IB Math Internal Assessment
What is the relationship between the Fibonacci Sequence and the Golden Ratio and what are the real world applications of the two concepts?
Bhavna L. Singh
Candidate Code: 001477-0174
Capuchino High School
IB Math HL 2
Session: May 2018
Word Count:2322
Introduction:
The topic of research is the Fibonacci sequence and the golden ratio, specifically it's appearance in various subjects ranging from biology to art to architecture. The Fibonacci sequence is referenced as 'Fibonacci' because it is the common name of the Italian mathematician, Leonardo of Pisa, who introduced it to europeans in his work, Liber Abaci, for the popularization of the Hindu-Arabic Number system in the 1202 AD. The sequence is referred to as 'Misrau Cha' , which in Sanskrit means two are mixed, in ancient Indian texts by Pingala 200 BCE reasserted by Virahanka in 700 CE, Halayudha 950 AD, Gopala 1135 AD, and about fifty years before Fibonacci introduced it Hemachandra but is hinted at in Hindu scripture from much earlier, so India is where the Fibonacci sequence is its origin despite its Italian name. The sequence itself starts from one, one is added to itself to get two and it continues with the term being added to the previous term to get the next, therefore bringing about the Fibonacci sequence. The sequence is often depicted in a spiral which can allow the the Fibonacci sequence to be applied to a physical structure. The golden ratio, also known as phi, is closely related to the Fibonacci sequence. It makes a major appearance in geometry which motivated Ancient Greek mathematicians and scholars to explore this mysterious concept which is evident because the first known written definition of the golden ratio was in Euclid's Elements 300 BCE and in the features of Phidias' Parthenon built around 430 BCE. The golden ratio is calculated through division of a term with its previous term throughout the Fibonacci sequence as it oscillates towards the golden ratio, as demonstrated with 144/89 being 1.6179 and 233/144 is 1.61805 which we can see are extraordinarily close to the golden ratio 1.61804…, which therefore shows that as the limit of the Fibonacci sequence's division approaches infinity the division is equal to phi, making the range of application of the ratio the same as the sequence. This paper will explore the extent of these mathematical concepts appeal, because frankly the Fibonacci sequence and the golden ratio are incredibly fascinating to observe and apply, as well as them having immense informational depth.
Body:
Fibonacci Sequence:
Natural Structures
There is so much to go for the application of the Fibonacci sequence as it shows up in various natural structures like in botany, I learned of the connection when I was looking at random Khan Academy videos when I was younger, and even in human biology, which i only learned when I explored more of it this year. For example in flowers the number of petals in a normal flower is most often a Fibonacci number like buttercups and wild roses having five petals each, corn marigolds have thirteen petals and some daisies have up to eighty nine petals. It's fascinating because these plant share only so many genes in common yet they still mimic the same pattern as one another. The way the petals are organized is incredible too since they are organized in Fibonacci spirals, which are just spirals but the number of those spirals is a Fibonacci number, like pine cones, for instance, have a few different types of spirals on them but the number of those individual spirals are a Fibonacci number. For example a pine cone could have thirteen positive steep spirals, thirteen negative steep spirals, twenty one positive gradual spirals and twenty one negative gradual spirals. Though some plants do not conform to this type of pattern, the fact that numerous plants do conform to this pattern is incredibly cool and definitely not randomly done. In biological structures it gets even more unusual. We can see the most simple patterns of singles or pairs with a pair of ears, eyes, hands and feet as well as a single nose, mouth, navel, or even more in fives with five fingers and five toes, but it's even seen in microstructures like DNA we can see that a full cycle of DNA is twenty one angstroms wide and thirty four angstroms long. This repeated pattern of Fibonacci numbers cannot be a coincidence with the vast difference between plants and humans to think that this pattern transcends the categories of living beings is incredible.
Man-made Structures:
It's not just natural structures that mimic the Fibonacci sequence, man-made structures, like art , do as well. For instance, the Fibonacci spiral shows up in art throughout the world, the Fibonacci sequence can be depicted as a spiral that goes through squares with the dimensions of Fibonacci numbers, from least to greatest.
Wikipedia
The curve is liberally used in art like in photography, most often photos have the subject in the center but the most appealing of the photos who do not have it center most often conform to the rule of thirds which utilizes the Fibonacci spiral for a more eye-catching photo. The parts where the curve is tighter is where the photo has the most detail and as the curve loosens there is less detail in the photo therefore carrying the eye along the curve. We can observe this quite clearly in the following photos, my friends in photography were quite surprised by the relationship between the Fibonacci sequence, a mathematical concept, and photography since they had not known of it before.
Pinterest Jake Garn
The way that it fits into photography is fascinating but it also shows up in classic paintings like The Girl with Pearl Earring by Johannes Vermeer from the 17th century. We can see that similarly to photography most of the detail is in the tight curve of the spiral.
