Fibonacci Sequence

2202 words (9 pages) Essay

28th Feb 2017 Mathematics Reference this

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How Does the Fibonacci Sequence Relate to Nature and Other Math Processes?

Nature is all around us, and because I spend a lot of time outside I have been able to enjoy and observe all that nature has to offer. Due to the fact that I love science and discovering how everything around me functions and relates to everything else, I decided to investigate the relation that Fibonacci has with other math processes—as well as with the environment. I wanted to understand how plants know the best way to form their seeds or outer shell, and why some patterns may repeat in nature in different plants and organic materials. Thus, this exploration looks at two seemingly unrelated topics—Fibonacci and the golden ratio—both of which produce the same number, phi. While this could be mere coincidence, that possibility is negated when the fact that the number produced is irrational is introduced. It was this peculiar discovery, as well as the abundant appearances of Fibonacci in nature, that led me to choose this exploration topic.

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To begin, I should start by identifying what initially sparked my curiosity in this subject: a pinecone. As with many other plants, as well as fruits and vegetables, pinecones display the golden ratio. In order to better understand what I am talking about I have included a picture of a pinecone similar to the one that I first inspected.

Labeled below is the noticeable spiral pattern on the pinecone. Counting the number of spirals in that direction produces the number eight, and in the other direction it produces the number thirteen while a third and tighter spiral produces twenty-one. These numbers are situational to the pinecone in the pictures, but the Fibonacci numbers as a whole are far more complex than they first appear to be.

To understand the importance of these numbers it is crucial to understand the fundamentals of the Fibonacci sequence itself. The sequence usually begins with the numbers 1, 1, 2, 3, 5, 8, 13 and follows an easily definable pattern.

1, 1, 2, 3, 5, 8, 13

Start with the number 5, or the nth number in the sequence. We’ll call it n. 5 equals the two numbers before it added together: 2 + 3. Or, in broader terms, a number in the sequence is the sum of the two numbers preceding it.

1, 1, 2, 3, 5, 8, 13n = n-1 + n-2

An interesting idea comes up at the mention of this formula though.

=

This ratio just so happens to equal a number often notated as, or phi.

> 1/11Phi is greater than one,

< 2/12but less than two.

> 3/21.5Phi is greater than three halves,

< 5/31.666but less than five thirds.

> 8/51.6Phi is greater than eight fifths,

< 13/81.625but less than thirteen eights.

1.6180339988…

You’ll notice that each fraction listed above is made up of numbers from the original seven number sequence, in other words, each pair of Fibonacci numbers creates a ratio that gets closer and closer to phi as the numbers increase. This is better shown on a graph I created, displayed below.

The ratio created by these sequences as they approach phi is called the golden ratio. The golden ratio, however, is not as important to this study as the lesser known concept of the golden angle. Below is a representation of the golden ratio in relation to the golden angle, the smaller portion of the circle notated using alpha, or α.

α = 137.507764° 137.5°

The reason this conversion is necessary is because the golden angle is present in the next discussion topic: sunflowers. Or, more specifically, their seeds. Sunflowers are another great example of the appearance of Fibonacci in nature, and also led me to an interesting discovery.

In order to plot the distribution of a sunflower’s seeds we need an X and a Y coordinate pair. Using the square roots from an index numbered from one to one thousand and multiplying them by the cosine of the radian of the angle alpha gives us a formula to find x, dependent on the index number used. Y can be calculated with a very similar formula, using sine instead of cosine. The equations are listed in their entirety below.

When these formulas are used and input into Microsoft Excel they produce a graph similar to the following.

Wow! That graph bears a striking resemblance to the original Fibonacci spirals that appeared in the pinecones, and as mentioned earlier it is not mere coincidence.

While the use of the golden ratio is apparent, there is another aspect of it that I wish to address, the golden spiral. Its formulae are given by the following equations, and are readily apparent in nature as well (nautilus shells for example).

In these equations is the undetermined scaling factor and is the growth factor of the spiral. In the instance of the golden spiral, is equal to the operation below.

At first, these formulae appeared to be a strange smattering of numbers, and one I didn’t understand at all. However, upon noticing the appearance of a natural log in the formula for I made a connection to the letter , better known as Euler’s number, that is present in both the X and Y formulae. After thorough searches of many sources I discovered another math process that bares resemblance to the above formulae.

This is Euler’s formula. It becomes increasingly apparent that its resemblance is not coincidental when the formula is transformed into the final formula shown below.

