The amazing Golden Ratio
Artists reckon that the "Golden Ratio", also called the "Golden Section Phi" and nature's most astonishing number, is the ratio that controls the proportions of all beautiful objects, writes Assem Deif*
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Clockwise from top: Giza Pyramids; leaf arrangements around a stem and plant branching in a fibonacci formation; golden spiral; pascal triangle; panel of Khesi-Re
It is said that a well-proportioned face must lie in what is called a "golden rectangle" of dimensions in the ratio of approximately 1 to 1.6. Not only living forms, but also works of art and buildings, including the splendid domes of Persia and the Athens Parthenon, are found to adhere to this rule. The ratio became even more pronounced during the European Renaissance, when Leonardo Da Vinci studied the physical proportions of man and portrayed them in his unfinished canvas of St Jerome along with other works such as the "Mona Lisa" and the "Vitruvian Man".
So, quite apart from the other mathematical constants, Pi = 3.14, e = 2.718 (Euler Number), Gamma = 0.577 (Euler Constant), and i = sqrt(- 1) which possess mathematical properties only, the Golden Ratio Phi = 1.618... has an additional aesthetic feature. Since mathematicians in ancient times were often poets and philosophers who believed in the uniformity of nature, mathematics served to satisfy their need to understand the world around them and to resolve its secrets. Hence, they started tracking this constant in everyday objects such as plants and animals, and discovered that the proportions of many conformed with this ratio. This was why, by the time of the Renaissance, it had become known as the "Divine Proportion".
Mathematically, the Golden Ratio appears as the limit of several sequences, the most important of which are those that satisfy the recurrence relation s(n+1) = s(n) + s(n-1) where s (n) is the nth term of the sequence, whereas s(n- 1) and s(n+1) are respectively the preceding and succeeding terms, as in the famous Fibonacci sequence
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ....
in which any term is the additive sum of its two preceding terms, or in any other sequence satisfying the above recurrence, even if we start with any two numbers such as 5 and 2; that is 5, 2, 7, 9, 16, 25... We always have that the quotient of two successive terms s(n+1)/s(n) approaches a certain number called Phi. To calculate the value of Phi from the above recurrence relation written as s(n+1)/s(n) = 1 + s(n-1)/s(n), it turns out, in the limiting situation, that Phi = 1 + 1/phi, or that Phi equals half of the quantity 1+sqrt(5) = 1.618...
Historians trace the Golden Ratio back to Euclid, yet it appears that even before him it was governing the dimensions of monuments in ancient Egypt. The most pronounced of these is the Great Pyramid. The dimensions of the inner triangle (the so-called "Egyptian triangle") of Khufu's Pyramid, for instance, in Royal cubits (one cubit equalling roughly 0.524 metres), are (220c, 280c, 356c) i.e. in the ratios 1 : sqrt(Phi) : Phi. That the foregoing relation is not a matter of coincidence is discussed elsewhere. However, the Great Pyramid is not the only structure from ancient Egypt that complies with constants like Pi or Phi; Schwaller De Lubicz, who studied the temples of Upper Egypt from 1937 to 1952, collected massive amounts of evidence to show that the Egyptians used the Golden Ratio in many ways both in the architecture of their temples and in their drawings. So whereas, prior to De Lubicz's research, the discovery of the "golden rule" was generally credited to the Greeks (although some historians denied this), the findings of such Egyptologists as De Lubicz and Fliders Petrie produced irrefutable proof that the Egyptians had a mathematical understanding of these constants, the ratios, not the symbol, 1000 earlier. Petrie, for example, noticed that the dimensions of many Egyptian tombs, especially those of a parallelepiped structure, adhered to the ratios 1 : Phi : Phi square. The same ratio also appears in a grid surrounding a human body depicted in the royal tomb of Amenhotep III in the Valley of the Kings.
There was much cross-culture between the Egyptian and Greek civilisations in the cities of the north coast during the Hellenistic era, particularly in Alexandria where Egyptian and Greek scientists studied together at the Mouseion. Among them, in the third century BC, was one who was considered, par excellence, the most reputed scientist in antiquity, the great Euclid. Historians call him Euclid of Alexandria without precluding the possibility he might have been Egyptian. It was in Alexandria that Euclid wrote his opus The Elements, still the most famous mathematical work ever written. Since the invention of the Gutenberg machine this work, compiled in 13 volumes, has been printed more than any book apart from the Bible. (Euclid introduced the ratio, obtained from extreme and mean ratio of three collinear points, in volume VI). The Arab mathematician Al-Haggag produced the first translation of The Elements into Arabic, and as the original Greek work was subsequently lost it was only through the Arabic translation that the book became known to the rest of the world.
Euclid was only one of the scientists who performed research at the Mouseion. The great intellects of the day flocked to Alexandria, and among them we encounter such names as Eratosthenes, Archimedes, Apollonius, Menelaus, Heron, Nicomachus, Ptolemy, Diophantus, Pappus, Galen, Theon and his daughter Hypatia. Greek scholars were visiting Egypt even before the Mouseion was founded, including Thales, Socrates, Plato and Aristotle, and above all Pythagoras who spent 22 years in Egypt about 600 BC and announced his theory only after leaving. Records tell us that the Egyptians were aware of the triangle 3:4:5 which Pythagoras himself called the "Holy Triangle". Eight pyramids from the fourth and fifth dynasties have their inner triangle conforming to these ratios.
