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The Fibonacci Series: When Math Turns Golden

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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.

The Greeks used the 1.618 proportion to construct The Golden Rectangle. It was a rectangle with sides measuring one and 1.618 (or with side measuring to consecutive Fibonacci Numbers). This was considered the most mathematically beautiful structure, and frequently used in architecture. The Parthenon incorporates a number of Golden Rectangles into its structure and decoration. What's more, the Pyramids have their own Divine Proportions. If the base of the Pyramids is considered one unit, the sloping sides are 1.618 units, and the height is the square root of 1.618 units high.


Today, many photographs and paintings use golden proportions. Take a Golden Rectangle, or a rectangle in which the two sides are Fibonacci Numbers. That rectangle can be chopped up into smaller rectangles and squares that all also have Fibonacci proportions. Many works of art contain objects that fit within these proportions.


But what about curves? That's where the Fibonacci sequence really shines. Draw an arc from one corner of those nested squares to the opposite corner. Do it enough, with increasingly nested squares, and it makes a Golden Spiral. The spiral is used in art, but it's seen even more often outside of a gallery.

Fibonacci Spirals in Nature:

So far, the Fibonacci Series has been popping up solely as a result of humans going crazy for a certain series of numbers. Although the original problem was illustrated with rabbits, anyone knows that population doesn't actually expand that way. Rabbits don't always get born in pairs, and even though they're famous for their fertility, they don't conceive every time they try.


In fact, the best examples of real world Fibonacci Series are found in the plant kingdom. Many plants that branch outwards towards the sun do so in branches equal to Fibonacci numbers. The original sprig comes up from the earth. For the first period of time, it just sprouts upwards. Then it develops meristem points - points from which new branches can form - and those sprout into two separate branches. Those branches push upwards for another period of time, and then develop two points of their own. The overall number of sprouting points develops in a Fibonacci Series.

The most celebrated example of the Fibonacci Series is the spirals it creates. Florets on a cauliflowers, fruitlets on a pineapple, seeds on a sunflower, they all spiral outwards. And each of those spirals contains a number of seeds, florets, bumps, leaves or tubercles that are equal to a Fibonacci number. Some might say that humans pick and choose, deciding to ignore the flowers that do not spin out their seeds in a Fibonacci series, but it turns out there's a reason for the repeated Fibonacci numbers on different species of spiraling plant - it's the perfect way to pack.


As a sunflower bulb develops seeds, it has to give each of its potential offspring equal space to do flourish. They need to be packed as evenly and equally as possible. But spirals aren't the best way to fit seeds into a space, so why do plants do it? Because they don't build a space and then pack it full of seeds like humans do a warehouse. They make seeds as the bulb that those seeds mature in expands.

Plants make these spiralling seedpods by maturing seeds at the center and the stretching the space the seeds inhabit outwards. If they deposited the seeds one directly beneath another, the seeds would be squished on top and bottom and have space to the sides - the flower pod would become an elongated seed holder and would strain its stem. And so the flower matures seeds in a circular pattern, setting the growing seeds at an angle to each other and letting them expand outwards. But what's the most efficient way to do that? If the flower made four seeds for every completely circular 'turn', the fourth seed would be deposited right underneath the first seed - making four rows of seeds that push outwards. That's better than a line, but still not an efficient use of space. It turns out that the best number of seeds to deposit per turn is around 1.618. Since flowers can only make whole numbers of seeds, this means that no seed is lodged directly under its predecessor. Instead, the seeds spiral outwards from the center.


Since these spirals have the Divine Proportion of 1.618 seeds per turn, counting the seeds any spiral will result in getting a Fibonacci Number. The Fibonacci Series is also seen in plants that put out leaves from one stalk. As the plant grows upwards, the number of leaves per completed by the end of each 'turn' around the stem will be a Fibonacci number.


Fibonacci For Fun:

When math people get ahold of something they never let go. The Fibonacci Series has a whole lot of strange, and interesting, quirks. For example, the sum of any 10 consecutive numbers in the Fibonacci Series is divisible by 11. The square of a Fibonacci Number, minus the square the Fibonacci Number two terms before it, will yield another Fibonacci Number. Piano keys in an octave are made up of Fibonaccie Numbers; eight white, five black, and thirteen in all.


If you are subscription fee curious about the many Fibonaccis silently lurking around the world, The Fibonacci Quarterly is a paper that is devoted entirely to the Fibonacci Series. Don't worry about them running out of material. The editor says they have a backlog of waiting articles. The series is everywhere.

Via Google, Scientific American Book Club, Time, Math Is Fun, World Mysteries, Maths Surrey,Again, and Again, and Again.