Patterns In Nature: The Fibonacci Sequence
The Fibonacci sequence is clearly visible in some of natures most exquisite structures, and it is a fascinating subject for photographic exploration.
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Photography By Numbers
If you're looking for a summer photo project then why not base it around the Fibonacci sequence? From the spiralling patterns in a sunflower seed head to the exquisite arrangement of leaves on an aloe vera plant - the structures that have the Fibonacci clearly written into them are some of the most photogenic there are.
A series of images that capture flowers with 3, 5, 8, 13, 21 and 34 petals, for example, could be a great starting point...
The Fibonacci Sequence
The Fibonacci sequence is named after a 13th-century Italian mathematician Leonardo of Pisa, who became known as Fibonacci. He is credited with spreading throughout much of Europe the use of the Hindu-Arabic numerical system including the digits 0-9 and place value, the way in which the value of a digit depends on its position (units, tens, hundreds and so on).
As well as explaining how to use the new numerical system, Fibonacci's book, Liber abaci (1202) addresses a number of mathematical problems. One of these problems relates to how quickly rabbits could breed, supposing none of the rabbits died and that the female always produced a new pair (one male and one female). Rabbits can reproduce at the age of one month, so at the end of its second month, a female can produce another pair of rabbits. The total number of pairs of rabbits at the beginning of each month followed a pattern: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, and so on. Each number is the sum of the previous two. The numbers get large very quickly, and the sequence is infinite.
It wasn't until much later that the significance of these numbers was understood after French mathematician Edouard Lucas studied them in closer detail in the 19th century while working on his own, similar, sequence of numbers. He gave Fibonacci's series of numbers its name. Following Lucas's research, the numbers were increasingly observed in structures in the natural world - from the spiralling bracts of a pine cone and the florets on a cauliflower to the number of petals on a sunflower.
It transpired that the sequence described something much more complex than the breeding pattern of rabbits.
A Natural Growth Pattern
The Fibonacci sequence is all about growth; you take the information you have beforehand to get the next piece of information. This is a very simple way of generating growth quickly and explains why the Fibonacci numbers appear in nature so often. The sequence is applicable to the growth of all living things, from a single plant cell to a honey bee's family tree; nature relies on simple operations to build immensely complex, often beautiful, structures, and the Fibonacci sequence reflects this.
Only as recently as 1993 were the Fibonacci numbers scientifically proven to exist in nature, and there is still much to learn.
The Golden Ratio
Closely related to the Fibonacci sequence is the golden ratio, or phi - a number that has facinated mankind for many centuries.
The golden ratio can be derived by dividing a line segment at the unique point where the ratio of the whole line to the largest segment is the same as the ratio of the large segment to the small segment. This is illustrated in the diagram below.
The golden ratio has a value of approximately 1.618034.
The fundamental thing about the golden ratio is its mathematical properties. It is an irrational number, meaning that it can't be expressed as a fraction (0.25 is ¼, 0.5 is ½, and so on). It is in fact as far as you can get from a fraction, which is why it is important for biological growth.
Take seeds in a flower head for example. Small seeds are produced at the centre of the flower and then spread outwards. In order to fill the space efficiently, each new seed appears at an angle offset from the preceding one. As the process repeats, a spiral naturally occurs. if the turn were a simple fraction the seeds would eventually stack up in lines, leaving gaps in between, so the plant uses the golden ratio.
Furthermore, each new growth is scaled down each time; it must be big enough so that it doesn't die out yet small enough it doesn't overlap. Only a reduction in size ratio on 0.618 - the inverse of the golden ratio will work.
If you divide each number in the Fibonacci sequence by the preceding one, the new sequence converges towards the golden ratio.
1/1 = 1, 2/1 = 2, 3/2 = 1.5, 5/3 = 1.66, 8/5 = 1.6 13/8 = 1.625, 21/13=1.615, 34/21 = 1.619, 55/34 = 1.618
Fibonacci numbers are whole number approximations of the golden ratio, which is one of the reasons why they crop up in nature so often. Pine cones, for example, have two sets of spiralling bracts; eight in one direction and 13 in the other - two consecutive Fibonacci numbers.
The Fibonacci Sequence in Nature
The leaves of a plant are arranged in such a way that the maximum number can spiral around the stem before a new leaf grows directly above it. This ensures that each leaf receives the maximum amount of sunlight and catches as much rain as possible. As each new leaf grows, it does so at an angle offset from that of the leaf below. The most common angle between successive leaves is 137.5°. This is called the golden angle, and it divides the complete 360° in the golden ratio, 1.618034...
It is estimated that 90% of all plants arrange their leaves in a way that involves the Fibonacci numbers.
Many flowers have three, five or eight petals - all Fibonacci numbers. Corn marigolds have 13 petals; some asters have 21 petals; daisies can be found with 34, 55 or even 89 petals.
While the number of petals on some flower species, such as five petal butter cups, is very exact, the number on many species can vary, with the average being a number in the Fibonacci sequence. Double Fibonacci numbers appear in flowers too; for example, six-petal daffodils. There are of course exceptions to this rule, such as rock stonecrop, which has seven petals. And, of course, petals may fall off as the plant grows.
Natures Building Blocks
The Fibonacci sequence can be used to create golden rectangles.
Place two squares of the same size together to create a rectangle and continue to add squares that are the same length as the longest side of the rectangle (1+2=3, 2+3=5 and so on).
If you join the corners of each square you end up with a logarithmic spiral. This motif is found frequently in natural forms, including seed heads, shells and fern fronds.
The Fibonacci spiral can also be used as a composition guide in photography. Key elements can be aligned around the spiral in much the same way as the rule of thirds. This form of composition is called the Divine Composition with Fibonacci’s Ratio. Fibonacci’s Ratio is a powerful tool for composing your photographs, and it shouldn’t be dismissed as a minor difference from the rule of thirds.
Last updated on: Tuesday 24th October 2017
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