Leonardo Pisano, also known as Leonardo Bonacci and Leonardo Bigollo (“the traveler”), is best known for introducing the Fibonacci sequence:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55,…
in which every term, starting from the third, is a sum of two previous terms.
He was never actually called Fibonacci – this name was made up in the 19th century from filius Bonacci (“son of Bonacci”). His book Liber abaci (“The Book of Calculations”), written after Fibonacci studied and traveled extensively in the Arab world, discussed various calculations, algebraic methods, and their applications, including applications to commerce. This book was also hugely influential in bringing the Hindu-Arabic place-valued decimal system and numerals to Europe. Before this, European used Roman numerals. While you may have hated long multiplication and division drills in elementary school, doing the same calculations with Roman numerals would have been a lot more painful. The decimal system made business accounting easier and may have played a role in the growth of commerce in medieval Europe. So Fibonacci was a much more important figure in the history of mathematics than he’s usually given credit for.
However, let’s go back to the Fibonacci numbers now. Here’s the problem Fibonacci used in his book to introduce the sequence:
A certain man had one pair of rabbits together in a certain enclosed place, and one wishes to know how many are created from the pair in one year when it is the nature of them in a single month to bear another pair, and in the second month those born to bear also.
Let’s look at what happens with the rabbits. Remember only adult rabbits can have babies and every month every adult pair produces a pair of babies, which grow to be adults next month.
| Month | Adult rabbit pairs | Baby rabbit pairs | Total pairs |
| 1 | 1 | 0 | 1 |
| 2 | 1 | 1 | 2 |
| 3 | 2 | 1 | 3 |
| 4 | 3 | 2 | 5 |
| 5 | 5 | 3 | 8 |
| 6 | 8 | 5 | 13 |
There are many other problems in Fibonacci’s book. Here are a couple more examples. Try to solve them.
1. A certain man buys 30 birds which are partridges, pigeons, and sparrows, for 30 denari. A partridge he buys for 3 denari, a pigeon for 2 denari, and 2 sparrows for 1 denaro, namely 1 sparrow for ½ denaro. It is sought how many birds he buys of each kind.
2. A certain man proceeding to Lucca on business to make a profit doubled his money, and he spent there 12 denari. He then left and went through Florence; he there doubled his money, and he spent 12 denari. Then he returned to Pisa, doubled his money, and spent 12 denari, and it is proposed that he had nothing left. It is sought how much he had at the beginning.
This type of problems can be done fairly easily using algebra. In Fibonacci’s times, algebra it its modern form was not known, so he wrote the solutions in words, and this took a lot more space and effort than writing equations. So next time when you struggle with a system of linear equations, be thankful you don’t live in Middle Ages.
Anyway, let’s go back again to Fibonacci sequence or we’ll be here all day. Well, we could be here all day, or in fact a whole lot of days if we just talk about a Fibonacci sequence, because it has a tendency to pop up in many unexpected places, not only in mathematics but also in computer science, physics, and even biology (outside of rabbit breeding).
Many flower heads, pine cones, vegetable heads and other plant parts show spiral patterns, and often if you count the number of spirals in two different directions they will turn out to be two consecutive Fibonacci numbers (see picture). In some cases there are mathematical models explaining how this happens.
Johannes Kepler, a great 17th century astronomer and mathematician, noticed that the ratio of two consequent Fibonacci numbers approaches the golden ratio (also called the golden section) as the numbers grow larger. The golden ratio itself was known since Euclid. It appears if you try to divide a line segment into two parts so that their ratio is equal to the ratio of the whole segment’s length to the longest part. In algebraic terms,
\[ \frac{a + b}{a} = \frac{a}{b} \]
Setting \(x = b/a\), this converts to a quadratic equation \(x^2 = x + 1 \) with solutions:
\[ \frac{1 + \sqrt{5}}{2} = 1.618… \textrm{ and } \frac{1 – \sqrt{5}}{2} = -0.618… \]
The positive solution is the golden ratio and is usually denoted by \( \phi\). This ratio was also thought to be the most pleasing to the eye, especially in the golden rectangle, in which the ratio of the longer side to the shorter is the golden ratio. It is believed that the Greek Parthenon temple, as well as many other buildings and works of art use the golden ratio in their proportions. Unfortunately, neither of these beliefs is confirmed by modern research, so you’re free to choose any other rectangle as your favorite.
Just for fun, here are a couple ways you can write the golden ratio:
\[ \varphi = 1 + \frac{1}{1+\frac{1}{1+\frac{1}{1+…}}} \]
\[ \varphi = \sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+…}}}} \]
Now remember we briefly talked about different kinds of spirals in Geometry of Aikido? Turns out there is a special spiral, the golden spiral associated with the Fibonacci numbers. It’s a logarithmic spiral which grows outward by a factor of the golden ratio for every quarter turn. This can be easier to see if we look at the “Fibonacci spiral” which is an approximation of the golden spiral. This picture shows how it’s constructed:
We start with two squares of side 1, which make a 2×1 rectangle. Then we add a 2×2 square to the longest side of this rectangle to get a 3×2 rectangle, and continue adding squares in this manner. Then connect the opposite corners of each square by a quarter-circle as shown to get the Fibonacci spiral – which is not a true spiral, but is a decent approximation of the golden spiral.
Finally we can talk about martial arts. This video demonstrates a spiral movement of the arm in martial arts which seems to follow a golden or Fibonacci spiral. It’s pretty clear to me that the spiral movements in aikido do have some kind of logarithmic structure. We often use the words “extension”, “expansion” to describe what nage (the defender) is doing during a throw. Aikido techniques always start in your center. This is the center of gravity of the human body, and while it varies between individuals, it’s usually thought to be inside the body a little below the navel. Starting from the center, you expand out and uke (the attacker) is caught in this expanding movement, ultimately falling or rolling away. Logarithmic spiral seems a fairly natural model of this expansion. It’s not at all clear to me whether it’s actually close to a golden spiral or it depends on the technique and the nage.
It’s eternally tempting for a human mind to find universality in very different areas of life and the universe. Without this quality, science would have been impossible. However, there is always a danger to go too far and presume simple constants and simple explanations for complex things. Fibonacci sequence and the golden ratio are mathematically simple objects which nevertheless have many fascinating properties and applications, but they’re not necessarily everywhere.
Learn More
Professor Keith Devlin of Stanford wrote a book called The Man of Numbers: Fibonacci’s Arithmetic Revolution. He also has a lot of information about Fibonacci online, including an article about Liber abbaci and a video exploring myths and truth about Fibonacci and the golden ratio.
Dr. Ron Knott created an extremely informative and fun website about the Fibonacci numbers, the golden ratio, and related topics.