# Alfred S. Posamentier on the Fibonacci Numbers Dr. Alfred S. Posamentier

Alfred S. Posamentier is dean of the School of Education and professor of mathematics education at the City College of New York. He is the co-author, along with Ingmar Lehmann, of The Fabulous Fibonacci Numbers, the story of Italian mathematician Leonardo Pisano and the numerical sequence he discovered. What are the Fibonacci Numbers?
The Fibonacci numbers are the most ubiquitous numbers in all of mathematics and beyond. They are the sequence of numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …. The sequence, as you can plainly see, begins with two 1s and then each succeeding number is obtained by adding the two preceding numbers. The third Fibonacci number is 1 + 1 = 2, and then 1 + 2 = 3, 2 + 3 = 5, 3 + 5 = 8, and so on.
What other contributions to mathematics did Leonardo Pisano make?
Perhaps what Leonardo Pisano (a.k.a. Fibonacci) should be most remembered for is not only the Fibonacci numbers, rather for being the first person to publish a book in western Europe that used the Indian numerals (as he called them): 9, 8, 7, 6, 5, 4, 3, 2, 1 and the zero, which he said the Arabs – from whom he got these numbers — called a zephyr. It is curious to note that he presented these numerals at the very beginning of his seminal book, Liber Abaci, in the order from right to left, as the Arabs most likely had written them. This might be the reason that we essentially write our numbers in the ascending order from right to left – something we grow up on and take for granted, yet it is not consistent with our other writing from left to right.
Fibonacci was perhaps the greatest mathematician of his time. He wrote several books, which showed how he used algebraic methods to solve geometric problems and geometric methods to solve algebraic problems. Suffice it to say, Fibonacci, best know for the numbers that bear his name, provided many mathematical discoveries from which future generations have wisely benefited.
Where would we see the Fibonacci numbers in nature?
There is practically no end to the sightings of the Fibonacci numbers in nature. From the number of spirals of the bracts on a pineapple or pinecone, to the ancestral tree of the male bee, the Fibonacci numbers continuously appear. This is best seen on the pages of our recently published book, The Fabulous Fibonacci Numbers. For me to describe the many sightings here would do them, and the Fibonacci numbers, a disservice.
What is the golden ratio and how does it relate to the Fibonacci numbers?
The golden ratio is most often seen as the ratio of the length and width of the golden rectangle. It is the ratio in the proportion: width is to length as length is to width plus length. The amazing thing is the relationship of this ratio to the Fibonacci numbers. That is, as you begin to take the ratio of two successive number of the Fibonacci sequence, with progressively larger Fibonacci numbers, you will get ever so much closer to the golden ratio – the ratios approach the golden ratio as a limit!
In other words, the ratio of the successive Fibonacci numbers 89/55 = 1.18181818… , while the ratio of the larger Fibonacci numbers 610/377 = 1.618037135…., which is considerably closer to the golden ratio (1.61803398877…) than the smaller pair of consecutive Fibonacci numbers.
Where would we find the golden ratio in art and architecture?
The golden ratio is also quite ubiquitous in art and in architecture. We find it by placing a rectangle around the Parthenon (Athens, Greece) and the United Nations building (New York), as well as at the doors of the Cathedral of Chartres (France). Let’s not forget that the Pentagon building in Washington, D.C. must contain the golden ratio as do all regular pentagons. Although there are many artists who have deliberately based their works on the golden ratio – often as the golden rectangle – many allow us to discover it for ourselves. Since Leonardo da Vinci illustrated the Franciscan monk, Fra Luca Pacioli’s book, De divina proportione with an anatomical study of the “Vitruvian” man (1505), we know that he was aware of the golden ratio. Therefore when one studies the Mona Lisa one can find many golden ratios in the picture’s proportions and even a golden rectangle that can perfectly encase Mona Lisa’s head. Again, in the interest of time and because it is difficult to describe visual experiences, I feel compelled to refer you to our book, which devotes a chapter to this topic. The Parthenon with golden rectangles superimposed.

Finally, how do you view the Fibonacci numbers? Are they some sort of mystical or Platonic reality that we’re tapped into subconsciously and now consciously? Or are they nothing more than a mathematical pattern that we can overlay on some things but not on others?
There are some things we will never know. This is just one of them. Why the Fibonacci numbers show up in the most unexpected of places – places where oftentimes we believe there is nothing related to mathematics, as the general public perceives it – is a mystery. Perhaps the Fibonacci numbers just reflect a certain pattern or symmetry that is indigenous to our society. I think the wild variety of the many applications of the Fibonacci numbers is what motivated us to write the book and I hope that will do that same for the general public to read the book. Then everyone can be in a position to speculate about the ubiquity of the Fibonacci numbers.