Fibonacci And Nature Essay, Research Paper

The math project topic Eddy and I have chosen is the Fibonacci Sequence and it’s relation to nature. The Sequence is very popular and involves many aspects of life including animals, plants and other educational purposes. The topic is extremely interesting and will change the way students look at everyday things by considering Fibonacci and his famous numbering system.

The Fibonacci Sequence is a series of numbers first created in 1202 by Leonardo Fibonacci. It is a relatively simple series, but it’s ramifications and applications are practically limitless. It has fascinated mathematicians for over 700 years, and nearly everyone who has worked with it has added a new tidbit of information to the Fibonacci puzzle. The mathematics of the Sequence is a constantly expanding branch of number theory, with more and more people being drawn into the complex subleties of Fibonacci’s legacy.

The Sequence works by taking the last two numbers in the sequence and adding them to form the next number in the sequence. Thus, if we start with “0″ and “1″ and add them, we find the third Fibonacci number, which is 1(i.e., 0 + 1 = 1). Each successive number is found in the exact same manner. Therefore, the fourth number would be 2(i.e., 1 + 1 = 2) and the fifth number would be 3(i.e., 1 + 2 + 3). The Sequence will then continue in this manner…

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89……

In the year 1202, Fibonacci became interested in the reproduction of rabbits. He created an imaginary set of ideal conditions under which rabbits could breed, and posed the question, “How many pairs of rabbits will there be a year from now?”. The ideal set of conditions was as follows:

1. You begin with one male and one female rabbit. These rabbits have just been born.

2. A rabbit will reach sexual maturity after one month.

3. The gestation period of a rabbit is one month.

4. Once it has reached sexual maturity, a female rabbit will give birth every month.

5. A female rabbit will always give birth to one male rabbit and one female rabbit.

6. Rabbits never die.

So, how many male/female rabbit pairs are there after one year?

Month 0 – At the beginning of the experiment, there is one pair of rabbits (condition #1)

Month 1 – After one month, the two rabbits have mated but have not yet given birth. Therefore, there is still only one pair of rabbits.

Month 2 – After two months, the first pair of rabbits gives birth to another pair, makin two pair in all.

Month 3 – After three months, the original pair gives birth again, and the second pair mate, but do not give birth. This makes three pair.

Month 4 – After four months, the original pair give birth, and the pair born in month 2 give birth. The pair born in month 3 mate, but do not give birth. The is make two new pair, for a total of five pair.

Month 5 – After five months, every pair that was alive two months ago gives birth. This makes three new pair, for a total of eight pair.

This birth rate continues for each month, equalling the Fibonacci Sequence.

The series of numbers also relates to the formation of our fingers. Each single person has 2 hands. Each of those hands have 5 fingers. Those fingers are divided into three different sections, separated by 3 different knuckles. Each of these numbers are Fibonacci’s, and if we dug down deeper into the biology of fingers, there would be quite a few more Fibonacci numbers.

Other than animals, the Fibonacci Sequence also greatly affects plant life. The formation of leaves, seeds, and limbs usually involves the Fibonacci numbers. On many plants, the number of petals is a Fibonacci number.

3 petals: lily, iris

5 petals: buttercup, wild rose, larkspur, columbine (aquilegia)

8 petals: delphiniums

13 petals: ragwort, corn marigold, cineraria,

21 petals: aster, black-eyed susan, chicory

34 petals: plantain, pyrethrum

55, 89 petals: michaelmas daisies, the asteraceae family

The most fascinating appearances of the Fibonacci Series in nature are the spirals that can be seen in everything from sunflowers to pine cones to pineapples. This will explain that this phenomenon comes not from perfection through evolution (which is, in itself, an oxymoron!), but from the dynamics of plant growth. To understand how these spirals work out, you must go back to the beginning, to where flowers and fruits and seeds start: the apex. The apex is the tip of the shoot of a growing plant. It is the bud on the end of a stem on a tree and the bulb of a flower before it blooms. Around the apex grow little bumps called primordia. As more primordia develop, they are pushed farther and farther from the apex and they develop into the familiar features of a plant, be it a leaf, a flower, or parts of a fruit. Let us consider a sunflower with primordia growing from the center. The first primordia to develop end up being farther from the apex than later primordia.

Therefore, it can be deduced from this in what order the primordia appeared.

As it happens, if one took the first and second primordia and measured the angle between them with the center of the seed head as the vertex, the angle would be very close to 137.5 degrees. That angle is very important in describing how primordia form the spirals we see. It is, in fact, known as the

golden angle. Here’s where the Fibonacci Series comes in. Take two consecutive Fibonacci Numbers and divide the smaller by the larger. Then multiply by 360 degrees. If you try 55/89 * 360 = 222.472… . You can round

that degree measure to 222.5 degrees. Remembering from trigonometry that angles can be measured internally or externally, so if you subtract it from 360 degrees to convert it, you get 137.5 degrees, the golden angle. In other

words, 360(1-) = 137.5… . The mathematics of the famous Sequence makes sense.

Fibonacci’s Sequence is best described with diagrams, of which there are many. The rectangles, spirals, and birth diagrams make it much easier to understand how the Sequence works. There are also a great number of puzzles and games relating to the Sequence that show how coincidental and precise the amazing math works out.

The Sequence has been described as the “magical intersection” between the fields of math and beauty, and that’s exactly what it is. Fibonacci proved that everything has a reason for it’s formation and/or existance in nature and helped people understand why things are the way they are. The Sequence, being the pattern of life itself, is one of the most fascinating subjects in the world today and will never be forgotten, thanks to Fibonacci’s mathematics.