What do the pyramids in Egypt, Leonardo do Vinci's portrait of the Mona Lisa, sunflowers, the snail, the pine cone and your fingers all have in common?
The answer to this question lies hidden in a sequence of numbers discovered by the Italian mathematician Fibonacci. The characteristic of these numbers, known as the Fibonacci numbers, is that each one consists of the sum of the two numbers before it. (1)
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, …
Fibonacci numbers have an interesting property. When you divide one number in the sequence by the number before it, you obtain numbers very close to one another. In fact, this number is fixed after the 13th in the series. This number is known as the "golden ratio."
233 / 144 = 1.618
377 / 233 = 1.618
610 / 377 = 1.618
987 / 610 = 1.618
1597 / 987 = 1.618
2584 / 1597 = 1.618
THE HUMAN BODY AND THE GOLDEN RATIO
When conducting their researches or setting out their products, artists, scientists and designers take the human body, the proportions of which are set out according to the golden ratio, as their measure. Leonardo da Vinci and Le Corbusier took the human body, proportioned according to the golden ratio, as their measure when producing their designs. The human body, proportioned according to the golden ratio, is taken as the basis also in the Neufert, one of the most important reference books of modern-day architects.
The "ideal" proportional relations that are suggested as existing among various parts of the average human body and that approximately meet the golden ratio values can be set out in a general plan as follows: (2)
The M/m level in the table below is always equivalent to the golden ratio. M/m = 1.618
The first example of the golden ratio in the average human body is that when the distance between the navel and the foot is taken as 1 unit, the height of a human being is equivalent to 1.618. Some other golden proportions in the average human body are:
The distance between the finger tip and the elbow / distance between the wrist and the elbow,
The distance between the shoulder line and the top of the head / head length,
The distance between the navel and the top of the head / the distance between the shoulder line and the top of the head,
The distance between the navel and knee / distance between the knee and the end of the foot.
The Human Hand
Lift your hand from the computer mouse and look at the shape of your index finger. You will in all likelihood witness a golden proportion there.
Our fingers have three sections. The proportion of the first two to the full length of the finger gives the golden ratio (with the exception of the thumbs). You can also see that the proportion of the middle finger to the little finger is also a golden ratio. (3)
You have two hands, and the fingers on them consist of three sections. There are five fingers on each hand, and only eight of these are articulated according to the golden number: 2, 3, 5, and 8 fit the Fibonacci numbers.
The Golden Ratio in the Human Face
There are several golden ratios in the human face. Do not pick up a ruler and try to measure people's faces, however, because this refers to the "ideal human face" determined by scientists and artists.
For example, the total width of the two front teeth in the upper jaw over their height gives a golden ratio. The width of the first tooth from the centre to the second tooth also yields a golden ratio. These are the ideal proportions that a dentist may consider. Some other golden ratios in the human face are:
Length of face / width of face,
Distance between the lips and where the eyebrows meet / length of nose,
Length of face / distance between tip of jaw and where the eyebrows meet,
Length of mouth / width of nose,
Width of nose / distance between nostrils,
Distance between pupils / distance between eyebrows.
Golden Proportion in the Lungs
In a study carried out between 1985 and 1987 (4), the American physicist B. J. West and Dr. A. L. Goldberger revealed the existence of the golden ratio in the structure of the lung. One feature of the network of the bronchi that constitutes the lung is that it is asymmetric. For example, the windpipe divides into two main bronchi, one long (the left) and the other short (the right). This asymmetrical division continues into the subsequent subdivisions of the bronchi.(5) It was determined that in all these divisions the proportion of the short bronchus to the long was always 1/1.618.
THE GOLDEN RECTANGLE AND THE DESIGN IN THE SPIRAL
A rectangle the proportion of whose sides is equal to the golden ratio is known as a "golden rectangle." A rectangle whose sides are 1.618 and 1 units long is a golden rectangle. Let us assume a square drawn along the length of the short side of this rectangle and draw a quarter circle between two corners of the square. Then, let us draw a square and a quarter circle on the remaining side and do this for all the remaining rectangles in the main rectangle. When you do this you will end up with a spiral.
The British aesthetician William Charlton explains the way that people find the spiral pleasing and have been using it for thousands of years stating that we find spirals pleasing because we are easily able to visually follow them. (6)
The spirals based on the golden ratio contain the most incomparable designs you can find in nature. The first examples we can give of this are the spiral sequences on the sunflower and the pine cone. In addition to this, an example of Almighty God's flawless creation and how He has created everything with a measure, the growth process of many living things also takes place in a logarithmic spiral form. The curves in the spiral are always the same and the main form never changes no matter their size. No other shape in mathematics possesses this property. (7)
1- Guy Murchie, The Seven Mysteries of Life, First Mariner Boks, New York, pp. 58-59.
2- J. Cumming, Nucleus: Architecture and Building Construction, Longman, 1985.
3- Mehmet Suat Bergil, Dogada/Bilimde/Sanatta, Altin Oran (The Golden Ratio in Nature/Science/Art), Arkeoloji ve Sanat Yayinlari, 2nd Edition, 1993, p. 87.
4- A. L. Goldberger, et al., "Bronchial Asymmetry and Fibonacci Scaling." Experientia, 41 : 1537, 1985.
5- E. R. Weibel, Morphometry of the Human Lung, Academic Press, 1963.
6- William Charlton, Aesthetics: An Introduction, Hutchinson University Library, London, 1970.
7- Mehmet Suat Bergil, Doðada/Bilimde/Sanatta, Altýn Oran (The Golden Ratio in Nature/Science/Art), Arkeoloji ve Sanat Yayinlari, 2nd Edition, 1993, p. 77.