How Do the Golden Ratio and Fibonacci Numbers Shape the World Around Us?
What is the Golden Ratio?
The golden ratio is a number which is obtained by dividing a line into two parts so that the longer part divided by the smaller part is also equal to the whole length divided by the longer part. It is represented by Greek letter phi, Φ. The golden ratio is also known as the golden mean or golden section. Other names include divine proportion, divine section, golden proportion, golden cut, nature golden ratio fibonacci, and golden number. The golden ratio in nature can be seen if observed carefully.
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Golden Ratio in Nature with Explanation
According to the definition of the golden ratio,
\[\frac{a+b}{a}\] = \[\frac{a}{b}\] = Φ
Then, \[\frac{a}{a}\] + \[\frac{b}{a}\] = Φ
⇒ 1 + \[\frac{1}{\phi }\] = Φ
⇒ Φ\[^{2}\] - Φ - 1 = 0
On solving, we have
Φ = \[\frac{1+\sqrt{5}}{2}\] and Φ = \[\frac{1-\sqrt{5}}{2}\]
Since we have defined Φ as the ratio of two positive numbers, therefore
Φ = \[\frac{1+\sqrt{5}}{2}\] ≈ 1.618
Fibonacci sequence and its relation with fibonacci golden ratio in nature.
The golden ratio is closely related with the Fibonacci sequence too. Mathematician Leonardo Fibonacci discovered the unique properties of Fibonacci sequence.
The Fibonacci sequence starts from 0, 1, 1, 2, 3, 5, 8, 13, 21 and so on. It follows the rule
F\[_{n}\] = F\[_{n-1}\] + F\[_{n-2}\] (for n>1) , where F\[_{n}\] is n\[^{th}\] Fibonacci number
This magical sequence ties directly with the Golden Ratio because if we take any two successive Fibonacci numbers, their ratio is very close to the Golden Ratio. As the numbers get higher, the ratio becomes closer to 1.625.
Fibonacci Spirals in Nature
The golden spiral in nature or divine ratio in nature can be observed in flower petals, shells, tree branches and in several other objects. Some of the examples golden section in nature are -
Shells:
The rectangle in which the ratio of the sides a/b is equal to the golden mean (phi) is known as a golden rectangle. The unique properties of the Golden Rectangle can be observed in shells. This shape can result in a nesting process that can be repeated into infinity — and which takes on the form of a spiral. It is called the logarithmic spiral, and it is also present in nature. The cochlea of the inner ear also follows this logarithmic spiral. It can also be seen in the horns of certain goats, and in the pattern of the spider's webs.
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Flower Petals:
The number of petals present in different flowers is another example of the Fibonacci sequence present in nature. Famous examples include the lily, which has three petals, buttercups, which have five, the chicory's 21, the daisy's 34, and so on. The explanation for this phenomena is the ideal packing arrangement of petals as selected by Darwinian processes. Each petal which is placed at 0.618034 per turn (out of a 360° circle) are the best possible exposure to sunlight and other factors.
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Tree Branches:
The Fibonacci sequence can also be seen in tree branches form or split. The main trunk will grow until it produces a branch, which further creates two growth points. Then, one of the new stems branches into two, while the other one lies dormant. This pattern of branching is repeated for each new stem. A good example is sneezewort. Root systems and even algae exhibit this pattern.
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Family Tree of a Rabbit:
Suppose a newborn pair of rabbits, one male and one female, is put in the wild. The rabbits’ mate at the age of one month. By the end of its second month, a female can produce another pair of rabbits. Suppose that the rabbits never die and also each female always produces one new pair, with one male and one female, every month from the second month on. How many pairs will form in one year?
The answer is found in a series of numbers now known as the Fibonacci series. Consider pair A of rabbits giving birth to pairs B, C, D and E. Each of these new pairs gives birth to other pairs B1, B2, B3, C1, and C2, who in turn give birth to B11, etc. At the end of each month, the total population of rabbits forms the Fibonacci series.
