Explaining The Golden Ratio Because no one Else Will

The compilation I presented to my Calculus Teacher as a High Schooler.

Have you seen this image?

Many people, even those not interested in either math or the arts, have seen this image. This is no simple Mona Lisa, because what is that strange figure surrounding it? That, my friends, is the “Golden Ratio,” and today I’m going to aim to explain not only its origin, but its real appearances.

What Even is the Fibonacci Sequence?

In simple terms, the Fibonacci sequence is a mathematical sequence where each number that conforms to it is the sum of the two numbers preceding it. The sequence starts with 0 and 1, and goes on following the equation:

where n is greater than 1 (n>1).

Why Fibonacci?

Leonardo Pisano Bigollo, better known as Fibonacci or Leonardo Fibonacci, was an important mathematician due to his role in reviving ancient mathematics. He was the Son of Guilielmo Bonacci and often used the term "bigollo " to refer to himself as a good-for-nothing man or a traveler. He received an education in North Africa, in what is now known as Bejaia in Algeria. He was taught mathematics there and studied accounting to help his father with his business. Soon enough, he fell in love with mathematics and traveled throughout different countries to learn more. In 1200, he returned to Pisa, and in 1202, he wrote his book Liber abaci. The book was based on the arithmetic and algebra that Fibonacci had accumulated during his travels. One of the most important contributions this book made to society, other than the Arabic numerals, was that of the Fibonacci sequence, which, despite being mentioned briefly through a problem involving rabbits, granted him great attention in the future.

It is important to note that, despite the name and the attention that it drew, especially during the 19th century, the Fibonacci series had already been present in different Sanskrit texts, and Fibonacci only took upon himself to teach it in Europe.

Leonardo would later on receive the nickname of Fibonacci as a way of allegedly distinguishing him from another Leonardo Pisa.

Golden Ratio:

It originated thousands of years ago in Ancient Greek, where mathematicians would study it because of its frequent appearance in geometry, where it would receive the name “extreme and mean ratio.” Despite its simplicity, it has led many mathematicians, including Pythagoras, Euclid, Leonardo Pisa, Johannes Kepler, and many others, to spend endless hours over this simple ratio and its properties. Its first definition appeared when Euclid provided various propositions of proofs employing the golden ratio. Despite its mystery, it wasn’t until Johannes Kepler discovered the work of Simon Jacobs in 1608 that the relationship between the Fibonacci numbers and the golden ratio was drawn, which would lead Michael Maestlin to, for the first time, find a decimal approximation of the number: about 0.6180340.

Phi, or the Golden Ratio, is a mathematical constant that follows the ratio: (a/b) = (a+b)/b. Which is equal to the symbol φ, which also satisfies the quadratic equation φ² = φ +1. φ is an irrational number with value (1+ (5)undefined )/2.

Despite this being the formal definition of what the Golden ratio is, it is not very clear. For this reason, before looking at the equations, it is important to understand the concept behind it, and what it really represents. Basically, imagine you have a line segment, and you cut it somewhere; it doesn’t matter where. Subsequently, you name the shorter side a, and the longer side b. Now, what if there was a cut so perfect that the ratio of the whole line to a equals the ratio of a to b. That is what the Golden Ratio is, and that ratio is 1.618033988.

Golden Ratio’s Crazy Properties

The Golden Ratio is a very curious number. As mentioned before, it satisfies the equation φ² = φ +1, which may seem unimportant or minor until one starts to analyze it. Usually, when numbers are squared, they become much bigger. If you square 2, you get 4; if you square 3, you get 9. However, this doesn’t happen with φ; when 1.618033998 is squared, it becomes 2.618033998, which is φ+1. It didn’t grow by much; it’s almost as if it absorbed the squaring and shrugged. This means that every power of φ can be expressed in terms of simpler φ values: It keeps reappearing inside itself.

Now, let’s go back to the initial equation mentioned above. ( a+b)a = a/b = φ. To determine φ as a number, we can divide the numerator and denominator of the fraction on the left-hand side by b, and then later substitute a/b by φ to obtain:

Which we would later use to approximate the number through an equation of continued fractions. This is where we can notice why it is an irrational number.

Most irrational numbers, when you write them as continued fractions, involve all sorts of different integers; nevertheless, φ is not the case, as its continued fraction representation is very simple. Too simple even:

This would subsequently make it a very difficult, if not one of the hardest, numbers to approximate with fractions.

What Simon Jacobs Found and What Fibonacci Never Noticed

Centuries after Leonardo’s rabbit problem. German mathematician Simon Jacobs noticed something regarding the numbers that were part of the sequence: if you divide any Fibonacci number by the one before it, the answers keep getting closer to φ or 1.618:

Fibonacci Sequence:

1,1,2,3,5,8,13,21,34,55,89,144

Ratios:

5/3=1.667

8/5=1.6

13/8=1.625

55/34=1.618

144/89=1.618

The further you go, the more it approximates toward φ.

This could be represented as the lim as n approaches infinity of the ratio of the terms of the Fibonacci sequence:

The Fibonacci sequence, therefore, turned out to be a very slow calculator of the Golden Ratio.

The Legends and Myths

Though the Golden Ratio is present in many art pieces, the most famous examples being the Great Wave of Kanagawa or The Sacrament of the Last Supper. It has been constantly surrounded by many legends and myths regarding its presence in various architectural monuments, as well as in many other art pieces, like those of the Mona Lisa or the Last Supper. The truth is that the idea that the golden ratio is a universal “secret to beauty” used by all great artists is a myth. While some creators use the proportion, it is heavily exaggerated. Most historical masterpieces do not adhere to the ratio, and any claims of its presence are often a result of retrospective fitting.

The Reality: Nature at is finnest

Nevertheless, the reality can seem much more compelling than those myths surrounding the proportion, as in reality, it appears in some plant growth patterns due to evolutionary advantages; one example of this is that of sunflowers. If you count the spirals going clockwise, then counterclockwise. You’ll probably get 34 and 55. Or 55 and 89. Or 89 and 144, depending on how large the sunflower is. Consecutive Fibonacci numbers, every time. Pine cones do it. Pineapples do it. Romanesco broccoli does it. A lot of natural beings follow the ratio because, despite not doing it for aesthetic reasons, they are still following something brutally beautiful. Solving a problem.