Leaves and the Golden Ratio

When we look at the plants and trees around us, we see that their branches are covered in large numbers of leaves and in season, flowers. Looking at them from a distance, we might imagine that these branches and leaves are arranged haphazardly, at random. The fact is, however, that where the branches emerge from the stem or trunk, the sequence of the leaves on those branches and even the symmetric shapes of flowers are all established beforehand by means of fixed laws and miraculous measures. Plants have been implementing these laws to the letter ever since the day they were first created. In other words, no leaf or flower comes into being by coincidence.

Approximately how many branches a tree will have, where they will emerge, how many leaves there will be on a branch and the way these will be arranged are all determined beforehand. In addition, every plant has its own rules regarding branching and leaf order. Scientists can describe and classify plants solely in terms of these characteristic sequences. The extraordinary thing is that a poplar in China, for example, is aware of the same measures and rules as a willow in England, and implements them in the same way. Of course, it is not chance that creates these mathematical calculations unique to each plant, and in the most aesthetic manner. It is Omniscient Allah Who creates this beauty and this order with flawless calculation.

As is revealed in the Qur’an:


He to Whom the kingdom of the heavens and the earth belongs. He does not have a son and He has no partner in the Kingdom. He created everything and determined it most exactly. (Surat al-Furqan:2)

These sequences, which vary according to each variety of plant, may be circular or spiral. One of the most important consequences of this special arrangement is the way that one leaf does not cast shade on any other. The order in the arrangement of the leaves around the stem is set out in specific numbers according to what in botany is known as leaf divergence. This sequence in leaves is based on a complex calculation. If N is the number of turns from one leaf around the stem until we come to another leaf on the same plane, and if P is the number of leaves on each turn, then P/N is referred to as leaf divergence. These levels are ½ in grasses, 1/3 in marsh plants, 2/5 in fruit trees (for example, apples), 3/8 in species of bananas, and 5/13 in bulbous plants. 1

The way that every tree from the same species implements the ratio set out for its own species is a great miracle. How, for example, does a banana tree know about this ratio and act upon it? According to this calculation, when you start from one leaf and take 8 turns around it, you will come to another leaf on the same plane. And you will encounter three leaves on these turns. Wherever you may go, from South Africa to Latin America, the ratio will remain the same. Just this ratio in the arrangement of leaves is significant evidence that living things did not come into existence by coincidence, but that they were created in line with an exceedingly complex calculation, plan and system. It is Allah, the Lord of sublime knowledge and wisdom, Who encodes such a ratio in the genetic make-up of living things and creates this information and feature for them.

One of the most frequently encountered arrangements in trees is pairs of leaves and branches that emerge exactly opposite one another. After the seed sprouts, it opens up two rudimentary leaves, which leaves are set out 180 degrees opposite one another. The two leaves that sprout above these first two also grow opposite one another and at right angles to the first pair, for the greatest possible distribution between them. In this way, there are now four leaves spaced every 90 degrees on the stalk. In other words, if we look at this branch from above, we see that the leaves are so arranged as to constitute a square, and that the upper leaves thus do not shade the lower two. 2

This is a sight we are quite familiar with. However, most people never think about why it is that seeds germinate up in this way. The fact is, however, that this is the result of planning and order. The aim behind it is to prevent leaves from shading one another and to enable them all to make maximum use of sunlight.

The more complex spiral form is also often to be seen. To observe this spiral action in plants, tie a thread to the base of one leaf and then extend the thread to the branches and knots, make a loop around the base of every leaf you come to, and keep the curves as regular as possible. Using this method, you will see that each leaf on an elm or lime tree is 180 degrees away from the neighbouring leaf on the branch; thus the thread will turn halfway round the branch for every next leaf. Leaves on beech trees are set 120 degrees apart, with 1/3 turns per leaf. The ratio for apple trees are 144 degrees and 2/5 turns, and for black pine, 5/13 turns. If you have an interest in mathematics, you will see that these exact figures cannot be the work of chance, and that each unit is the total of those preceding it (as shown below). Each two numbers exhibit the same simple calculation: 1, 1, 2 (1+1), 3 (1+2), 5 (2+3), 8 (3+5), 13 (5+8), 21 (8+13), 34 (13+21), 55 (21+34), 89 (34+55), 144 (55+89), 233 (89+144), 377 (144+233), and so on.

