Interesting talk about the Fibonacci series

Fibonacci series of fun from ---------- http://bzhang.lamost.org/website/index.php? P = 137

It is generally believed that the Fibonacci series is based on the rabbit breeding problem: if there was a rabbit at the beginning, they would have a rabbit every month, how many rabbits will be born one month after a rabbit is born and the breeding condition is the same as that of the first rabbit?

The answer is: the total number of rabbits per month can be expressed in the following series: 89,144,233, and so on .... This series was written by Leonardo Fibonacci, an Italian number expert in the early 13th century.Liber Abaci. If the first two elements of the sequence are 1, the recursive relationship is:

Of course, at one time, mathematicians used 0 as the first entry (or 0th items) of the Fibonacci series ).

This series looks quite simple, but it hides some interesting things.

　

Sequence elements

People have found many interesting things about the elements of the Fibonacci series.

Prime Number and Union number: the prime number element of the Fibonacci series is also the prime number of the series. The only exception is 4th element 3. However, this rule, in turn, is not true. The prime number element of a series may also be a combination of numbers. This "rule" can provide clues for people to search for large numbers. But does it still have this rule after a considerable number of elements? No one knows.

If we plot the first 511 elements of the Fibonacci series in binary format (Pegg 2003, from Wolfram Research ):

Is it a fragment?

Item 10n:, 209,2090, 20899,208988, 2089877,20898764 .... (Sloane's a068070) That is to say, this number keeps approaching 208987640249978733769... The first few items. While 208987640249978733769... Related to such a number:

Binet formula: This formula is not a commonly used formula of the same name in orbital mechanics, but another formula for the n entry of the Fibonacci series, which was found by Jacques Philippe Marie Binet in 1843:

What do you see? Does the two numbers in brackets seem to be related to the Golden split?

　

Fibonacci series and golden Division

The Scottish Robert simson proves that when the number of items tends to be infinite, the ratio of the post term of the Fibonacci series to the previous term is close to the Golden division, that is, 1.61803398875 .... This may indicate the natural connection between the Fibonacci series and the golden split.

For example, the Fibonacci spiral is the most direct example. If the number of the clockwise spiral is two adjacent items in the Fibonacci series, it can be called the Fibonacci spiral or the golden spiral. Such a spiral can best use the circumference, the most uniform density. It is not difficult to construct a method. You only need to first construct the golden rectangle type with the same number related to the Fibonacci series (the ratio of length to width is gold division), and then draw a 1/4 arc in each rectangle, connect each arc. Such a golden rectangle can also be found in some famous artistic works, such as the famous works of da Vinci called Mona Lisa.

The Fibonacci spiral drawn by the computer

Fibonacci spiral and golden moment

　

Fibonacci series in nature

The most typical example is the leeches or leaves arranged in the Fibonacci spiral. Ji, chrysanthemum, sunflower, pine fruit, pineapple ...... They all grow in this way. The reason for this is simple: Such a layout can make plants grow properly and make full use of the sun and air, so many plants have become what they are today in the evolution of years. Of course, due to the influence of climate or pests and diseases, real plants often do not have a perfect Fibonacci spiral.

The number of branches on each layer also forms the Fibonacci series.

I once saw the following figure on the Internet. It is just a coincidence that the number of petals matches the various elements of the Fibonacci series?

In addition, the crystal structure is often related to the Fibonacci sequence.

BTW, Mathematica, comes with a toolkit named Fibonacci.