# Appreciating the elegance of math and nature

November 23 is celebrated as Fibonacci day because when the date is written in the mm/dd format (11/23), the digits in the date form a Fibonacci sequence: 1,1,2,3.

The sequence displays a simple pattern: every two numbers, when added together, equal the following number. Another uncanny pattern is that the ratio of any two sequential Fibonacci numbers approximates the value of 1.618, which is also commonly known as the golden ratio phi (φ).

#### Nature’s order and mystery

From the shape of the galaxy to the eye of a hurricane, from a nautilus shell to a flower petal, various arrangements of natural elements follow surprising mathematical regularities related to the Fibonacci sequence. For example, it plays a vital role in phyllotaxis, the study of the arrangement of branches, leaves, flowers or seeds in plants.

In fact, the very program of life itself – the DNA molecule – contains the golden ratio. The DNA molecule measures 34 angstroms long by 21 angstroms wide for each full cycle of its double helix spiral. Both 34 and 21 are Fibonacci numbers and the ratio 34/21 is a close approximation of phi.

“There is something to do with optimality in nature. For example, flower petals grow up in these spirals to avoid overlapping and the Fibonacci patterns allow an optimal way to space them out,” says Nahid Walji, lecturer and research associate at the Department of Mathematics at the University of British Columbia (UBC).

He adds that the Fibonacci sequence also shows up in the study of fractals, a relatively recent branch of mathematics that is also abundantly found in nature.

“Something that nature seems to do in general, it is this idea of self-similarity. Self-similarity in my opinion is a way for nature to create complexity from simplicity. You are encoding something in the genetics that is easy to encode, but that creates complexity. Because complexity is important to nature to survive and to evolve, and that is a beautiful notion.”

Despite the widely observed patterns, scientists and mathematicians don’t have an exact explanation why. There is still a lot of ongoing research and even an entire scientific journal, The Fibonacci Quarterly, dedicated to topics related to the Fibonacci numbers.

Sujatha Ramdorai, mathematics professor and Canada Research Chair at UBC, acknowledges that mathematicians sometimes don’t have an answer for nature’s mysteries.

“‘Mathematics is the language in which God has written the universe,’” she says, quoting Galileo Galilei. “We just have to go with an open mind and try to investigate why something is happening and try to give reasoning or philosophy to understand the bigger picture. Along the way, we usually come up with other questions and more interesting connections to other areas.”

#### The appeal of the golden ratio

Just as fascinating as the golden ratio’s appearance in nature is its wide applications in art, architecture and music.

The ratio has been linked to beauty and elegance and is found embedded in the dimensions of many ancient monuments. It is speculated that the Parthenon in Athens, built between 447 and 438 BCE, was constructed based on the golden ratio.

The genius renaissance man Leonardo Da Vinci has also long been associated with the golden ratio. Da Vinci created the illustrations for De Divina Proportione (On the Divine Proportion), a book about mathematics written by Luca Pacioli around 1498. In the book, Pacioli writes specifically about the mathematics of the golden ratio and its application in art and architecture. Da Vinci’s use of the divine proportion is evident in some of his own artworks such as The Last Supper and Mona Lisa.

The golden ratio can also be found in classical music. Mozart, for instance, based many of his works on the Golden Ratio – especially his piano sonatas. Mozart arranged his piano sonatas so that the number of bars in the development and recapitulation divided by the number of bars in the exposition would equal approximately 1.618.

“A lot of times we see patterns in science, and we want to see if it will work in music. I still see people using Fibonacci to make music – it is very beautiful, it is really beautiful to see the echo of the Fibonacci in all facets of life,” Walji says.

#### Who really discovered the sequence?

In the west, the Fibonacci sequence first appears in the book Liber Abaci (1202) by Leonardo of Pisa, known as Fibonacci.

Though clear evidence has shown that the sequence has already been described much earlier in Indian mathematics, as early as 200 BC in work by Pingala on enumerating possible patterns of Sanskrit poetry formed from syllables of two lengths. Pingala’s work was central to the understanding of the composition of the Vedas, the oldest scriptures of Hinduism.

“With math that is this old, there is always some uncertainty where it originated and from which culture,” says Walji. “There is clear evidence that it was in India before. But maybe there are other cultures that came up with it as well and got lost in the midst of time as well.”

Ramdorai adds that there is also more evidence that Fibonacci was also known in the African civilization earlier from the patterns and constructions of their textile, architecture and music.

“This is what I like to call cultural mathematics. Some believe that everything was discovered during the renaissance. I feel people should learn more about how different civilizations have all contributed,” she says.