Patterns in Nature: The Beauty of Fibonacci Numbers and the Golden Ratio

Nature and mathematics are very similar and closely interlinked. Most patterns in nature occur in various mathematical sequences, and the evidence of this is unbelievably fascinating.     

Everything you see and experience around you is the result of several natural factors coming together. In the same way, several numbers, formulae, and theories need to come together in mathematics to arrive at an answer. 

The link between mathematics and nature is not surprising when you really think about it. For instance, nature, like mathematics, can be extremely precise in its creations or become susceptible to errors when there is something less or more due to an imbalance or faulty calculations. 

However, in this article, we will be exploring the beautiful and precisely balanced results of these two forces coming together in the form of the Fibonacci series and the golden ratio. These patterns have appeared in nature to improve several shapes and forms. In some cases, the patterns improve efficiency, while in others, they are critical to an organism’s survival and growth. 

Fibionacci Series – The Numbers of Growth 

The pattern of the Fibonacci series follows a simple rule. To achieve endless growth, increase, or to move forward, a number has to combine with its preceding number. This results in an endless calculation that begins with nothing, which is zero. Then, one is added to zero, resulting again in one. Now, this one is added with the one preceding it to form two, and two is then combined with the one preceding it to form three. Then, three combines with the two before it and grows to five, and the calculation endlessly continues, following the same rule of addition, increasing to 8, 13, 21, and so on.¹  

It all looks something like this: 0, 0+ 1=1, 1+1=2, 2+1=3, 3+2=5, 5+3=8, 8+5=13…  

At first glance, they may look like a random set of numbers that are a result of this simple little rule of calculation, but this rule is the beginning of several beautiful things in nature and is especially closely related to the flora and fauna around us. 

Fibonacci in the Plant World

Many flowers and trees have petals and leaves occurring in Fibonacci numbers, and as these numbers increase, they also create patterns called Fibonacci spirals. Some examples of Fibonacci numbers and patterns in nature around us:  

Flower Petals: 

3 petals:  Lilies and iris

5 petals: Buttercups, wild rose, columbine, larkspur, parnassia 

8 petals: Delphiniums, cosmos

13 petals: Marigold, asters 

21 petals: Black-eyed Susan, asters, daisies, spoon mum 

34 petals: Plantain, pytethrum, daisies 

55 & 89 petals: Daisies

Sunflower: Individual flowers within a sunflower are arranged in a clockwise and counterclockwise fibonacci spiral. 

Trees: Elm, cherry, linden, lime, grasses, beech, hazel, blackberry, oak, apple, holly, plum, common groundsel, poplar, rose, pear, willow, almond, and several other trees grow leaves that follow the fibonacci spiral from the initial stages of their growth. 

Vegetables: The florets of cauliflower and romanesque broccoli form a fibonacci spiral. Lettuce leaves are arranged in a fibonacci spiral as well.

Fruit: Bananas and apples when cut in half, not lengthwise, show ridges that appear in the fibonacci sequence, that is, 3 or 5, respectively.

Wildlife: Reproductive patterns of honeybees and rabbits. The fibonacci numbers in five-armed starfish and five pointed sand dollar

In flowers, plants, and trees, the pattern appears for several reasons, such as: 

• To make use of the space for packaging and producing as many seeds as possible
• To expose as many leaves as possible to the sun
• To capture the rain water and allow it to flow down to the soil around the roots of the trees

As the fibonacci numbers keep increasing, they also start converging into another pattern that is often found in nature⏤the golden ratio.¹  

The Numbers of Perfection – The Golden Ratio

Also referred to as “Phi,” this pattern is formed by a ratio of 1:1.6, and can be found in Fibonacci numbers as they keep increasing. When the numbers in the fibonacci series are divided by their preceding numbers, we consistently get 1.6 after the first few numbers.²

The golden ratio of 1:1.6 can be understood more clearly with a straight line divided into two halves, in which one half of the line is slightly longer than the other. For example, if the line is 2 centimeters, divide it into two halves measuring 1 centimeter each, and extend the second half by 6 millimeters, making it 1.6 centimeters. This is considered the golden ratio.² 

Now we have a line that is divided into a ratio of 1:1.6 and the same measurements also create a rectangle that has a side that is 1 centimeter long and another that is 1.6 centimeters long. When this shape is placed beside itself over and over again, with increasing measurements of the same ratio, it creates a flourishing spiral that can efficiently support several activities such as growth, evolution, movement, and more, as you’ll see from the examples below.²  

Storms and Seas: Tornadoes, hurricanes, and waves display movement and growth patterns that follow the golden ratio. 

Plantlife: The golden ratio can also be found in the spiral arrangement of a pinecone and the spikes of a pineapple. 

Outer Space: It can even be seen in the alignment of planets, spirals of galaxies, black holes, and several other celestial bodies.⁴

Wildlife: It is found in the proportions of the bodies of dolphins and ants. The shape of snail shells is a good example of a golden ratio. The flight patterns of some predatory birds, such as the hawk, follow the golden ratio spiral.⁴

Human Body: The golden ratio has also been called the “divine ratio” as it is not only found all around us but also in us! From the shape of the human brain, the proportions of various parts of our body to our very DNA are made of the proportions of the golden ratio.³

We don’t yet clearly know whether these patterns are simply a result of our inherent intelligence to adapt to the most efficient shapes and forms that aid growth and evolution, or if they exist because math is at the core of our very existence. But, in the meantime, we can continue to explore math and the world around us in the hopes of finding an answer someday.   

Come explore the fascinating world of math with BYJU’S FutureSchool’s math courses designed by experts to ensure the best teaching and learning practices that help students connect concepts to real-world applications. 

References

Dr R Knott: fibandphi (AT) ronknott.com. (n.d.). The Fibonacci Numbers and Golden section in Nature – 1. R-Knott. Retrieved April 11, 2022, from https://r-knott.surrey.ac.uk/Fibonacci/fibnat.html

Davidson Art Online. (2018, June 4). Golden Ratio = Mind Blown! [Video]. YouTube. https://www.youtube.com/watch?v=c8ccsE_IumM

BYJU’S. (2020, August 22). Mind-Boggling Facts About The Golden Ratio I Class 6 I Learn With BYJU’S [Video]. YouTube. https://www.youtube.com/watch?v=ReS0Fj9osY0

Dvorsky, G. (2013, February 20). Mathnasium of Pflugerville. Mathnasium. Retrieved April 11, 2022, from https://www.mathnasium.com/examples-of-the-golden-ratio-in-nature