29/09/2021

The mysteries of the Fibonacci sequence

Present in various elements of nature, the Fibonacci sequence remains shrouded in mystery and surprising facts more than eight centuries after its discovery. The progression is found in leaf petals, chameleon tails, snail shells, and even the structure of galaxies, for example. To this day, the limits of this simple numerical pattern are unknown, and its applications continue to be investigated.

Leonardo Pisano, known as Leonardo Fibonacci, lived between 1170 and 1250. In 1202, he published his greatest work, "Liber abaci," or "The Book of the Abacus" in English, which brings together much of the knowledge acquired from Arab and Jewish mathematicians, with particular emphasis on the decimal numbering system, still used today.

But the book's merit goes beyond that. Fibonacci described the sequence that would later bear his name. The famous numerical pattern, whose progression is always the result of the sum of the two previous numbers (Fn = Fn -1 + Fn -2 ), was illustrated with an exercise: “A man placed a pair of rabbits in an enclosed space. How many pairs will be produced in a year, if we assume that each pair produces another per month starting from their second month of life?” The number of pairs follows the sequence F1 = 1, F2 = 1, F3 = 2 , F4 = 3, F5 = 5, F6 = 8 , F7 = 13, F8 = 21, F9 = 34, F10 = 55 , F11 = 89, F12 = 144, F13 = 233.

The Fibonacci sequence is also noteworthy for its relationship with the so-called golden ratio. If you divide a number by the previous one, the result will be increasingly closer to 1.618… (F <sub>n</sub> / F <sub>n-1 </sub> ≅ 1.618). Also known as the golden ratio, the golden number is represented in mathematics by the Greek letter φ (phi), and is frequently seen in various fields, whether in mathematics, physics or chemistry, as well as in art, architecture and design, for example. In relation to the Fibonacci sequence, the larger the numbers chosen, the closer the result will be to the golden ratio.

Other relationships can be made with this numerical pattern. For example, the sum of 10 consecutive numbers in the sequence is always equal to 11 times the seventh number. If we add the numbers from F4 to F13 ( 3 + 5 + 8 + 13 + 21 + 34 + 55 + 89 + 144 + 233), we will get 605, which is equal to the seventh number (F10 = 55) multiplied by 11.

This is just one of the properties of the Fibonacci sequence, which continues to fascinate mathematicians to this day. Even with the doubts that still remain about the Italian's invention, there is no doubt that he was the greatest mathematician of medieval Europe, largely responsible for popularizing the numerical system that is still used today.

BBC Brasil has made a video about the history of Fibonacci, the mysteries of the sequence, and its relationship to the golden ratio, which is available on their YouTube channel . It's worth checking out!