While the visual similarities may be obvious when the formula is displayed as it is above, the importance of each variable can be clarified with simple explanations. is the arbitrary scaling factor, responsible for determining the scale of the spiral. dictates the rotation of the spiral, and remains constant. The in dictates the growth of the spiral, and the dictates the speed—together representing the speed of the growth of the spiral. More simply put, any given ordered pair can be found by multiplying the growth of the spiral by its rotation (as shown in the originally given formulae for finding said coordinates.)

What is produced, however, after inputting over two thousand pieces of data, derived from the coordinates calculated using the formulae above, into Microsoft Excel, is shown in the graph below.

After putting in the Fibonacci squares (using the original golden ratio) into the spiral its appearance and relation to Fibonacci become even clearer.

Very simply put, my investigation yielded the result that the Fibonacci sequence, the golden spiral, and Euler’s number are all related to one another in nature. The results are eye opening for me, as I am beginning to realize just how much of the world is made up of math—rather than my previous belief that everything natural occurred randomly. My exploration only stemmed into plants, and while that may only have practical use in fields such as botany, all three have great value in many fields. To begin with, Fibonacci appears in bee populations, proportions of the human body, formation of cells, and possibly more practically in code and the stock market. Any of these fields could present an interesting extension to my exploration, and because they all stem from Fibonacci they all have roots in combinatorics and number theory. The implications of this are staggering! Simply the thought that all of these vastly different fields are related to one another by one sequence of numbers discovered by Leonardo of Pisa, better known as Fibonacci himself, is baffling considering that he discovered them while looking at the breeding patterns of rabbits. There are so many other areas in nature that Fibonacci appears in, and I’m so excited that I have the opportunity to discover and study them now that I know more about them.

Works Cited

Azad, Kalid. “Intuitive Understanding Of Euler’s Formula.” Better Explained. N.p., n.d. Web. 23 Feb. 2015. <http://betterexplained.com/articles/intuitive-understanding-of-eulers-formula/>.

“Nature by Numbers.” Eterea. N.p., n.d. Web. 3 Feb. 2015. <http://www.etereaestudios.com/docs_html/nbyn_htm/about_index.htm>.

“Spirals.” < http://faculty.smcm.edu/sgoldstine/pinecone/spirals2.gif>

Wolverson, Tim. “Plot a Fibonacci Spiral in Excel.” Reviews and Guides. WordPress, 08 Feb. 2014. Web. Jan.-Feb. 2015. <https://timwolverson.wordpress.com/2014/02/08/plot-a-fibonacci-spiral-in-excel/>.

McDonald 1

How Does the Fibonacci Sequence Relate to Nature and Other Math Processes?

Nature is all around us, and because I spend a lot of time outside I have been able to enjoy and observe all that nature has to offer. Due to the fact that I love science and discovering how everything around me functions and relates to everything else, I decided to investigate the relation that Fibonacci has with other math processes—as well as with the environment. I wanted to understand how plants know the best way to form their seeds or outer shell, and why some patterns may repeat in nature in different plants and organic materials. Thus, this exploration looks at two seemingly unrelated topics—Fibonacci and the golden ratio—both of which produce the same number, phi. While this could be mere coincidence, that possibility is negated when the fact that the number produced is irrational is introduced. It was this peculiar discovery, as well as the abundant appearances of Fibonacci in nature, that led me to choose this exploration topic.

To begin, I should start by identifying what initially sparked my curiosity in this subject: a pinecone. As with many other plants, as well as fruits and vegetables, pinecones display the golden ratio. In order to better understand what I am talking about I have included a picture of a pinecone similar to the one that I first inspected.

Labeled below is the noticeable spiral pattern on the pinecone. Counting the number of spirals in that direction produces the number eight, and in the other direction it produces the number thirteen while a third and tighter spiral produces twenty-one. These numbers are situational to the pinecone in the pictures, but the Fibonacci numbers as a whole are far more complex than they first appear to be.

To understand the importance of these numbers it is crucial to understand the fundamentals of the Fibonacci sequence itself. The sequence usually begins with the numbers 1, 1, 2, 3, 5, 8, 13 and follows an easily definable pattern.

1, 1, 2, 3, 5, 8, 13

Start with the number 5, or the nth number in the sequence. We’ll call it n. 5 equals the two numbers before it added together: 2 + 3. Or, in broader terms, a number in the sequence is the sum of the two numbers preceding it.

1, 1, 2, 3, 5, 8, 13n = n-1 + n-2

An interesting idea comes up at the mention of this formula though.

=

This ratio just so happens to equal a number often notated as, or phi.

> 1/11Phi is greater than one,

< 2/12but less than two.

> 3/21.5Phi is greater than three halves,

< 5/31.666but less than five thirds.

> 8/51.6Phi is greater than eight fifths,

< 13/81.625but less than thirteen eights.

