Fibonacci and the Golden Ratio

Leonardo of Pisa, known as Fibonacci (ca. 1170-1240), was one of the most significant mathematicians of the Middle Ages. His most important work is the Liber abbaci, in which he also elaborates on the so-called Fibonacci sequence. The sequence usually begins with one, one. Each subsequent number is the sum of the two preceding numbers:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ...

Many growth patterns in nature can be explained by the Fibonacci sequence. This is particularly evident when creating the so-called Fibonacci spiral based on the sequence. For example, it illustrates the growth of a snail shell or the arrangement of seeds in a sunflower. The Fibonacci sequence is directly related to the Golden Ratio. As one progresses further in the sequence, the quotient of consecutive numbers approaches the Golden Ratio more closely.

The Golden Ratio is the division ratio of a segment or other quantities in which the ratio of the whole to its larger part equals the ratio of the larger part to the smaller part. (a+b):a = a:b = 1.618033… The division ratio of the Golden Ratio calculated as a number by dividing these quantities is an irrational number with infinitely many digits, much like Pi.

The ratio of the Golden Ratio is often found in nature, such as in the arrangement of leaves and in the inflorescences of some plants. Through repeated rotation by the Golden Angle, new positions are continually created, such as for the leaf attachments of a flower or plant stems. As with any irrational number, exact overlaps never occur, minimizing the overlap of leaves, which impedes photosynthesis, overall.

The Golden Ratio is also valued as an ideal principle of aesthetic proportioning in artistic, architectural, and craftsmanship practice, such as in Greek temple construction or Leonardo da Vinci's Mona Lisa.