This special progression is known as a Fibonacci series, named after the mathematician Fibonacci who discovered it. This rule embodies aesthetic perfection, and is used as a basic measure in such disciplines as painting, sculpture and architecture. This same sequence is frequently encountered in nature, and serves as an important key to understanding the fine calculation and order in plants.

Ratios beyond 3/8 can be found in seaweed, cabbage, or in the arrangements of seeds on the head of a sunflower, which go in spirals in both directions. The florets of these plants turn in spirals as they circle around the center from right or left, and the number of seeds per turn in the spirals is determined according to the Fibonacci series. For example, the center of a daisy uses three consecutive fractions: 13/34, 21/55 and 34/89. In other words, the number of florets in each rotation around the center, and the angles involved, are all determined beforehand. 30The Fibonacci series appears very frequently in nature. The fractions produced using these numbers give us what is known as the Golden ratio. In other words, when we write down the consecutive fractions in the Fibonacci numbers, as shown below, the divisions that result possess this Golden ratio, signifying complete aesthetic perfection: 1/1, 1/2, 2/3, 3/5, 5/8, 8/13, 13/21, 21/34, 34/55, 55/89 . . . .

As we have seen, the sequence obtained by this means matches the consecutive numbers in the Fibonacci series. We see this sequencing in pine cones (5/8, 8/13), on pineapples (8/13), in the centers of daisies (21/34) and in sunflowers (21/34, 34/55, 55/89) in the numbers of righthand and lefthand spirals. The ratios emerging as a result imparts aesthetic beauty to flowers, trees, seeds, sea shells and a great many other living things in nature.

The place occupied in nature by the Golden ratio is by no means limited to this, but also manifests itself in the ideal leaf angles. As we know, plant leaves are arranged to make the maximum use of solar rays. For example, the angle between the leaves in a plant with a 2/5 leaf divergence is:

2 x 360 degrees / 5 = 144 degrees. 31

There are more numerical miracles in leaves. The surfaces of leaves also have structures that can be understood as the result of specific mathematical calculations. The vein from the center of a leaf (or midrib), and the smaller vessels extending from it to the outer edges of the leaf, and the tissues in between that are nourished by these, all endow the leaf with a distinctive shape and structure. Although leaves come in countless different shapes, they still preserve these same precise measurements.

The fact that leaves are arranged and shaped according to specific mathematical formulae is one of the most convincing proofs that they have been specially created. The sensitive measurements and balance we see in a plant’s molecules and in its DNA also appear in the plant’s external appearance. In addition to providing such vital functions as receiving maximum benefit from sunlight, these formulae bestow great beauty on the plant, and present an extraordinary picture when combined with the colors resulting from combinations of specific arrangements of molecules. This Golden Ratio is an aesthetic rule well known to and used by artists. Works of art produced in line with it possess an aesthetic appeal. The plants, flowers and leaves designed in accordance with this rule—in turn, imitated by human artists—are all examples of Allah’s sublime creative artistry.


Allah reveals in the Qur’an that He has created all things to a measure. Some of the relevant verses are:

As for the Earth, We stretched it out and cast firmly embedded mountains in it and made everything grow in due proportion on it. (Surat al-Hijr:19)

. . . Allah has appointed a measure for all things.(Surat at-Talaq: 3)

. . . Everything has its measure with Him. (Surat ar-Ra‘d: 8)

. . . Allah takes account of everything. (Surat an-Nisa’: 86)

1. Dr. Sara Akdik, Botanik, (“Botany”) İstanbul: Şirketi Mürettibiye Publishing, , 1961, p. 106.
2. Guy Murchie, The Seven Mysteries Of Life, 1978, Abd, Boston: Houghton Mifflin Company, p. 57


2010-09-16 17:27:51

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