There are many applications for the Golden Ratio varying from mathematical optimisation to architecture. We find it in other sciences too. In biology it controls the distribution of the leaves around the stem of a plant such that they receive the maximum amount of light. It was found that the distribution of their angle of rotation or distances from one another followed terms as in the sequence 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34, in which the numerator a(n) of the quotients represents the number of turns we climb, say, around the stem in such that any one leaf returns to a position exactly above the point where it started, whereas the denominator b(n) represents the number of leaves in between. This ratio differs from plant to plant, yet it varies on average between 1/2 and 1/3. The reader will notice that in the sequence both the numerators and denominators are terms in the Fibonacci sequence, in which a(n) and b(n) satisfy b(n+1) = b(n) + a(n+1). Since both a(n+1)/a(n) and b(n+1)/b(n) tend to Phi, it follows that a(n+1)/b(n) tends to Phi-1 = 1/Phi. As to the distribution of the leaves around the stem b (n)/a(n), it tends to Phi square for at each turn the leaves organise themselves on the stem.
We can elaborate further to calculate the divergence angle of any one leaf such that no two leaves would be above one another; allowing for maximum light exposure. Having that each complete turn of any one leaf in order to return to its original position travels an angle of a(n)x360 while passing by b(n) leaves in between, it follows that each leaf occupies an angle of a(n)x360/ b(n) = 360/(Phi square) almost 137 degrees.
Thus, if there are Phi square leaves per turn (or equivalently 1/Phi square turns per leaf), then each leaf gets the maximum exposure to light while casting the least shadow on the others. This also gives the best possible area exposed to falling rain, or, in the case of flowers, the best possible exposure to attract insects for pollination.
Still in biology, the Fibonacci sequence, of which the ratio of two successive terms tend in the limit to Phi, often appears in the reproduction of animals and plants. The figure depicts the increase in the number of leaves in some plants from row to row.
Likewise, in rabbit reproduction, say, starting with one pair of rabbits, at the end of the first month they will have mated but there is still one only pair. At the end of the second month they produce a new pair, so now there are two pairs of rabbits. At the end of the third month the original pair produces a second pair, making three pairs. At the end of the fourth month the original pair has produced yet another new pair, while the pair born two months before produces their first pair, making five pairs. And so on (assuming that each pair born consists of a male and a female rabbit).
A fascinating phenomenon associated with the Golden Ratio is its regular appearance in such objects of nature as nautili and other shells. Such shells imitate a curve called in mathematics the "logarithmic spiral", first investigated by Jacob Bernoulli in the 18th century. This was the second curve in history, next to the circle, to have its length calculated, since although it has an infinite number of loops, its length approaches a finite value.
One extremely amazing appearance of the logarithmic spiral in nature is associated with raptors. Since these predatory birds must keep the prey in sight all the time, and since their eyes are on the side of the head, a hawk or eagle swivels its head to one side at an angle of about 40 degrees and fixes its prey in this eye. Keeping its head fixed at that 40-degree angle, the bird then dives in a way which keeps the prey in sight in that one eye. The fixed angle of the head results in the bird's following a logarithmic spiral path that converges on its prey.
A special case of the logarithmic spiral is the "golden spiral". This is drawn either from the outside or the inside. In the first option, we start from a square of unit side-length, then extend it into a "golden rectangle" of base unity and height 1.618. Then we draw a quarter of a circle inside the square, and another one in the extension of radius 0.618 = 1/Phi, etc... making sure that each time we isolate a square from a golden rectangle. An interesting feature that I found with this spiral is that its inner area can almost be filled up (I said almost) by a set of adjacent Egyptian triangles.
As to the mathematical properties of Phi, we summarise them as follows: Phi = 1.618..., 1/ Phi = 0.618, Phi square = 2.618... and so on. For instance, Phi raised to the 11th power equals 199.00502... and 1/Phi raised also to the 11th power equals 0.00502. Also Phi has the peculiar continued fraction representation
Again, in terms of nested radicals:
Also, by writing Phi square = 1 + Phi, Phi cube = 1 + 2Phi, Phi power 4 = 2 + 3Phi, Phi power 5 = 3 + 5Phi, Phi power 6 = 5 + 8Phi, etc... we notice that the coefficients represent two Fibonacci sequences. The interesting properties of the golden ratio led the academic circle to issue a regular periodical called the Fibonacci Quarterly.
We can still deduce further properties: Phi power 10 = Phi power 9 + Phi power 8 = Phi power 8 + 2Phi power 7 + Phi power 6 = Phi power 7 + 3Phi power 6 + 3Phi power 5 + Phi power 4 = Phi power 6 + 4Phi power 5 + 6Phi power 4 + 4Phi power 3 + Phi power 2 = .... in which the coefficients in the polynomial represent successively rows of a "Pascal Triangle" , known to the Arabs and which represent the coefficients of a binomial expansion of any two numbers raised to a positive integer n.
The applications of the Golden Ratio in the physical sciences are endless, so we have restricted ourselves in this short article to a few of the most famous which yet have no complicated mathematical features.
* The writer is a professor of mathematics at Cairo University and Misr University for Science and Technology.