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The Golden Ratio
Apr 1, 2004

It is very obvious that there is an amazing system at work in the universe. Words are usually insufficient to explain this perfection. Therefore, one must refer to the different language and approach of mathematics. Characteristics found in events and structures that are similar, but seem unconnected with one another indicate that there is a Creator who is the Absolute Ruler of the entire universe. In this article, we will discuss a unique number in mathematics, the Golden Ratio, and its place in the universe, as well as its history, its usage in art and in aesthetics.

The appeal of the Golden Ratio to the human eye and brain has been scientifically tested. When subjects are presented with a range of rectangles, people invariably pick out as most pleasing ones those whose sides are of the Golden Ratio. This golden number is the basic number at work in aesthetics; there are even claims that the Golden Ratio was used by Leonardo da Vinci when painting the Mona Lisa, and by the Greeks in building the Parthenon. But the surprising thing is that a number deemed aesthetically pleasing by human beings also crops up in nature and science. In a newly published article in Physical Review B, it is stated that the Golden Ratio appears in the structures of some metals. The Golden Ratio is seen in the arrangement of seeds on flower heads, in the spirals of sea shells and galaxies, even in black holes. This ratio can be found almost everywhere in the universe.

Although the Greek mathematician Euclid first defined the Golden Ratio in around 300 BC, the followers of Pythagoras probably knew of it two centuries earlier. Euclid defined it as a line that can be divided into two unequal parts (Figure 1), where the ratio of the smaller part of the line to the longer part is the same as the ratio of the longer part to the whole. This ratio is 1.6180339887..., the Golden Number.


Figure 1: If you take a Golden Rectangle and take out a square, what remains is another, smaller Golden Rectangle.

What makes the Golden Ratio special is the number of mathematical properties it possesses. The Golden Ratio is the only number whose square can be produced simply by adding 1 and the reciprocal of which can be arrived at by subtracting 1. If you take a Golden Rectangle – that is a rectangle where the length-to-breadth ratio is equal to the Golden Ratio, and take out a square, what remains is another, smaller Golden Rectangle. Also, think of any two numbers. Make a third by adding the first and second, a fourth by adding the second and third, and so on. If you start with 7 and 11, then what you have is:

7, 11, 18, 29, 47, 76... When you have written down approximately 20 numbers, calculate the ratio of the last to the penultimate: the answer should approximate the Golden Number.

In mathematical terminology it is (an/an-1) equals to the Golden Ratio.


Figure 2: The ratio of each bone at the top of the hand to the bones at the bottom of the fingers is the GR.

Another example of the Golden Ratio is the Fibonacci Numbers, that is a number series where each number is simply the sum of the previous two numbers. These numbers are 1, 1, 2, 3, 5, 8, 13, 21, 34, 55... The ratio between any two successive Fibonacci numbers approaches the Golden Ratio as the numbers get larger. We can find the Fibonacci series particularly in the spirals of sea shells and in the arrangement of seeds on sunflower heads.

It was the elusive nature of the Golden Ratio that led the Italian friar and mathematician Luca Pacioli to equate it with the incomprehensibility of God. In the 15th century, he wrote a three-volume treatise, Divina Proportione (Divine Proportion), that was crucial in the dissemination of the Golden Ratio beyond the world of mathematics. After him, many artists, architects, and musicians used the Golden Ratio in their works; for example, musicians such as Debussy and Bartok and the architect Le Corbusier.


Figure 3: Pine-cones show the Golden Ratio spiral clearly.

Let’s examine the arrangement of leaves on the stem of the plant Phyllotaxis. As each new leaf grows, it does so at an angle offset from the leaf below. The most common angle between successive leaves is 137.5 degrees – the Golden Angle; 137.5=3d 360-360/G, where G is the Golden Ratio. Why does the Golden Ratio play a role in the arrangement of leaves? It is all down to the irrationality of the number. A new leaf must collect sunlight without throwing too much of a shadow on the leaves below. A plant must arrange its leaves in such a way that the greatest number can spiral around the stem before a new leaf can sprout immediately above a lower one – that is at 360 degrees. If the leaves were arranged at an angle of 120 degrees, then the leaves would grow as 3 separate columns with large gaps between them. This would effectively block out the sunlight to the lower leaves. If the angle were 50 degrees, then there would be spaces between the leaves. But, with an angle of 137.5, the maximum amount of leaves can be arranged with a minimum of space being left between the leaves.

The Golden Ratio also crops up in hard sciences. Let’s take a look at the growth of “quasi-crystals.” These maintain a five-fold symmetry, which means that they make a pattern that looks the same when rotated by multiples of one-fifth of 360 degrees. Since the time when these crystals were discovered in 1984, many physicists have been researching their properties. In Brookhaven National Lab in New York State, Tanhong Cai imaged the microscopic terrain of the surface of such crystals made from alloys of aluminum-copper-iron and aluminum-palladium-manganese. It is found that flat terraces are punctuated by abrupt vertical steps. The steps come in two predominant sizes, with the ratio of the heights of these two steps being the Golden Ratio. This fact was discovered in 2002.


Figure 4: A cauliflower has a center point where the florets are smallest, and they are organized in spirals around this center in both directions

The most surprising place where the Golden Ratio appears is in black holes, a discovery made by Paul Davies of the University of Adelaide in 1989. Black holes and other self-gravitating bodies, such as the sun, have a negative specific heat. This means that they get hotter as they lose heat. In a spinning black hole there is an outward centrifugal force acting to prevent any shrinkage of the hole. The force depends on how fast the black hole is spinning. It turns out that at a critical value of the spin (when the ratio between the square root of the mass value and the square root of the spinning parameter is equal to the golden ratio), a black hole flips from having negative to positive specific heat. In other words, the Golden Ratio determines the character of the black hole.

Figure 2 shows the finger bones of a hand. The ratio of each bone at the top of the hand to the bones at the bottom of the fingers is the Golden Ratio. Pine-cones show the Golden Ratio spiral clearly (Figure 3). If one looks carefully at an ordinary cauliflower, one can see a center point where the florets are smallest, and the florets are organized in spirals around this center in both directions (Figure 4). The flower, Echinacea Purpura, has the same spirals (Figure 5).


Figure 5: The flower, Echinacea Purpura, has the same GR spirals.

With the help of developing science, new examples of the Golden Ratio are waiting to be discovered in the universe. The latest discoveries demonstrate that by using this ratio in technology, new products which make our lives easier will soon be available to all. This mystery that is spread throughout the universe may be an opportunity to renew and change our points of view on life.


• Chown, Marcus, “Why Should Nature Have a Favorite Number,” New Scientist, 21-28 December 2002, 55-56.