If you are curious enough to observe reality from a mathematical perspective, then patterns begin to emerge from everywhere. The construction of nature, the petals of roses, the size of the human itself and its relationship between its parts, even the structure of snails and the succession with which rabbits reproduce. So Plato’s famous phrase “God sometimes geometrizes” does not seem so far from reality to us. Geometry and shapes are present everywhere you look, which is why when it is described to us that a golden ratio exists, it is more intriguing. But what is this golden ratio and what is its relationship with the Fibonacci sequence?
Different names have been assigned to it; the golden, golden, golden number, golden ratio, the Fibonacci number and even the divine proportion. This number is represented by the letter phi of the Greek alphabet (Φ, φ). And it has been a concept widely studied throughout history by mathematicians, but also by artists, biologists, architects and musicians. Everyone who loves the puzzle called reality and the universe has ever heard about this proportion. But what really is the golden ratio?
What is the golden ratio?
It is an irrational algebraic number, meaning that its decimals tend to infinity and it is not periodic. Which means that it does not result from the repetition of decimals as in the case of dividing 1 by 9, which results in 0.11111… and so on to infinity. The golden ratio was discovered since ancient times, with the Greeks, and was used more as a proportion than as an arithmetic expression. It was used to describe the relationship between two segments on a line, therefore, it was more of a geometric construction than an irrational number. For this reason it is associated more with aesthetic aspects, although of course mathematicians have also studied it in depth.
The most interesting thing about the golden ratio is that it is present naturally in the world, its proportion applies to the thickness of the branches, the shell of the snails, the parts of the body, the characteristic shape of the snails and in many other aspects. This is why it enchants mathematicians, biologists and artists alike who try to emulate the aesthetics of nature.
Algebraically it is expressed as the sum of the unit plus the square root of five, divided by two. The result is 1.618033988749894… until continuing to infinity. But it is not a number full of meaningless numbers, but rather it has a large number of properties that mathematicians have been able to take advantage of. For example, we know that the golden number is the only one whose square (Φ²=2.61803398874988) and its reciprocal (1/ Φ= 0.61803398874988…) contain exactly the same decimals.
What is the relationship between the golden ratio and the Fibonacci sequence?
First, it must be clarified that Phi does not receive its name from Fibonacci, nor was it the mathematician Leonardo Bigollo Pisano (known as Fibonacci) who discovered the number. This had already been used previously by the ancient Greeks. Phi is named after the Greek sculptor Phidias, author of great architectural riches such as the Parthenon in Athens and who used the golden ratio in his works.
However, to understand the golden ratio it is necessary to go back to the Fibonacci sequence. This sequence begins with the numbers 0 and 1, from there the following elements are obtained by adding their two predecessors. Thus we have 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89… But within this sequence that if graphed results in the famous Fibonacci spiral, the golden ratio can be clearly seen.
This succession also has its intrinsic properties and can be observed in the patterns in which the leaves of plants grow or the succession with which rabbits reproduce. But in addition to these surprising properties, it also has a very close relationship with the golden ratio. It turns out that the golden ratio is hidden in the direct relationship of two consecutive terms of the proportion and the further away from zero they are, the closer they are to the golden ratio. For example, 5/3= 1.666, 13/8= 1.625… 89/55= 1.6181 and so on until you get closer to phi.
The golden ratio in nature
For some strange reason nature loves geometry and there are patterns of proportions in all aspects of nature. The clearest cases where you can see how this happens is to admire the growth of the . In the following images you can see different flowers with their petals numbered and surprisingly they follow the Fibonacci sequence.
And like this clear example many more, it is enough to raise your curiosity towards the most obvious details of nature to discover that Plato was right, God likes to geometrize.