Part of the beauty of mathematics is its mystique – and I don’t necessarily mean the incomprehensive terrains of math understood only by professional mathematicians. Many number sequences are delightfully magical; they follow beautiful patterns that seem to have been given by the gods. Best of all, we can visualize these patterns in pictures, and what stunning pictures they are!
Take, for example, the Fibonacci sequence, namely after the Italian mathematician Leonardo of Pisa, later known as Fibonacci, who introduced the sequence to Western European mathematics in his 1202 book Liber Abaci.
Perhaps the most pleasing pattern of the Fibonacci sequence in nature is that of the nautilus shell. Imagine that whoever created the nautilus shell gave it the following instructions for its growth – start with a little 1 x 1 square house. Then, each time it outgrows its house, it adds another room to it, with a dimension that is the sum of the two previous rooms. In other words, follow the number sequence 1, 1, 2,3, 5, 8, 13, 21. After the first two “ones”, the next number is simply obtained by adding the previous two numbers. The result of this growth is a spiral, like the spiral of the famous nautilus shell. It is undoubtedly beautiful and simple.
These numbers seem fundamental to the way nature grow things. Apart from the shell, other examples include the branching in trees, the arrangement of leaves on a stem, the fruit sprouts of a pineapple, the flowering of an artichoke, the uncurling fern, the arrangement of a pine cone’s bracts and so on.
Fibonacci numbers are also useful in many areas of science and engineering, including computer algorithms such as the Fibonacci search technique and the Fibonacci heap data structure, and graphs called Fibonacci cubes used for interconnecting parallel and distribution systems.
In art, the Fibonacci sequence frequently manifests in the form of the Golden Ratio, 1: 1.618033988. This is the ratio that consecutive pairs of Fibonacci numbers tend toward as the Fibonacci numbers gets bigger. For example, 21/13 is 1.6153846, and seven numbers forward, the ratio is 610/377 = 1.618037. The Golden Ratio appears regularly in art because it leads to aesthetically pleasing proportions. Here are two famous examples.
The first example is the Vitruvian Man by Leonardo Da Vinci.
On the right-hand side, you will see that somehow, Leonardo figured out that proportioning the man’s physique using the “nautilus spiral” gives rise to pleasing results. And the nautilus spiral is implicitly defined by a series of Golden Ratios (see note 1).
You can see too, the Golden Ratio was used by famed 19th century Japanese painter Katsushika Hokusai in his masterpiece, The Great Wave.
 To see the connection between the Fibonacci sequence and the Golden Ratio, we revisit the Fibonacci numbers that define the curl of the nautilus shell. Begin with a rectangle with sides in the 1: 1.618033988 ratio. Partitioning that rectangle into a square and new rectangle gives that new, smaller rectangle in the same ratio. As you continue this partitioning inside each new rectangle, your rectangles get smaller and smaller, but are still in keeping with the ratio. The result is the nautilus spiral (or more precisely, the Golden Spiral).