# The Fibonacci Series

We are sure the wonders of the Fibonacci series will not be news to most of you, but this wonerdful video might. Please take a look and delight (if the video doesn’t play in your browser just click the link ‘Watch on YouTube’) – Deskarati

The Fibonacci Series, a set of numbers that increases rapidly, began as a medieval math joke about how fast rabbits breed. But it’s became a source of insight into art, architecture, nature, and efficiency. This mathematical game explains the structures of leaves and lungs, is replicated in paintings and photographs, and pops up as the basis for the pyramids, the Parthenon, and packing efficiency. Find out where the Fibonacci Sequence comes from and why it keeps eerily showing up.

The Origin of the Series:

The Fibonacci Series gets its name from Leonardo Fibonacci, who lived in the twelfth century. He wanted to calculate the ideal expansion of pairs of rabbits over a year. He assumed that each pair would produce another pair as soon as they matured at one month. In January, a new pair of rabbits would be born (1) they would reach maturity in a February (1) and breed, producing a new pair in March (2). They would then breed again, and produce a new pair in April (3), and another pair in May. Meanwhile, they rabbits born in March would reach maturity in April so in May would see two new pairs of bunnies produced, bringing it to a total of 5 pairs. Now the rabbits born in January, March, and April would all be adding new pairs, bringing June’s total to 8 pairs..

The expansion would carry forward, with each new pair coming to maturity and starting their own little Fibonacci Series to be added to the whole. Over the months, with no deaths, the rabbit pair expansion would look like this:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, . . . .

Anyone can see that by December the poor owner would be inundated with rabbits. Sharp-eyed readers can also see that each new number in the sequence is the combination of the two numbers before it. Five plus eight makes thirteen. Eight plus thirteen makes twenty-one, and so on.

Fibonacci Goes Gold in Art and Architecture:

Many would respond to this with a shrug and a mental note to not let Fibonacci near any of their rabbits. It turns out, though, that he was really on to something. Mathematicians and artists took this sequence of number and coated it in gold. The first step was taking each number in the series and dividing it by the previous number. At first the results don’t look special. One divided by one is one. Two divided by one is two. Three divided by two is 1.5. Riveting stuff. But as the sequence increases something strange begins to happen. Five divided by three is 1.666. Eight divided by five is 1.6. Thirteen divided by eight is 1.625. Twenty-one divided by thirteen is 1.615.

As the series goes on, the ratio of the latest number to the last number zeroes in on 1.618. It approaches 1.618, getting increasingly accurate, but never quite reaching that ratio. This was called The Golden Mean, or The Divine Proportion, and it seems to be everywhere in art and architecture.

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### 3 Responses to The Fibonacci Series

1. alfy says:

THE GOLDEN MEAN OR THE GOLDEN SECTION

There are more fundamental methods of producing the incommensurable number r, the “golden ratio” or “golden section” generated by the Fibonacci series.

In one dimension, imagine a line divided into two parts, a smaller part, length a, and a longer part, length b. In two dimensions, imagine a rectangular sheet of paper, the short side, length a, and the long side, length b.

The line is “divided in the golden section” if the ratio of the longer to the shorter equals the ratio of the longer to the whole.
In the case of the rectangle the sides are “in the golden section” if the ratio of the longer to the shorter equals the ratio of the longer to the sum of both.

Algebraically, b/a = (a+b)/b, and cross multiplying b2 = a2 +ab

This is a quadratic a2 +ab-b2 = 0, and as we already know it to be incommensurable we can apply the formula for solving the roots of a quadratic, namely:

y = -B +- root (B2 -4AC) where in this case A = 1, B = 1 and C = -1
2A

Hence, y = (-1+root(1+4))/2 or (-1-root(1+4))/2 = (root 5-1)/2 or (-root 5-1)/2 = either

0.618033988749895………or -1.618033988749895……………….As the negative value has no meaning as a ratio we stick to the positive value for the “golden section” that is the smaller over the larger. Of course, the larger over the smaller is the positive value 1.618033988749895.

I have always been fascinated by the incredibly widespread occurrence of the golden section throughout the natural world not only of plants and animals but of inanimate matter as well. However, I understand from mathematicians that it is not seen as very interesting at all mathematically, which is rather disappointing because it does appear to be at the very heart of things.