What is the Fibonacci Sequence?

The Fibonacci Sequence is a mathematical formula that frequently appears in nature. It also makes some really interesting spiral shapes. The sequence was discovered by an Italian mathematician named Leonardo Fibonacci around the 13th century. This might sound strange, but he discovered the phenomenon by studying rabbits. So, what is the Fibonacci Sequence and how is it used?

Leonardo Fibonacci

Up close view of spiraling aloe vera plant, showing the golden ratio design
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Leonardo Fibonacci, also known as Leonardo of Pisa, studied math from a young age. He was sent by his father to study under an Arab mathematician, which took him all over the world. During his career, he wrote several books and even became friends with the Holy Roman Emperor, Frederick II.

In the 12th century, not much was known about Arabic numerals (the number system we use today) in Europe. In his travels to Egypt, Syria, Greece, and Sicily, he was able to study a wide variety of numerical systems and see how they were used. He wrote his first book called “Liber abaci” about how the Arabic numeral system worked, including basic arithmetic; how it could be used with money, weights, and measures; and how to use a few tips for calculations like the 'rule of three' and the 'rule of five' method for finding proportions. It was a hit and pushed him to the forefront of the mathematical world.

Discovering the sequence

Spiraling stained glass windows rising in the golden ratio at a chapel in Dallas, Texas
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After his initial successes, he began to study math in nature, which led him to his most famous discovery: the Fibonacci sequence. Around 1202, Fibonacci came up with an interesting thought experiment. He wanted to see if he could mathematically calculate how many pairs of rabbits could be produced in one year. Because that’s something everyone wonders at some point, right?

He knew that rabbits could reproduce at one month old, had a one-month gestation, and typically produce two offspring, one male and one female. So, theoretically, if he started with one pair of rabbits, a male and a female, they would have matured by the second month, and in the third month would give birth to another pair. The original pair could mate again in the third month, while the younger pair is maturing. In the fourth month, the original pair would give birth to yet another set of rabbits while the second pair was mating. In the fifth month, both pairs would reproduce while the youngest pair matures, resulting in five sets of rabbits.

If you look at it in terms of numbers by month, the sequence goes like this: 1, 1, 2, 3, 5. Each consecutive number is the sum of the two numbers before it. If you continue the pattern, it would go: 8, 13, 21, 34, 55, 89, 144. So, in Fibonacci’s original experiment, he would have 144 pairs of rabbits in one year. What he would do with all those rabbits, who knows, but since his initial discovery, the Fibonacci sequence has become known as the Golden Ratio.

The golden spiral

Up close view of spiraling fern stem in the golden ratio Fibonacci sequence
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The Fibonacci sequence can also be depicted in the form of a spiral that grows wider as defined by the Golden Ratio. The first arc is in a 1x1 square. A second arc is connected to the original in an adjacent 1x1 square. The next portion of the arc is in a 2x2 square. The next in a 3x3 square, and so on up through the sequence. The result is a spiral that spreads out and gets wider as it goes.

The Fibonacci sequence in nature

Cross section view of nautilus shell interior, spiraling in the Fibonacci sequence
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Most impressively, this ratio can be found throughout the natural world. The number of petals that grow on flowers is almost always a number in the Fibonacci sequence. Lilies have three petals, buttercups have five, and daisies have 34, just to name a few.

The golden spiral can also be found throughout nature. If you look at the spiral pattern on a shell belonging to a hermit crab, you’ll notice that the spiral starts at a point in the center and widens as it goes outward. Sunflowers and pinecones also follow the golden spiral with their seed patterns. On the larger scale, hurricanes and even galaxies follow the golden spiral pattern as well. These spirals are an exact depiction of the Fibonacci sequence.