also be divided into a similar square and a even smaller rectangle, and the process can keep on going. The sides of all these rectangles as shown in the diagram above are proportional. The relationship between each rectangle and its next rectangle is a golden ratio or proportion. I have come up with a hypothesis based on the golden rectangles as shown in the diagram above. I have separated seven rectangles as shown in the diagram below from the above diagram. Each smaller rectangle is divided from the previous larger rectangle. My hypothesis is that one of the smaller rectangles has a proportion of the largest rectangle close to 5%, which is similar to the proportion between the a side and b side in the Müller-Lyer illusion. 

       

        Now it is the time to have a close look at the proportion between the a side and b side in the diagrams below. As I have said before, the length of the a side is approximately 5% of that of the b side. But I did not get the accurate value for the proportion. For the purpose of confirming the hypothesis, we need to get an accurate value for the proportion between the sides a and b. I think that it is relatively easier to find the exact degree for ∠A than to measure the side a. Then we can calculate the length of the side a by means of the equation: a² = b² +  c² - 2bc Cos A. The value I have got for ∠A is slightly less than 2.5°. Since I do not have the precise measurement, I am going to calculate the proportions between the sides a and b for 2.3°, 2.4°, 2.5° respectively.  First, when ∠C = 45^°, we have the following results: the side a is 5.4321% of the side b when ∠A = 2.3^°; 5.5574% when ∠A = 2.4^°; and 5.6852% when ∠A = 2.5^°. And when ∠C = 135^°, we have the following results: the side a is 5.5422% of the side b when ∠A = 2.3^°; 5.6745% when ∠A = 2.4^°; and 5.8094% when ∠A = 2.5^°.

               

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