# Fibonacci: Capturing a Mathematical Enigma in Jewelry

## Jana Gumovsky

I have always found fascination in numbers and becoming a mathematician has been one of my secret aspirations. Alas, my proficiency in higher mathematics has never quite panned out the way I had hoped but my fascination remained. In fact, my grandfather is an esteemed mathematician in Moldova, who well in his 80s still reads lectures at the capital's polytechnic university. Perhaps it's genes, or maybe it's the endless possibilities and the excitement of the unknown but I will always remain drawn to numbers.

Introducing: The Fibonacci Bracelet - my homage to the world of numbers.

These beaded bracelets follow a specific number pattern : 1, 1, 2, 3, 5, 8, which are the first 6 numbers of the Fibonacci sequence. But what does that mean? Mathematics, like art, has an inherent beauty and often that beauty transpires when equations and number patterns are discovered in our world, hiding in plain sight. But first, a bit of history:

Leonardo Bonacci (c. 1170 – c. 1250), popularly known as Fibonacci or Leonardo of Pisa, was the first great Western mathematician after the decline of Greek science.

His well-known moniker “Fibonacci” is derived from the Latin words “filius Bonacci”, literally translated to “son of Bonacci”. Fi'-Bonacci could be considered the equivalent of the English William-son or John-son.

Fibonacci was born in Pisa, Italy, to Guglielmo Bonacci, a wealthy merchant who directed a trading post at a major port located in present day Algeria. As a boy, Fibonacci accompanied his father on his commercial trips to the Orient. It was during his travels along the Mediterranean coast that the budding mathematician became acquainted with the Hindu-Arabic number system and discovered its enormous practical advantages compared to the Roman numerals, which were still current in Western Europe.

Fibonacci ended his travels around the year 1200 and returned to Pisa. Upon his return, inspired by his interactions with the foreign merchants he met while under the tutelage of his father, Leonardo wrote a number of influential texts that played an important role in reviving ancient mathematical skills. His works garnered him recognition among his contemporaries and high esteem from the reigning Holy Roman Emperor, Frederick II.

His most well-known published book is *Liber Abaci* (1202), literally translated as “Book of Calculations” or “Book of the Abacus”. The book, which went on to be widely copied and imitated, was based on the arithmetic and algebra that Fibonacci had accumulated during his travels. In it, Fibonacci introduced the so-called *modus Indorum* (method of the Indians), today known as Arabic numerals and the Hindu-Arabic place-valued decimal system. The book showed the practical importance of the new numeral system by applying it to commercial bookkeeping, conversion of weights and measures, the calculation of interest, money-changing, and other applications. Furthermore,* Abaci* contains a large collection of problems aimed at merchants. They relate to the price of goods, how to calculate profit on transactions, how to convert between the various currencies in use in Mediterranean countries, and problems which had originated in China. The book was well received throughout educated Europe and had a profound impact on European thought.

A math problem included in *Liber Abaci* led to the introduction of the Fibonacci sequence for which Fibonacci is best remembered today: *A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair which from the second month on becomes productive?* The following assumptions accompanied this problem: no rabbits died during the year and in each pair there was always one male and one female rabbit.

The answer can be derived by a number sequence in which the next number is decided by adding the two numbers immediately preceding it.

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, etc.

This sequence, and its accompanying equation, has proved extremely fruitful and appears in many different areas of mathematics and science.

Now this is where it gets (even more) interesting.

Although Fibonacci's *Liber Abaci* contains the earliest known description of the sequence outside of India, the sequence has its roots in Indian mathematics. It has been noted by Indian mathematicians as early as the sixth century in connection with Sanskrit prosody, which is the study of metre in Sanskrit poetry. In the Sanskrit oral tradition, there was much emphasis on how long (L) syllables mix with the short (S), and counting the different patterns of L and S within a given fixed length results in the Fibonacci numbers.

Furthermore, if a Fibonacci number is divided by the number that directly precedes it in the sequence (particularly when dividing the larger numbers), the ratio is consistent. And when that ratio is transformed into a number we end up with 1.618033…*ad infinitum.* This magical number is commonly known as φ (Phi) or as the “Golden Ratio” and it was a matter of great fascination to the ancient Greeks. When the Fibonacci sequence is displayed as a series of adjacent squares they form a rectangle known as the “Golden Rectangle” and if you run a line through their corners you will have a spiral where the ratio at each juncture equals 1.618.

As with many other mathematical concepts, the purview of the Fibonacci numbers and the golden ratio spans well beyond the realm of pure mathematics. The golden spiral mirrors the spiral of the nautilus shell, snail shell, the cochlea of the inner ear in human anatomy, the horns of many animals, the arrangement of flower petals, of leaves on a branch, of seeds in a pinecone. The proportions within human and animal anatomy also exemplify the golden ratio. This ratio permeates our makeup to its smallest component- our DNA, with 21 angstroms in width and 34 angstroms in height, the double helix of our genetic makeup includes two consecutive Fibonacci numbers. The golden rectangle also holds mysterious aesthetic properties and for centuries its proportions have been incorporated in great works of art (such as Dali’s The Sacrament of the Last Supper (1955)) and in architectural structures such as the Greek Parthenon, the Great Pyramid of Giza and the great mosque of Kairouan, among others.

And now, “jewelry designers” can be added to the lengthy list of professions that share a fascination with Fibonacci numbers.

Love,

Jana

References:

- http://scienceworld.wolfram.com/biography/Fibonacci.html
- http://www-history.mcs.st-and.ac.uk/Biographies/Fibonacci.html
- http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibBio.html
- http://scienceworld.wolfram.com/biography/Fibonacci.html
- http://io9.com/5985588/15-uncanny-examples-of-the-golden-ratio-in-nature
- http://www.urban-flavours.com/2014/06/la-giaconda/
- Fibonacci Images via Google Images