The spiral of seeds in a pine cone, the fruitlets of a pineapple. What do they have in common? They both conform to the Fibonacci Sequence.
As anyone who has read Dan Brown's thriller The Da Vinci Code or seen the movie knows, the Fibonacci Sequence is a sequence of numbers created by adding two sequential integers together, starting at 0.
The sequence can be described by the equation:
Fn = Fn - 1 + Fn - 2, where n > 1 so,
F0 = 0, F1 = 1 and F2 = F1 + F0 = 1.
The sequence of numbers comprising the Fibonacci Sequence is: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 ...
The person who brought the Fibonacci Sequence to Western audiences is Leonardo of Pisa, who was born around 1170 A.D. and died around 1250 A.D. He was later nicknamed Fibonacci, from Filius Bonacci, which means 'son of Bonacci'. The sequence had actually been deduced by Indian and Arab mathematicians a thousand years earlier.
In 1202, Fibonacci described the sequence in his Liber Abaci ('Book of Calculation'), which was intended as a math guide for tradesmen, so they could calculate profit and loss, and loan balances.
In Liber Abaci, Fibonacci introduced the sequence with a problem involving rabbits. The problem starts with one male and one female rabbit. After a month, they mature and produce a litter of one male and one female rabbit. A month later, those rabbits reproduce and have a litter of one male and one female rabbit, and so on. The question Leonardo asked was, how many rabbits would you have after one year? The answer, it turns out, is 144 — and the formula used to get arrive at that answer is what's now known as the Fibonacci sequence.
Squares and arcs
During the 19th century, mathematicians began examing the Fibonacci Sequence again, and they realized that if you drew squares of the Fibonacci Numbers, then placed the sides of the squares together, the new side of a larger square was formed. This can be repeated infinitely.
They then realized that if you drew circular arcs connecting the opposite corners of the squares, you get a spiral called a logarithmic spiral. This spiral is seen in many natural phenomenon, such as in the arrangement of leaves on a stem or seeds on a pinecone.
But, that's not all. Fibonacci Numbers show up in all sorts of places in nature. Some flowers have 3, 5, 8 or 13 petals, where each petal is placed to allow for maximum exposure to sunlight. The rows of seeds in sunflowers and pinecones often add up to Fibonacci Numbers, because that is the most efficient way to pack as many seeds as possible into a small space.
The Golden Ratio
If you divide any Fibonacci Number by the one before it in the sequence, you get a ratio of approximately 1.618033..., which is called the Golden Ratio. As the Fibonacci Numbers get higher, the ratio becomes even closer to 1.618. For example, the ratio of 3 to 5 is 1.666, the ratio of 13 to 21 is 1.625, and the ratio of 144 to 233 is 1.618.
The Golden Ratio is found by dividing a line into two parts, a and b, so that the longer part divided by the smaller part is also equal to the whole length divided by the longer part. That is:
The Greek letter "phi" represents the Golden Ratio, which is also known as the golden mean, golden section, divine proportion, and divine section. It is 1.6180339887..., an irrational number that is also equal to the solution to the quadratic equation:
x2 - x - 1 = 0, with a value of
The Golden Rectangle is a rectangle whose sides are Fibonacci Numbers, such as in the image below. For example, a = 8 and b = 5, so that a + b = 13 and the ratios yield: 1.6180339887498948420… The Golden Rectangle is considered one of the most visually satisfying of all geometric forms, and it is commonly used in art, especially in Renaissance paintings and sculptures.
Leonardo Da Vinci used the Golden Ratio in the proportions of his "Last Supper," in his "Vitruvian Man", and in the "Mona Lisa". Michelangelo, Raphael, Rembrandt, Georges Seurat, and Salvador Dali also incorporated the Golden Ratio into their works.
The Golden Ratio can perhaps even be seen in the Great Pyramid of Giza, where the length of each side of the pyramid's base is 756 feet, and its height is 481 feet. The ratio of the base to the height is roughly 1.5717, which is close to the Golden Ratio.
Ancient Greek sculptor Phidias (500 B.C. - 432 B.C.) is said to have applied phi to the design of the sculptures he created for the Parthenon. Plato (428 B.C. - 347 B.C.) celebrated the Golden Ratio, and Euclid (365 B.C. - 300 B.C.) linked it to the construction of a pentagram, a five-sided figure.
In the 1970s, British physicist Roger Penrose included the Golden Ratio in his Penrose Tiles, which allowed surfaces to be tiled in five-fold symmetry. In the 1980s, phi was theorized to have appeared in quasicrystals, a then-newly discovered form of matter.
Beauty and the Nautilus
Studies have shown that when test subjects view a series of faces, the ones they deem the most attractive have Golden Ratio proportions between the width of the face and the width of the eyes, nose, and eyebrows.
The Golden Spiral is found frequently in plants most likely because, in order for plants to maximize the exposure of their leaves to the Sun, they need to grow them at non-repeating angles. The easiest way to guarantee this is to have an irrational value for the number of leaves, and many of the spirals we see in nature are a consequence of this behavior. The distributions follow logarithmic spirals, the general mathematical form of a golden spiral.
Finally, have you ever noticed that the covers of many high school mathematics textbooks display a nautilus shell? The shell can be described as having a spiral that expands by the Golden Ratio every 180 degrees. Although this is just an approximation, it is often cited as a sign of the appearance of the Golden Ratio in nature, and that's why it's on the cover of math textbooks.