Is there some kind of magical formula for the universe? Perhaps not, but in nature we find some very common patterns over and over again. For example, the Fibonacci sequence. This is a series of progressively increasing numbers, each of which (Fibonacci numbers) is the sum of the first two numbers. (We'll go into more detail about this math formula later.) ）
The Fibonacci sequence is equally applicable in nature, and as a corresponding proportion, it reflects a variety of patterns in nature – such as the near-perfect spiral of a nautilus' shell and the daunting vortex of a hurricane.
Humans may have known about the Fibonacci sequence for thousands of years – the mathematical concept of this interesting pattern dates back to ancient Sanskrit texts dating back between 600 and 800 BC. But in modern times, we associate it with all sorts of things: a medieval man's obsession with rabbits, computer science, and even the seeds of sunflowers.
1. Fibonacci numbers and how rabbits reproduce
In 1202, Italian mathematician Leonardo Pisano (also known as Leonardo Fibonacci, meaning "son of Bonaci") wanted to know how many baby rabbits a male and female rabbit could bring. More precisely, Fibonacci asks the question: how many pairs of rabbits can a pair of rabbits breed in a year? This thought experiment assumes that female rabbits always give birth to a pair of rabbits, and that each pair includes a male rabbit and a female rabbit.
Imagine this: two newborn rabbits are placed in an enclosed area and then begin to breed like large rabbits. Rabbits must be at least one month old to give birth, so in the first month, there is only one pair of rabbits. By the end of the second month, the female rabbit gives birth to a new pair of rabbits, for a total of two pairs.
By the third month, the original pair of rabbits gave birth to a pair of newborns, and their previous offspring had grown into large fertile rabbits. This leaves three pairs of rabbits, two of which will give birth to two new pairs of rabbits in the next month, for a total of five pairs of rabbits.
So how many rabbits will there be in total a year? This is where mathematical formulas come in. As complicated as it sounds, it's actually quite simple.
The first few numbers in the Fibonacci sequence are: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, and then increase to infinity.
The mathematical formula to describe this sequence goes like this: Xn+2 = Xn+1 + Xn, basically, each integer is the sum of the first two numbers. (You can also apply this to negative integers, but we're only talking about positive integers here.) ）
To get 2, add the two numbers in front of it (1+1)To get 3, add the two numbers in front of it (1+2)
This set of infinitely summed numbers is known as the Fibonacci sequence or Fibonacci sequence. The ratio between numbers in the Fibonacci sequence (1.6180339887498948482...) ) is often referred to as the golden ratio or golden number. The ratio of consecutive Fibonacci numbers approaches the golden ratio as the number approaches infinity.
Want to see how these fascinating numbers manifest themselves in nature? You don't need to go to your local pet store; It's just a matter of looking around.
2. How the Fibonacci sequence works in nature
Although the seeds, petals, and branches of some plants follow the Fibonacci sequence, it does not mean that everything in nature grows according to this law. Just because a series of numbers can be applied to a staggering variety of objects, doesn't mean that the numbers have any relevance to the real world.
It's like a number superstition, like a celebrity dying in a group of three, and sometimes a coincidence is just a coincidence.
However, although some would argue that the universality of continuous Fibonacci numbers in nature is exaggerated, the frequency with which they occur is enough to prove that they reflect patterns of natural existence. You can spot these patterns by looking at how various plants grow. Here are a few examples:
Observe the arrangement of seeds in the center of the sunflowers, and you will notice that they take on a golden spiral shape. The amazing thing is that if you count these spirals, the total will be a Fibonacci number. Divide the spiral into a left and right spiral, and you will find that the number of spirals on the left and right is exactly the two adjacent numbers in the Fibonacci sequence.
You can find similar spiral patterns in pine cones, pineapples, and cauliflowers, which also reflect the Fibonacci sequence in this way.
Some plants show a Fibonacci sequence at their point of growth, where branches form or fork. A tree trunk grows until it produces a branch, forming two growth points. The main trunk then produces another branch, resulting in three growth points. Then, the main trunk and the first branch produce two more growth points, for a total of five growth points. This continuous pattern follows the Fibonacci sequence.
Also, if you count the number of petals of a flower, you will usually find that the total is a number in the Fibonacci sequence. For example, lilies and irises have three petals, buttercups and wild roses have five petals, delphinium has eight petals, and so on.
A honey bee colony consists of a queen, several drones, and many worker bees. Female bees (queen and worker) have a pair of parents: a drone and a queen. On the other hand, drones only hatch from unfertilized eggs. This means that they have only one mother. Thus, the Fibonacci number can represent the drone's family tree, i.e. it has one mother, two grandparents, three great-grandparents (maternal grandmother has two parents, maternal grandfather has only one mother), and so on.
Storm systems like hurricanes and tornadoes often follow the Fibonacci sequence. The next time you see a hurricane hovering on the weather radar, pay attention to the clouds on the screen for a noticeable Fibonacci spiral.
Take a good look at yourself in front of the mirror. You will notice that most of your body parts follow the numbers one, two, three, and five. You have a nose, two eyes, three segments per limb, and five fingers per hand. The proportions and measurements of the human body can also be divided by the golden ratio. The DNA molecule also follows this sequence, with each double helix cycle being 34 angstroms long and 21 angstroms wide.
Why do so many patterns in nature reflect Fibonacci sequences? Scientists have been exploring this question for centuries. In some cases, this correlation may be just a coincidence. In other cases, this ratio exists because this particular growth pattern is the most efficient. In plants, this may mean the maximum proportion of exposure to light-loving leaves or the maximum space utilization of seed arrangement.
Misconceptions about the golden section
While experts agree that the Fibonacci sequence is common in nature, there is more controversy as to whether the Fibonacci sequence is represented in certain artistic and architectural examples. Although some books claim that the Great Pyramid and the Parthenon (as well as some of Leonardo da Vinci's paintings) were designed according to the golden ratio, upon examination, this was found to be false.
Mathematician George Makworski pointed out that both the Parthenon and the Great Pyramid had parts that did not conform to the golden ratio, which was overlooked by those who rushed to prove that the Fibonacci number existed in everything. In ancient times, the term "golden average" was used to denote something that avoided extremes in either direction, and some people confuse the golden average with the golden ratio, which is a newer term that only appeared in the 19th century.
Something fun
November 23 was established as Fibonacci Day, not only in honor of the forgotten mathematical genius Leonardo Fibonacci, but also because when the date is written as 11/23, four numbers make up a Fibonacci sequence. Leonardo Fibonacci is also widely credited as one of the people who contributed to our shift from Roman numerals to the Arabic numerals we use today.
作者：Robert Lamb & Jesslyn Shields
Translation: Meyare
Reviewer: Small line
原文：Why Does the Fibonacci Sequence Appear So Often?
Edit: There is a lot of interest
The views expressed in the translation are solely those of the author
It does not represent the position of the Institute of Physics of the Chinese Academy of Sciences