Fibonacci Sequence & Golden Ratio: Math in Nature

James Ng
4 min readOct 26, 2023
A plant vine, curled in a shape of a spiral, traced by the Golden ratio spiral
Is the spiral a random occurrence or a grand design?

You always hear people say “Math is boring!” or ask “What is the point of Math?” You do not have to love or hate Math to appreciate it. Simply put, Math is the science of structure, order, and relation used to make sense of the world around us. If you have ever marveled at nature, then you can at least appreciate Math because there might be some sort of grand design…

Leonardo Bonacci (Fibonacci), 1170–1250, was an Italian mathematician who was studying the growth of the rabbit population and asked the following question: “How many pairs of rabbits will one pair produce in one year, given that naturally they produce a new pair every month?”

From his observations, Bonacci came up with the Fibonacci sequence, which is 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, …, etc. It is a simple, math sequence in which each number is the sum of the previous two numbers, starting at 2. The picture below shows this sequence in the formation of tree branches.

Fibonacci sequence in tree branches.
Fibonacci sequence in tree branches.

The Fibonacci sequence is significant because of its connection to the Golden Ratio, 1.618033988749894. Two quantities a and b are said to be in the Golden Ratio if

Golden ratio equals the sum of a and b divided by a, which also equals a divided by b, where a > b > 0

where a < b < 0. The exact value of the ratio is expressed as the Greek Letter φ (phi),

Greek letter phi equals the sum of 1 and the square root of 5, divided by 2.

Those who are wish to learn more about the calculation of this value can check it out on Wikipedia.

Notice how the Fibonacci numbers perfectly fit into the Golden Ratio in the schematic below, which forms the Golden rectangle, starting with numbers 1 and 1. A square of side length 2 (sum of 1 and 1) is added to form another rectangle, maintaining the Golden Ratio of a + b and a as each rectangle’s longer side and shorter side, respectively (see image below). From such observation, smart mathematicians were able to draw a logarithmic spiral through each square to form the Golden spiral. How golden!

Picture of a spiral that shows the Fibonacci sequence.
Gold Spiral with Fibonacci Sequence
Golden Ratio in a rectangle

Spirals in Nature

Enough with the math. Here comes the cool part!

If the Golden Ratio is applied as a growth factor in some math equation, we would get a logarithmic spiral — the Golden spiral. Interestingly, this can be found throughout nature. Here are some examples:

Spirals in botany. Top-left: spiral aloe, bottom-left: flower bud, top-right: seed head, bottom-right: pinecone.
Spirals in botany. Top-left: spiral aloe, bottom-left: flower bud, top-right: seed head, bottom-right: pinecone.
Spirals in nature. Top: ocean wave, bottom-left: hurricane, bottom-right: whirpool.
Spirals in nature. Top: ocean wave, bottom-left: hurricane traced with the spiral, bottom-right: whirpool.
The Golden ratio and spiral traced onto a human face and ear.
The Golden ratio and spiral can be traced in a human face and ear.
“The universe is within you.” Left: golden ratio in human hand. Right: gold spiral in a galaxy
Left: Human hand. Wrist and knuckle joints form the Golden ratio. Right: Spiral in a galaxy — approximation.

Everything in the universe is within you. — Rumi

It is important to note that these are approximate calculations of the Golden spiral being applied to natural phenomena. Whether you are a believer or non-believer, the patterns appear frequently enough that its significance cannot be understated.

The Fibonacci sequence and Golden Ratio has captivated mathematicians, artists, designers, and scientists for a long time. Despite the chaotic nature of the universe, this suggests an important characteristic of how nature follows some sort of a blueprint in the grand design of physical matter and attempts to organize it in the most efficient way possible.

What do you think, random or not?

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James Ng

Software engineer, math & physics educator, mentor