Grade 12 Applied Mathematics Project

The Fibonacci Sequence

Index

Abstract
Introduction
The History
The Rabbit Problem
Maths and Fibonacci
Presence in Nature
Conclusion
Bibliography

Introduction

The Fibonacci Sequence is a series of numbers in which each number is the sum of the two that precede it. Starting at 0 and 1, the sequence looks like this: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so on forever. The Fibonacci sequence can be described using a mathematical equation: Xn+2 = Xn+1 + Xn.

The sequence can theoretically continue to infinity, using the same formula for each new number. Some resources show the Fibonacci sequence starting with a one instead of a zero, but this is fairly uncommon.

The Fibonacci sequence exhibits a certain numerical pattern which originated as the answer to an exercise in the first ever high school algebra text. This pattern turned out to have an interest and importance far beyond what its creator imagined. It can be used to model or describe an amazing variety of phenomena, in mathematics and science, art and nature. The mathematical ideas the Fibonacci sequence leads to, such as the golden ratio, spirals and self- similar curves, have long been appreciated for their charm and beauty, but no one can really explain why they are echoed so clearly in the world of art and nature.

The History

The story began in Pisa, Italy in the year 1202. Leonardo Pisano Bigollo was a young man in his twenties, a member of an important trading family of Pisa. In his travels throughout the Middle East, he was captivated by the mathematical ideas that had come west from India through the Arabic countries.

Ancient Sanskrit texts that used the Hindu-Arabic numeral system first mention it in 200 B.C. predating Leonardo of Pisa by centuries.

When he returned to Pisa he published these ideas in a book on mathematics called Liber Abaci, which became a landmark in Europe. Leonardo, who has since come to be known as Fibonacci, became the most celebrated mathematician of the Middle Ages. His book was a discourse on mathematical methods in commerce, but is now remembered mainly for two contributions, one obviously important at the time and one seemingly insignificant.

The important one: he brought to the attention of Europe the Hindu system for writing numbers. European tradesmen and scholars were still clinging to the use of the old Roman numerals; modern mathematics would have been impossible without this change to the Hindu system, which we call now Arabic notation, since it came west through Arabic lands.

The other: hidden away in a list of brain-teasers , Fibonacci posed the following question:

If a pair of rabbits is placed in an enclosed area, how many rabbits will be born there if we assume that every month a pair of rabbits produces another pair, and that rabbits begin to bear young two months after their birth?

This apparently innocent little question has as an answer a certain sequence of numbers, known now as the Fibonacci sequence, which has turned out to be one of the most interesting ever written down. It has been rediscovered in an astonishing variety of forms, in branches of mathematics way beyond simple arithmetic. Its method of development has led to far-reaching applications in mathematics and computer science.

But even more fascinating is the surprising appearance of Fibonacci numbers, and their relative ratios, in arenas far removed from the logical structure of mathematics: in Nature and in Art, in classical theories of beauty and proportion.

"Liber Abaci" first introduced the sequence to the Western world. But after a few scant paragraphs on breeding rabbits, Leonardo of Pisa never mentioned the sequence again. In fact, it was mostly forgotten until the 19th century, when mathematicians worked out more about the sequence's mathematical properties. In 1877, French mathematician Édouard Lucas officially named the rabbit problem "the Fibonacci sequence," .

The Rabbit Problem

As stated previously, Fibonacci presented the given problem:

“If a pair of rabbits is placed in an enclosed area, how many rabbits will be born there if we assume that every month a pair of rabbits produces another pair, and that rabbits begin to bear young two months after their birth?”

It is solved as follows:

Start: At the start no rabbits are born, as the initial pair has not had time to be pregnant and born (0).
The first month: One pair of rabbits are born (1).
The second month: Again, one pair of rabbits are born as the new rabbits have not yet matured to bear young (1).
The third month: Two pairs of rabbits reproduce, and one pair is not ready, so two pairs of rabbits are born (2).
The fourth month: Three pairs of rabbits reproduce and 2 pairs of rabbits are not ready, so three pairs of rabbits are born (3).
The fifth month: Five pairs of rabbits produce and three are not ready, so five pairs of rabbits are born (5).
And so on.

