The Golden Ratio and Fibonacci Sequences in Nature Art and Architecture

THE GOLDEN RATIO & FIBONACCI SEQUENCES IN NATURE, ART & ARCHITECTURE What connects spiral sea-shells, pine cones, the Mona Lisa and the Parthenon? It’s a mysterious number (called Phi), that has intrigued mathematicians and artists since the time of the Greeks. The number Phi is related to two mathematical ideas: that of the Golden Ratio, and that of Fibonacci Sequences. First, we’ll look at these two mathematical ideas, and then look at where they appear in the natural world, and in art and architecture. We’ll also look at how the Fiboancci sequence and the Golden ratio are closely related! The Golden Ratio In mathematics, the Golden Ratio is a proportion represented by the number Phi, or 1.61803398… This number is infinite, like the number Pi; that’s why it has the dots after it! You can see these proportions on the left. The line BC is divided in such a way that the proportion of the larger section to the smaller is the same as that of the whole line to the larger section (in other words 1.61803398… to 1). You can make a rectangle based on these proportions. Here’s one pictured below. In a golden rectangle, the longer side (A) is 1.61803398… times as long as the shorter side (B). The combined lengths of side A and B are 1.61803398… times the length of side A alone.

The Fibonacci Sequence Leonardo of Pisa (c. 1170 – c. 1250), also known as Fibonacci, was a medieval mathematician. As well as popularising the Arabic numeral system in Europe (at the time, Roman numerals were still used), he published a book called Liber Abaci, which contained the first mention of the mathematical sequence that bears his name. The Fibonacci sequence is one which starts with 0 and 1, and continues by adding the previous two numbers to get the next number in the sequence: 0,1,1,2,3,5,8,13,21,34,55,89,144,233 and so forth.. The Golden ratio in Architecture Many artists, philosophers and architects over the centuries, from the ancient Greeks onwards, have believed that the proportions of the Golden Ratio are uniquely beautiful and pleasing to the eye. In other words, the rectangle above looks more beautiful than one where the longer side is twice the length of the shorter side, or so it is said. Many of the artists and architects have even gone so far as to claim that these proportions were ordained by God; hence they are sometimes referred to as the Divine Ratio. They can certainly be

seen in nature, as we’ll see later. They can also be seen in many buildings and artworks. In the picture of the Parthenon on the left, for example, a rectangle drawn around the exterior of the building would exhibit the properties of the Golden Ratio. The relative heights of the pediment (the triangular crown of the building) and the frieze (the rectangular bit just below the pediment) are also in the proportions of the Golden Ratio (in other words the pediment is 1.61803398… times the height of the frieze). So are the heights of the columns, compared to the combined heights of the frieze and the pediment. There are lots of other examples on the exterior of the Parthenon.

Other buildings that exhibit these proportions include Paris’s Notre Dame Cathedral. See the picture on the right.

The Golden Ratio in art This system of proportioning also seemingly appears in many works of art, especially from the Renaissance, when there was a resurgence of interest in art and science. For example, see Da Vinci’s Mona Lisa on the left. There are many other examples of this proportioning in the Mona Lisa; her face, for example would fit neatly into a rectangle with sides based on the Golden Ratio. The series of rectangles that you can see superimposed on the Mona Lisa is linked to another aspect of the Golden Ratio, and points up its relationship with Fibonacci numbers. Spirals and the Fibonacci sequence If you were to draw a curved line connecting up the consecutive rectangles (as we’ve done on the left), you would get a spiral. These spirals appear often in nature; in Nautilus shells, for example, and in pine cones; see the Pathfinder Pack on Fibonacci Sequences for more information.

So what is the connection between spirals and Fibonacci numbers? Take a look at the diagram below. It shows (approximately) how the squares above are derived. Do you notice the numbers? Can you see that this is the start of the Fibonacci sequence? (1,1,2,3,5,8,13,21,34,55..)

The next square in the sequence would be 13 units to a side, the one after that 21 units, and so on. If you look at the individual numbers in the Fibonacci sequence and find the ratio between consecutive numbers, you’ll notice that as the sequence progresses, the figure gets closer and closer to the magical Golden ratio of 1 to 1.61803398…! For instance: 8 divided by 5 is 1.6 13 divided by 8 is 1.625 21 divided by 13 is 1.61538 34 divided by 21 is 1.61904 55 divided by 34 is 1.61764 So, the Golden Ratio is linked to Fibonacci Sequences, and both of these inform aspects of art, architecture and the natural world. It’s not surprising, perhaps, that artists and architects throughout history considered these mysterious numbers to be “divine”, considering that they often appear in nature!