The Golden Ratio

It is a numerical ratio found in Nature, Music, Art, Biology and many more aspects of life. Plants arrange their structure according to this number. Art is said to be more pleasing to the eye when this ratio is used in its dimensions, Music is considered beautiful when composed with notes arranged to this ratio. The list is endless...

It is incredible that one single number occurs throughout our lives daily - and almost all of us are unaware of it.

How can we mathematically derive it?

It's simple. Consider a sequence of numbers starting with 1. Add 1 to itself to get 2. Then, add 1 to 2 to get 3, and continue adding the latest two numbers of the sequence to get the next number. Your sequence will then look like this: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, . . .

This sequence is called the Fibonacci series (after its discoverer). The Golden ratio is simply any term in the sequence divided by the one before it. I.e. 2/1 = 2.0 3/2 = 1.5 5/3 = 1.67 8/5 = 1.6 13/8 = 1.625 21/13 = 1.615 34/21 = 1.619 55/34 = 1.618 89/55 = 1.618

As we travel through the sequence, and our numbers get higher, notice that the ratio converges to a single value. This is the Golden Ratio - it is an irrational - a number with an infinite number of decimal places - and it never repeats itself. The ratio is generally expressed as 1.618.

Where can we find it in life? . The first example is flowers. There are two basic ways that a flower will grow: flowers will either always have the exact same number of petals as other flowers in their species, or they will have a random number. As far as the first type of flower is concerned, many of those flowers with exact numbers of petals have a Fibonacci number of petals. In the other type, the average number of petals is often a Fibonacci number. . Fibonacci numbers also appear in pinecones. If you look at the bottom of one, you can see swirls. If you count the number of little scales in one swirl, it`ll be a Fibonacci number. The same pattern is found in the centre of sunflowers. . In the 1930`s, New York`s Pratt Institute laid out rectangular frames of different proportions, and asked several hundred art students to choose which they found most pleasing. The winner? The one with Golden Ratio proportions. . Consider this well known question:"How many pairs of rabbits will be produced in a year, beginning with a single pair, if in every month each pair bears a new pair which becomes productive from the second month on?"

In the first month, there is one pair of rabbits. In the second month, there is still only one pair of rabbits, which reach maturity. In the third month, there are two pairs of rabbits. In the fourth month, there are five pairs. This creates a sequence: 1, 1, 2, 3, 5, 8, 13, 21... Looks familiar doesn't it?