The pattern we see here is that each cohort or generation remains as part of the next, and in addition, each grown-up pair contributes a baby pair. The number of such baby pairs matches the total number of pairs in the previous generation. Symbolically

fn = number of pairs during month n
fn = fn-1 + fn-2

So we have a recursive formula where each generation is defined in terms of the previous two generations. Using this approach, we can successively calculate fn for as many generations as we like.

So this sequence of numbers 1,1,2,3,5,8,13,21,... and the recursive way of constructing it ad infinitum, is the solution to the Fibonacci puzzle. But what Fibonacci could not have foreseen was the myriad of applications that these numbers and this method would eventually have. His idea was more fertile than his rabbits. Just in terms of pure mathematics - number theory, geometry and so on - the scope of his idea was so great that an entire professional journal has been devoted to it - the Fibonacci Quarterly.

Maths and Fibonacci

A closer inspection of the numbers making up the Fibonacci sequence brings to light all sorts of fascinating patterns and mathematical properties. Fibonacci himself makes no mention of these patterns in his book, but the following patterns are a few that have been brought to light over years of examination of the numbers in the sequence.

Any two consecutive Fibonacci numbers are relatively prime, having no factors in common with each other
Summing together any ten consecutive Fibonacci numbers will always result in a number which is divisible by eleven
Every third Fibonacci number is divisible by two, or F3. Every fourth Fibonacci number is divisible by three, or F4. Every fifth Fibonacci number is divisible by five, or F5. Every sixth Fibonacci number is divisible by eight, or F6, and the pattern continues. In general, every nth Fibonacci number is divisible by the nth number in the Fibonacci sequence, or Fmn is divisible by Fn.
Fibonacci numbers in composite-number positions are always composite numbers, with the exception of the fourth Fibonacci number. In other words if n is not a prime, the nth Fibonacci number will not be a prime.
Multiplying any Fibonacci number by two and subtracting the next number in the sequence will result in the answer being the number two places before the original.
Summing consecutive odd-positioned Fibonacci numbers, starting with the first odd-positioned number, F1, will result in a number that is the next Fibonacci number in the sequence after the last term in the sum.
A similar pattern emerges when summing consecutive, even-positioned Fibonacci numbers beginning with F2, only this time, the result is a number that is one less than the Fibonacci number following the last even number in the sum.
The product of any Fibonacci number multiplied by the number two places after it will be one more or one less than the square of the Fibonacci number between the two. When the number to be squared is an even-positioned Fibonacci number, one is added, and when it is odd-positioned, on is subtracted.
When the square of a Fibonacci number is subtracted from the square of the number two places after it, the result is a Fibonacci number.
When the squares of two consecutive Fibonacci numbers are added, the sum is also a Fibonacci number.
The sum of squares of Fibonacci numbers is simply the product of the last squared number in the sum and the number that would come after it in the Fibonacci sequence.
For any three consecutive Fibonacci numbers, subtracting the cube of the smallest from the sum of the cubes of the two greater will result in another Fibonacci number.
The sum of any number of Fibonacci numbers.

Presence in Nature

Fibonacci can be observed everywhere – from architecture and art to the stock market. But one of the most prominent applications of it are seen in nature, all around us. Some instances of this are:

1. Flower petals

The number of petals in a flower consistently follows the Fibonacci sequence. Famous examples include the lily, which has three petals, buttercups, which have five (pictured at left), the chicory's 21, the daisy's 34, and so on. Phi appears in petals on account of the ideal packing arrangement as selected by Darwinian processes; each petal is placed at 0.618034 per turn (out of a 360° circle) allowing for the best possible exposure to sunlight and other factors.

2. Seed heads

The head of a flower is also subject to Fibonaccian processes. Typically, seeds are produced at the center, and then migrate towards the outside to fill all the space. Sunflowers provide a great example of these spiralling patterns.

In some cases, the seed heads are so tightly packed that total number can get quite high — as many as 144 or more. And when counting these spirals, the total tends to match a Fibonacci number.

3. Pinecones

Similarly, the seed pods on a pinecone are arranged in a spiral pattern. Each cone consists of a pair of spirals, each one spiralling upwards in opposing directions. The number of steps will almost always match a pair of consecutive Fibonacci numbers. For example, a 3-5 cone is a cone which meets at the back after three steps along the left spiral, and five steps along the right.

