The Fibonacci sequence is a series of numbers where each number is the sum of the two
preceding ones. It begins with 0 and 1, and the sequence looks like this:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, ...
Here’s how the sequence is formed:
● Start with the numbers 0 and 1.
● Add them together to get the next number: 0 + 1 = 1.
● Add the last two numbers (1 + 1) to get the next: 1 + 1 = 2.
● Continue adding the previous two numbers in the sequence to find the next one.
The Fibonacci sequence is named after Leonardo of Pisa, who is more commonly known as
Fibonacci. He was an Italian mathematician who lived during the Middle Ages, around the 12th
and 13th centuries. Fibonacci introduced this sequence to the Western world in his 1202 book
Liber Abaci (The Book of Calculation).
In Liber Abaci, Fibonacci used the sequence to solve a problem about the growth of a
population of rabbits. The problem asked how many pairs of rabbits would be produced in a
year, assuming that each pair produces a new pair every month starting from the second month.
The resulting number of pairs each month follows the Fibonacci sequence.
An important consideration here is that Fibonacci was not the first to discover this sequence.
Similar sequences were studied in ancient India as early as the 6th century, where they
appeared in the context of Sanskrit poetry. The mathematician Virahanka used a similar
sequence to describe patterns in poetic meters. What Fibonacci did was to introduce this
mathematical concept to Europe, where it would eventually become a key part of mathematical
study.
The appearance of the Fibonacci sequence in nature is not just a coincidence. It often arises due to processes that involve growth, optimal packing, and efficient resource distribution. Manifestation of Fibonacci sequence into the nature:
One of the most famous examples of the Fibonacci sequence in nature is the arrangement of
petals in flowers. Many flowers have number of petals that is a Fibonacci number—3 petals on a
lily, 5 on a wild rose, 13 on a daisy, and so on.
The arrangement of these petals is not random but rather the result of phyllotaxis*, a pattern of
growth that optimizes exposure to sunlight and air. The phyllotaxis often follows the golden
angle (approximately 137.5 degrees), an angle that is closely related to the Fibonacci sequence.
By spacing each leaf or petal according to this angle, the plant ensures that each one is
optimally positioned for photosynthesis.
Similarly, the pattern of seeds in sunflower heads often follows a Fibonacci sequence. The
seeds arrange themselves in spirals that curve both to the left and to the right, with the number
of spirals in each direction usually being consecutive Fibonacci numbers. This arrangement is
the most efficient way to pack seeds in a circular area, maximizing space and ensuring that
each seed has equal access to nutrients.
Another striking example is the spiral shape of certain shells, such as those of nautilus
mollusks. These shells grow in a logarithmic spiral*, a shape that can be mathematically related
to the Fibonacci sequence. The logarithmic spiral allows the shell to grow without changing its
shape, which is essential for maintaining the structural integrity and balance of the organism.
Interestingly, this same spiral pattern can be observed on a much larger scale, such as in the
arms of spiral galaxies. The spirals of galaxies follow a logarithmic pattern that is consistent with
the Fibonacci sequence, suggesting that similar physical principles govern both the microscopic
and cosmic scales.
The arrangement of leaves around a stem, known as phyllotaxis, often follows a Fibonacci
sequence. This arrangement helps ensure that each leaf has optimal exposure to sunlight and
air, without shading its neighbors. If you examine the branching patterns of some trees or the
spiral arrangements in pinecones and pineapples, you'll notice Fibonacci spirals.
The reason for this lies in the efficiency of growth and resource distribution. Plants grow new
leaves, branches, and flowers in a way that minimizes competition between them for sunlight
and other resources. The Fibonacci sequence offers an ideal blueprint for this type of
arrangement, allowing for maximal exposure and minimal overlap.
The Fibonacci sequence is closely related to another mathematical concept known as the
golden ratio (approximately 1.618). As you progress further along the Fibonacci sequence, the
ratio of consecutive Fibonacci numbers approaches the golden ratio. This ratio, denoted by the
Greek letter φ (phi), is well-known for its aesthetic properties and its frequent appearance in art,
architecture, and design.
In nature, the golden ratio is often associated with structural stability and aesthetic appeal. The
proportions found in the Fibonacci sequence and the golden ratio are not only pleasing to the
eye but also provide a practical framework for growth and organization. This is why many
natural forms, from the shape of seashells to the spiral arrangement of seeds, align with these
mathematical principles.
1. Resource Optimisation
2. Waste Minimization
3. Efficient Growth
4. Self-Similarity
5. Aesthetic Appeal
The Fibonacci sequence is not just a mathematical curiosity or a natural phenomenon but it also
has practical applications in various fields:
1. Computer Science: The Fibonacci sequence is used in algorithms, particularly in data
structures like Fibonacci heaps including the analysis of the efficiency of certain
algorithms, especially in recursive functions and the calculation of the greatest common
divisor (GCD).
2. Stock Market Analysis: In finance, the Fibonacci sequence is used in technical analysis
to predict market movements. Traders use Fibonacci retracement levels to identify
potential support and resistance levels based on the idea that markets often move in
waves that reflect Fibonacci ratios.
3. Art and Architecture: The Fibonacci sequence and the golden ratio have been used by
artists and architects to create aesthetically pleasing compositions. The Parthenon in
Greece, the works of Leonardo da Vinci, and the design of modern buildings often
incorporate these mathematical principles to achieve balance and harmony.
4. Biology and Medicine: The Fibonacci sequence has been used in modeling population
growth, understanding patterns in genetics, and even in the study of DNA. The double
helix structure of DNA and the branching patterns of blood vessels and nerves show a
remarkable correspondence to Fibonacci ratios.
The presence of the Fibonacci sequence in nature is a reminder of the deep connections between mathematics and the natural world. It shows us that the patterns and numbers we study in abstract mathematical terms often have real-world applications that govern the beauty and functionality of the world around us. To conclude, the Fibonacci sequence is more than just numbers on a page—it’s a universal principle that guides the growth, structure, and beauty of life itself.
Phyllotaxis: The arrangement of leaves on a plant stem, often following specific
mathematical patterns such as the Fibonacci sequence.
Logarithmic Spiral: A curve that appears in many natural forms, where the distance
between successive turns of the spiral increases in a geometric progression. It is closely
related to the Fibonacci sequence and the golden ratio.