Seed Heads:
Fibonacci numbers can also be seen in the arrangement of seeds on flower heads. Seeds are produced at the centre and then migrate towards the outside to fill all the remaining space. Sunflowers is a great example of these spiralling patterns. The reason behind this arrangement forms an optimal packing of the seeds so that, no matter how large the seed head, they are uniformly packed at any stage. All the seeds being the same size, no crowding in the centre and not too sparse at the edges.
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FAQs on Golden Ratio & Fibonacci: Unlocking Mathematical Patterns in Nature
1. What is the Fibonacci sequence?
The Fibonacci sequence is a special series of numbers that starts with 0 and 1. Each number that follows is the sum of the two numbers before it. The sequence begins: 0, 1, 1, 2, 3, 5, 8, 13, 21, and continues indefinitely. This pattern is not just a mathematical curiosity; it frequently appears in natural phenomena.
2. What is the Golden Ratio and what is its value?
The Golden Ratio, often represented by the Greek letter phi (φ), is an irrational number with an approximate value of 1.618. It is defined as the ratio that exists when a line is divided into two parts, such that the ratio of the whole length to the longer part is equal to the ratio of the longer part to the shorter part. This proportion is considered to be particularly aesthetically pleasing.
3. How are the Fibonacci sequence and the Golden Ratio connected?
The connection between the Fibonacci sequence and the Golden Ratio is one of the most fascinating aspects of mathematics. If you divide any number in the Fibonacci sequence by the one that precedes it, the result gets progressively closer to the Golden Ratio (φ ≈ 1.618). For example, 5/3 = 1.666..., 8/5 = 1.6, 13/8 = 1.625, and so on. The higher the numbers, the more precise the approximation becomes, linking the additive sequence to this fundamental ratio.
4. What are some common examples of the Fibonacci sequence in nature?
The Fibonacci sequence is remarkably prevalent in the natural world, often appearing in the way living things grow. Some key examples include:
Flower Petals: Many flowers have a number of petals that is a Fibonacci number, such as lilies (3 petals), buttercups (5 petals), or delphiniums (8 petals).
Seed Heads: The seeds in a sunflower head are arranged in two sets of spirals, with the number of spirals in each set typically being consecutive Fibonacci numbers (e.g., 34 and 55).
Pinecones and Pineapples: The scales on pinecones and pineapples are also arranged in spirals that follow Fibonacci numbers.
Tree Branches: The way trees branch can follow a Fibonacci pattern, where a trunk grows and produces a branch, creating two growth points. Then, one of the new stems branches, and so on.
5. Why is the Golden Ratio often linked to beauty and design?
The Golden Ratio is frequently linked to beauty because it creates proportions that the human brain finds inherently balanced and harmonious. Since this ratio appears so often in nature—from the proportions of the human body to the shape of shells—we are naturally attuned to find it pleasing. For this reason, artists, architects, and designers have used it for centuries in famous works like the Parthenon and the Mona Lisa to create compositions with a sense of natural order and aesthetic appeal.
6. Is the appearance of these numbers in nature a coincidence, or is there a functional reason?
While it might seem like a coincidence, the prevalence of the Fibonacci sequence and Golden Ratio in nature is often due to efficiency. In plants, this pattern allows for the optimal arrangement of parts. For example, in phyllotaxis (the arrangement of leaves on a stem), growth patterns based on the Golden Angle (~137.5°) ensure that new leaves do not block the sun from older ones, maximizing sunlight exposure for the entire plant. Similarly, Fibonacci spirals provide the most efficient way to pack seeds in a flower head, ensuring no wasted space.
7. What is the main difference between how the Fibonacci sequence and the Golden Ratio appear in the world?
The primary difference lies in what they represent. The Fibonacci sequence typically appears as counts of discrete items—for instance, 5 petals on a flower, 8 spirals on a pinecone, or 3 main branches on a plant. It answers the question "how many?". In contrast, the Golden Ratio appears as a proportion or a growth factor. It governs the dimensions and scaling of shapes, like the ratio of lengths in a spiral or the proportions of a face. It answers the question "what is the relationship between the sizes?"