Johannes Vermeer
Though the spiral is incredibly evident in visual art it is also prominent in music like on a piano eight notes make up a scale, thirteen notes make up an octave, there are eight white keys and five black keys. There are questions about whether or not Fibonacci numbers are related to pleasing musical frequencies but there is not enough definitive or convincing proof of this. The golden ratio has similar applications as the Fibonacci sequence, which will be explored next.
The Golden Ratio:
The Golden Ratio also known as the golden mean or phi(?), as said before is related to the Fibonacci sequence through the continuous division of a Fibonacci number with its previous term,as the limit of that division approaches infinity it actually equals the golden ratio, as depicted below.
Story of Mathematics
Natural Structures:
The golden ratio can be applied to many natural structures and not just because of its relationship to the Fibonacci example, the optimal ratio between the consecutive parts of the finger is the golden ratio, and when the fingers are curled up they are the golden/Fibonacci spiral. It is not just the human hand but nearly everything about the human body can be related to the golden ratio, which is shown below.
The Golden Ratio in the Human body
The image shows the ratio of the navel to the bottom of the feet and the navel to the top of the head as well as several other examples of the golden ratio, but does not include the fact that the ratio between the length of the humerus and the radius equals phi as well as several other examples. All of these measurements are based on bodies formed in optimal conditions soif the body is formed in non-optimal conditions the golden ratio does not always apply, since anything other than optimal conditions would disrupt the formation of the body making the golden ratio inapplicable to the areas affected. I tested this on the features of my classmates and myself to see if it was true and I saw that most often it is the facial features that are distorted due to non-optimal conditions because those are the fine details that would not matter as much in comparison to the length of bones or protecting our immune system. The golden ratio is not just in the human, it also shows up in botany just like the Fibonacci sequence. When plants are growing leaves they grow them in spirals and the angle between the leaves in the spiral is the golden angle. Phi on a circle would be 222.5 degrees and for peace of mind everyone uses the angle left over after subtracting 222.5 from 360, 137.5, which is called the golden angle. One of the reasons why plants form their leaves like this is so that the leaves would get the maximum amount of sunlight possible without blocking or being blocked by another leaf. This use of the golden angle leaves me astounded at this trait in not one species of plant but almost all species of plant that grew in optimal conditions, One can theorize that it is an evolutionary trait, which indicates to us that any plant that did not conform to the golden ratio did not survive as it would not get the nutrients needed for survival because of its non-golden-ratio conforming nature. This shows us the vast extent of the golden ratio in nature, from human biology to plants, but it does not just appear in nature since it also makes appearances in various man-made structures.
Man-made Structures:
There are many man-made structures that exhibit the golden ratio, including art and architecture from around the world. The golden ratio is often included in art, mostly due it the proportion being aesthetically pleasing to the general public, as said earlier when discussing the Fibonacci spiral in art, there are many ways one can apply it to art from using the spiral to make the audience focus on a point of interest to making the dimensions of the object(s) in the art conform to the golden ratio. The golden ratio generally makes objects more appealing in any of its forms. Although the golden ratio has numerous applications in art it has great purpose in architecture. One of the oldest uses of the golden ratio is when Phidias used it in his construction, the Parthenon, in 430 BCE and was defined in Euclid's Elements 300 BCE and was included in architectural theory by Luca Pacioli in his Divina Proportione 1509 AD, a treatise made up of his three independent works involving the golden ratio, architecture, and geometry. The golden ratio is not only in Europe it also shows up in places from around the world. Places like the Taj Mahal 1684 AD use the golden ratio in the dimensions of the parts of the construction to have an aesthetic appeal rather than using the spiral to carry the eye. While some buildings like Toronto's CN tower still use it for aesthetic appeal with the total height of the tower compared to the height of the observation deck being equal to the golden ratio. Therefore showing the international extent of the golden ratio in man made structures.
The Taj Mahal CN Tower in Toronto
Conclusion:
The golden ratio and the Fibonacci sequence are frankly fascinating as they have so many applications in the real world both mechanically and aesthetically. Not only that but there are so many avenues of research that are either unexplored or not explored deeply enough because there is just so much depth to the Fibonacci sequence and the golden ratio because they are not just rules and concepts to do with man made structure but they are inherent rules for all natural things and that is frankly amazing as it shows how everything is interconnected from little daisies to tall skyscrapers. It is not only man-made structures that make use of Fibonacci numbers and the golden ratio either. They used in religion as well with the thirteen days of funeral rites in Hinduism as well as the vast use of three throughout the world including Christianity's three aspects of god; the father, the son, and the holy ghost, the numerous European mythologies' Three Fates, the list can go on forever showing to us that the Fibonacci sequence and the golden ratio are not just scientific or mathematical, they play a part in religions all over the world making it a sociocultural aspect of this world giving it so much more depth. It is beautiful how both nature and humans all over the world have followed this code for millenia, which makes me so astounded and curious about this subject and researching this for months has made the subject so personal to me now.
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