4. Fruits and Vegetables

A pineapple is covered in hexagonally shaped scales, known as bracts. These bracts form spirals in three different directions, each passing through opposing sides of the hexagon. Five spirals rise gradually in one direction, eight spirals rise at a medium rate in a second direction, and thirteen spirals rise steeply in the third direction, giving three consecutive Fibonacci numbers for the three different sets.

5. Tree branches

The Fibonacci sequence can also be seen in the way tree branches form or split. A main trunk will grow until it produces a branch, which creates two growth points. Then, one of the new stems branches into two, while the other one lies dormant. This pattern of branching is repeated for each of the new stems. A good example is the sneezewort. Root systems and even algae exhibit this pattern.

6. Shells

The unique properties of the Golden Rectangle provides another example. This shape, a rectangle in which the ratio of the sides a/b is equal to the golden mean (phi), can result in a nesting process that can be repeated into infinity — and which takes on the form of a spiral. It's call the logarithmic spiral, and it abounds in nature.

Snail shells and nautilus shells follow the logarithmic spiral, as does the cochlea of the inner ear. It can also be seen in the horns of certain goats and the shape of the spider’s webs.

7. Spiral Galaxies

Not surprisingly, spiral galaxies also follow the familiar Fibonacci pattern. The Milky Way has several spiral arms, each of them a logarithmic spiral of about 12 degrees. As an interesting aside, spiral galaxies appear to defy Newtonian physics. As early as 1925, astronomers realized that, since the angular speed of rotation of the galactic disk varies with distance from the center, the radial arms should become curved as galaxies rotate. Subsequently, after a few rotations, spiral arms should start to wind around a galaxy. But they don't — hence the so-called winding problem. The stars on the outside, it would seem, move at a velocity higher than expected — a unique trait of the cosmos that helps preserve its shape.

8. Hurricanes

Hurricanes follow the Fibonacci spiral, where each square’s side is a Fibonacci number and connecting the opposite vertices creates a spiral.

9. Faces

Faces, both human and nonhuman, abound with examples of the Golden Ratio. The mouth and nose are each positioned at golden sections of the distance between the eyes and the bottom of the chin. Similar proportions can been seen from the side, and even the eye and ear itself (which follows along a spiral).

It's worth noting that every person's body is different, but that averages across populations tend towards phi. It has also been said that the more closely our proportions adhere to phi, the more "attractive" those traits are perceived. As an example, the most "beautiful" smiles are those in which central incisors are 1.618 wider than the lateral incisors, which are 1.618 wider than canines, and so on. It's quite possible that, from an evo-psych perspective, that we are primed to like physical forms that adhere to the golden ratio — a potential indicator of reproductive fitness and health.

10. Animal Bodies

Even our bodies exhibit proportions that are consistent with Fibonacci numbers. For example, the measurement from the navel to the floor and the top of the head to the navel is the golden ratio. Animal bodies exhibit similar tendencies, including dolphins (the eye, fins and tail all fall at Golden Sections), starfish, sand dollars, sea urchins, ants, and honey bees.

11. Reproductive Dynamics

Fibonacci sequence is observed in not only rabbits, but honeybees too. They follow Fibonacci in other interesting ways. The most profound example is by dividing the number of females in a colony by the number of males (females always outnumber males). The answer is typically something very close to 1.618. In addition, the family tree of honey bees also follows the familiar pattern. Males have one parent (a female), whereas females have two (a female and male). Thus, when it comes to the family tree, males have 2, 3, 5, and 8 grandparents, great-grandparents, gr-gr-grandparents, and gr-gr-gr-grandparents respectively. Following the same pattern, females have 2, 3, 5, 8, 13, and so on. And as noted, bee physiology also follows along the Golden Curve rather nicely.

12. DNA Molecules

Even the microscopic realm is not immune to Fibonacci. The DNA molecule measures 34 angstroms long by 21 angstroms wide for each full cycle of its double helix spiral. These numbers, 34 and 21, are numbers in the Fibonacci series, and their ratio 1.6190476 closely approximates Phi, 1.6180339.

Conclusion

The Fibonacci sequence is truly a fascinating sequence. With its colluded history which originally dates back to India, it has been present throughout major parts of our lives.

It also has several mathematical properties and characteristics, which make it all the more interesting to learn about.

Though the applications explored here are vast, this is just skimming the surface. Fibonacci can be used in one way or the other in almost any field.

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