1.6180339988…

You’ll notice that each fraction listed above is made up of numbers from the original seven number sequence, in other words, each pair of Fibonacci numbers creates a ratio that gets closer and closer to phi as the numbers increase. This is better shown on a graph I created, displayed below.

The ratio created by these sequences as they approach phi is called the golden ratio. The golden ratio, however, is not as important to this study as the lesser known concept of the golden angle. Below is a representation of the golden ratio in relation to the golden angle, the smaller portion of the circle notated using alpha, or α.

α = 137.507764° 137.5°

The reason this conversion is necessary is because the golden angle is present in the next discussion topic: sunflowers. Or, more specifically, their seeds. Sunflowers are another great example of the appearance of Fibonacci in nature, and also led me to an interesting discovery.

In order to plot the distribution of a sunflower’s seeds we need an X and a Y coordinate pair. Using the square roots from an index numbered from one to one thousand and multiplying them by the cosine of the radian of the angle alpha gives us a formula to find x, dependent on the index number used. Y can be calculated with a very similar formula, using sine instead of cosine. The equations are listed in their entirety below.

When these formulas are used and input into Microsoft Excel they produce a graph similar to the following.

Wow! That graph bears a striking resemblance to the original Fibonacci spirals that appeared in the pinecones, and as mentioned earlier it is not mere coincidence.

While the use of the golden ratio is apparent, there is another aspect of it that I wish to address, the golden spiral. Its formulae are given by the following equations, and are readily apparent in nature as well (nautilus shells for example).

In these equations is the undetermined scaling factor and is the growth factor of the spiral. In the instance of the golden spiral, is equal to the operation below.

At first, these formulae appeared to be a strange smattering of numbers, and one I didn’t understand at all. However, upon noticing the appearance of a natural log in the formula for I made a connection to the letter , better known as Euler’s number, that is present in both the X and Y formulae. After thorough searches of many sources I discovered another math process that bares resemblance to the above formulae.

This is Euler’s formula. It becomes increasingly apparent that its resemblance is not coincidental when the formula is transformed into the final formula shown below.

While the visual similarities may be obvious when the formula is displayed as it is above, the importance of each variable can be clarified with simple explanations. is the arbitrary scaling factor, responsible for determining the scale of the spiral. dictates the rotation of the spiral, and remains constant. The in dictates the growth of the spiral, and the dictates the speed—together representing the speed of the growth of the spiral. More simply put, any given ordered pair can be found by multiplying the growth of the spiral by its rotation (as shown in the originally given formulae for finding said coordinates.)

What is produced, however, after inputting over two thousand pieces of data, derived from the coordinates calculated using the formulae above, into Microsoft Excel, is shown in the graph below.

After putting in the Fibonacci squares (using the original golden ratio) into the spiral its appearance and relation to Fibonacci become even clearer.

Very simply put, my investigation yielded the result that the Fibonacci sequence, the golden spiral, and Euler’s number are all related to one another in nature. The results are eye opening for me, as I am beginning to realize just how much of the world is made up of math—rather than my previous belief that everything natural occurred randomly. My exploration only stemmed into plants, and while that may only have practical use in fields such as botany, all three have great value in many fields. To begin with, Fibonacci appears in bee populations, proportions of the human body, formation of cells, and possibly more practically in code and the stock market. Any of these fields could present an interesting extension to my exploration, and because they all stem from Fibonacci they all have roots in combinatorics and number theory. The implications of this are staggering! Simply the thought that all of these vastly different fields are related to one another by one sequence of numbers discovered by Leonardo of Pisa, better known as Fibonacci himself, is baffling considering that he discovered them while looking at the breeding patterns of rabbits. There are so many other areas in nature that Fibonacci appears in, and I’m so excited that I have the opportunity to discover and study them now that I know more about them.

Works Cited

Azad, Kalid. “Intuitive Understanding Of Euler’s Formula.” Better Explained. N.p., n.d. Web. 23 Feb. 2015. <http://betterexplained.com/articles/intuitive-understanding-of-eulers-formula/>.

“Nature by Numbers.” Eterea. N.p., n.d. Web. 3 Feb. 2015. <http://www.etereaestudios.com/docs_html/nbyn_htm/about_index.htm>.

“Spirals.” < http://faculty.smcm.edu/sgoldstine/pinecone/spirals2.gif>

Wolverson, Tim. “Plot a Fibonacci Spiral in Excel.” Reviews and Guides. WordPress, 08 Feb. 2014. Web. Jan.-Feb. 2015. <https://timwolverson.wordpress.com/2014/02/08/plot-a-fibonacci-spiral-in-excel/>.

McDonald 1

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