Expository Essay on Golden Ratio and Fibonacci Numbers in Nature

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Most people fear Mathematics and evade it as soon as they can especially when specializing in their careers. However, with an open mind, an explorative mind, Math is one of the most interesting essentials, especially in Science. Mathematics is part of life; it is part of nature and our day-to-day activities. More specifically, in Mathematics, there exists the Golden Ratio and the Fibonacci Numbers which are major interesting and intriguing discoveries of Mathematics. For that specific reason, the two sets (Golden Ratio and Fibonacci numbers) are entities worth exploring. They are wonderful n that they exist in nature. They exist in plants, animals, and other constituents in nature. They are natural and they are a representation of how Math is part of nature and how Math helps us understand nature. Consequently, this essay will concentrate on the two sets of wonders and show how Math is fun and is part of nature.

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Fibonacci Numbers and the Golden Ratio in Nature

To begin with, the Fibonacci numbers are can also be termed as the Fibonacci series/sequence. The sequence is referred to as nature's numbering system (Parveen, 2016). It comprises a set of numbers starting from 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, etc. whereby each subsequent number is the sum of the preceding two numbers. Deriving ratios from the consecutive numbers in the series, the resulting ratio becomes very close to the Golden Ratio which is approximately 0.61803 which is equivalent to an angle referred to as the Golden angle that is equivalent to 222.4922 or 222.5 degrees in terms of rotations (Nature, The Golden Ratio and Fibonacci Numbers, 2016). The close relationship between the Fibonacci and the Golden ratio and the fact that they exist in nature attracts much curiosity from mathematicians and scientists in general.

The Golden ratio is also referred to as phi = 0.6183 which is the Greek letter f. The interesting thing about the ratio is that it cannot be written as a simple fraction. It is an irrational number and can even be said to be proficient at not being any fraction.' In other words, the ratio cannot be sufficiently represented as a fraction. In plants, the ratio is said to be the best solution through which the plant cells turn such that they are able to maintain a compact non-straight arrangement thus enabling the plant to obtain the round shape. On a broader view, the Golden ratio offers an arrangement of the cells which maximizes the number of packed seeds in a particular seed-head (Parveen, 2016). The golden ratio is observed in mostly observed in the flowers of most plants. Take a sunflower for instance; it naturally forms spirals for the arrangement of the seeds (Nature, The Golden Ratio and Fibonacci Numbers, 2016). The seeds form from cells whereby, each newly formed cell forms after a specific turn which is consistent for each cell. Interestingly, the turn can be related to the Golden Angle of about 222.5 degrees which directly means that the turn can be associated with the Golden ratio of 0.61803

Closely related to this turn in sunflowers is the number of clockwise turns and anticlockwise that make the spiral. Notably, the clockwise turning spiral cells make up curved lines that sum up to 34 turns while the anticlockwise turning spiral cells sum up to 21 turns. The interesting bit about the number of turns is that they are consecutive numbers in the Fibonacci series. Even more interesting, they sum up to 55 which is a number in the Fibonacci series. To add to that, the two numbers, 21 & 34, yield a ratio of, 0.618 almost equal to the Golden Ratio. Further research about the nature of the sunflowers spiral shows that the flower has the numbers 89 and 144 which also happen to be in the Fibonacci series (Dvorsky, 2016). The numbers are a result of the total number of seeds per spiral of a sunflower.

As can be noted so far, the sunflower is a very good example of plants with the Fibonacci numbers and the Golden Ratio how they exist in nature. Other examples of plants that are characterized by the presence of the Fibonacci numbers are pineapples and pinecones. Observing the tapered pineapples and pinecones, one can make out a spiral arrangement of which is either turning clockwise or anticlockwise. The clockwise spiral turns are 13 in number while the anticlockwise turns are 8 in number (Dvorsky, 2016). Just as in the case of sunflowers, the numbers are consecutive on the Fibonacci series, sum up to a Fibonacci number, and are divided to yield a number close to the Golden ratio. The arrangement of branches on a plant, the arrangement of petals for flowers such as the rose flower, the arrangement of leaves on a branch, the arrangement of leaves of a succulent plant such as the Aloe Vera, sisal, etc. are all but an indication of the Fibonacci numbers and sequence in nature and the golden ratio.

Focusing on the animals, there exists a Golden rectangle whose sides x and y, yield a ratio x/y that is equal to phi. The rectangles can be repeated to infinity and can be viewed as a spiral. The human has the nose and mouth placed at golden sections as the distance between the eyes and the bottom of the chin. The human hand is such that the ratio between the forearm and the hand is a golden ratio. The proportions of the sizes of the phalanges in a human hand are also a golden ratio (Parveen, 2016). The snails' shells also form spirals that are forms of a golden ratio. The broader nature, such as the spiral galaxies also has the golden ratio. The microscopic constituents of nature, and specifically the DNA molecule, have dimensions equal to 21 A wide and 34 Along which are both Fibonacci numbers that yield a ratio close to the Golden ratio.


To sum it all up to the Golden ratio and the Fibonacci numbers are natural constituents of the plants and the animals and other natural phenomena. All these are enough evidence to show that Math is a major and crucial component of nature. Even the slightest the beauty of a plant, the slightest aspect such as the arrangement of the cells and seeds are not just by chance. They are all governed by Mathematics. In other words, the arrangements do not just occur by chance; they are part of a greater law of nature that is important in Mathematics. Mathematics is fun; it is a defining and crucial of nature, the Fibonacci numbers and the Golden ratio are the major proof to that fact.


Dvorsky, G. (2016). 15 Uncanny Examples of the Golden Ratio in Nature. Io9.gizmodo.com. Retrieved 10 November 2016, from http://io9.gizmodo.com/5985588/15-uncanny-examples-of-the-golden-ratio-in-nature

Nature, The Golden Ratio, and Fibonacci Numbers. (2016). Mathsisfun.com. Retrieved 10 November 2016, from https://www.mathsisfun.com/numbers/nature-golden-ratio-fibonacci.html

Parveen, N. (2016). Fibonacci in Nature. Jwilson.coe.uga.edu. Retrieved 10 November 2016, from http://jwilson.coe.uga.edu/emat6680/parveen/fib_nature.htm

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