What Is the Fibonacci Sequence?

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Romanesque broccoli spirals resemble the Fibonacci sequence.
Credit: Shutterstock
▼  The Fibonacci sequence is one of the most famous formulas in mathematics.
Each number in the sequence is the sum of the two numbers that precede it. So, the sequence goes: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so on. The mathematical equation describing it isXn+2= Xn+1+ Xn
A mainstay of high-school and undergraduate classes, it's been called "nature's secret code," and "nature's universal rule." It is said to govern the dimensions of everything from the Great Pyramid at Giza, to the iconic seashell that likely graced the cover of your school math textbook.
And odds are, almost everything you know about it is wrong.
Scattered history
So then, what's the real story behind this famous sequence?
Many sources claim it was first discovered or "invented" by Leonardo Fibonacci. The Italian mathematician, who was born around A.D. 1170, was originally known as Leonardo of Pisa, said Keith Devlin, a mathematician at Stanford University. Only in the 19th century did historians come up with the nickname Fibonacci (roughly meaning, "son of the Bonacci clan"), to distinguish the mathematician from another famous Leonardo of Pisa, Devlin said.
But Leonardo of Pisa did not actually discover the sequence, said Devlin, who is also the author of "Finding Fibonacci: The Quest to Rediscover the Forgotten Mathematical Genius Who Changed the World," (Princeton University Press, 2017). Ancient Sanskrit texts that used the Hindu-Arabic numeral systemfirst mention it, and those predate Leonardo of Pisa by centuries.
"It's been around forever," Devlin told Live Science.
However, in 1202 Leonardo of Pisa published the massive tome "Liber Abaci," a mathematics "cookbook for how to do calculations," Devlin said.  Written for tradesmen, "Liber Abaci" laid out Hindu-Arabic arithmetic useful for tracking profits, losses, remaining loan balances and so on, Devlin said.
In one place in the book, Leonardo of Pisa introduces the sequence with a problem involving rabbits. The problem goes as follows: Start with a male and a female rabbit. After a month, they mature and produce a litter with another male and female rabbit. A month later, those rabbits reproduce and out comes — you guessed it — another male and female, who also can mate after a month. (Ignore the wildly improbable biology here.) After a year, how many rabbits would you have? The answer, it turns out, is 144 ­— and the formula used to get to that answer is what's now known as the Fibonacci sequence.
"Liber Abaci" first introduced the sequence to the Western world. But after a few scant paragraphs on breeding rabbits, Leonardo of Pisa never mentioned the sequence again. In fact, it was mostly forgotten until the 19th century, when mathematicians worked out more about the sequence's mathematical properties. In 1877, French mathematician Édouard Lucas officially named the rabbit problem "the Fibonacci sequence," Devlin said.

The Fibonacci sequence and golden ratio are eloquent equations but aren't as magical as they may seem.
Credit: Shutterstock
Imaginary meaning
But what exactly is the significance of the Fibonacci sequence? Other than being a neat teaching tool, it shows up in a few places in nature. However, it's not some secret code that governs the architecture of the universe, Devlin said.
It's true that the Fibonacci sequence is tightly connected to what's now known as the  golden ratio(which is not even a true ratio because it's an irrational number). Simply put, the ratio of the numbers in the sequence, as the sequence goes to infinity, approaches the golden ratio, which is 1.6180339887498948482... From there, mathematicians can calculate what's called the golden spiral, or a logarithmic spiral whose growth factor equals the golden ratio.
The golden ratio does seem to capture some types of plant growth, Devlin said. For instance, the spiral arrangement of leaves or petals on some plants follows the golden ratio. Pinecones exhibit a golden spiral, as do the seeds in a sunflower, according to "Phyllotaxis: A Systemic Study in Plant Morphogenesis" (Cambridge University Press, 1994). But there are just as many plants that do not follow this rule.
"It's not 'God's only rule' for growing things, let's put it that way," Devlin said.
And perhaps the most famous example of all, the seashell known as the nautilus, does not in fact grow new cells according to the Fibonacci sequence, he said.
When people start to draw connections to the human body, art and architecture, links to the Fibonacci sequence go from tenuous to downright fictional.
"It would take a large book to document all the misinformation about the golden ratio, much of which is simply the repetition of the same errors by different authors," George Markowsky, a mathematician who was then at the University of Maine, wrote in a 1992 paper in the College Mathematics Journal.
Much of this misinformation can be attributed to an 1855 book by the German psychologist Adolf Zeising. Zeising claimed the proportions of the human body were based on the golden ratio. The golden ratio sprouted "golden rectangles," "golden triangles" and all sorts of theories about where these iconic dimensions crop up. Since then, people have said the golden ratio can be found in the dimensions of the Pyramid at Giza, the Parthenon, Leonardo da Vinci's "Vitruvian Man"and a bevy of Renaissance buildings. Overarching claims about the ratio being "uniquely pleasing" to the human eye have been stated uncritically, Devlin said.
All these claims, when they're tested, are measurably false, Devlin said.
"We're good pattern recognizers. We can see a pattern regardless of whether it's there or not," Devlin said. "It's all just wishful thinking."

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mattersofindia

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