The Fibonacci Sequence In Nature - Photography InspirationThe Fibonacci sequence is a mathematical sequence which is echoed in nature and it is a fascinating subject for photographic exploration.
If you are in search of a summertime photography challenge, then why don't you base it around the Fibonacci Sequence? From the spiralling patterns in a sunflower seed head to the beautiful arrangement of leaves on an aloe vera plant - the structures that have the Fibonacci Sequence written into them are probably the most photogenic there are.
What is the Fibonacci Sequence?
The Fibonacci sequence is named after a Thirteenth-century Italian mathematician Leonardo of Pisa, who was referred to as Fibonacci. He is credited with spreading through Europe the use of the Hindu-Arabic numerical system together with the digits 0-9 and place value, how by which the value of a digit is dependent upon its position (units, tens, hundreds and so on).
As well as explaining the way to use the brand-new numerical system, Fibonacci's book, Liber Abaci (1202) addresses numerous mathematical problems. One of these problems pertains to how quickly rabbits may breed, supposing not one of the rabbits died and that the female at all times produced a brand-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 forth. 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 importance of those numbers has been understood after French mathematician Edouard Lucas studied them in closer detail within the 19th century while working on his own, similar, sequence of numbers. He gave Fibonacci's collection of numbers its name. Following Lucas's research, the numbers were more and more observed in structures within the flora and fauna - from the spiralling bracts of a pinecone and the florets on cauliflower to the number of petals on a sunflower.
It transpired that the sequence described something a lot more complicated than the breeding trend of rabbits.
Fibonacci Sequence as a Natural Growth Pattern
The Fibonacci sequence is all about growth; you're taking the information you could have beforehand to get the following piece of data. This is a very simple way of generating growth quickly and explains why the Fibonacci numbers appear in nature so frequently. The sequence applies to the growth of all living things, from a single plant cell to a honeybee's family tree; nature depends on simple operations to construct immensely complicated, often gorgeous, structures, and the Fibonacci sequence reflects this.
Only as recently as 1993 were the Fibonacci numbers scientifically proven to exist in nature, and there's still a lot to learn.
Relationship Between Fibonacci Sequence and the Golden Ratio
Closely associated with the Fibonacci sequence is the golden ratio, or phi - a number that has fascinated 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 beneath.
The golden ratio has a value of roughly 1.618034.
The fundamental factor concerning 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 as far as you can get from a fraction, which is why it's important for organic growth.
Take seeds in a flower head as an example. Small seeds are produced at the centre of the flower and then spread outwards. To fill the space efficiently, each new seed appears at an angle offset from the previous one. As the process repeats, a spiral naturally occurs. if the turn were a simple fraction the seeds would ultimately stack up in lines, leaving gaps in between, so the plant uses the golden ratio.
Furthermore, every new growth is scaled down every time; it must be large enough so that it doesn't die out yet sufficiently small it doesn't overlap. Only a reduction in dimension ratio of 0.618 - the inverse of the golden ratio will work.
Golden Numbers in the Fibonacci Sequence
If you divide every number in the Fibonacci sequence by the preceding one, the brand-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 among the reasons why they crop up in nature so frequently. Pinecones, as an example, have two sets of spiralling bracts; 8 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 quantity can spiral across the stem before a brand-new leaf grows directly above it. This guarantees that every leaf receives the maximum amount of daylight and catches as much rain as possible. As every 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 known as the golden angle, and it divides the entire 360° in the golden ratio, 1.618034...
It is estimated that 90% of all plants organize 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 discovered with 34, 55 and even 89 petals.
While the number of petals on some flower species, such as five-petal butter cups, is very precise, the quantity on many species can vary, with the average being a number within 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 might 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 dimension together to create a rectangle and proceed to add squares which 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 every square, you find yourself in a logarithmic spiral. This motif is found regularly in natural forms, including seed heads, shells and fern fronds.
The Fibonacci spiral may also be used as a composition guide in photography. Key components can be aligned across the spiral in much the same way as the rule of thirds. This type of composition is known as the Divine Composition with Fibonacci's Ratio. Fibonacci's Ratio is an impressive tool for composing your photographs, and it should not be brushed aside as a minor difference from the rule of thirds.
