What are the golden numbers?

"Golden Number" and Optimization Method

More than two thousand years ago, eudoxus, an ancient Greek mathematician, found that if a line segment (AB) is divided into two segments (AP and PB), if the ratio of the length of a small line segment to the length of a large line segment is exactly equal to the ratio of the length of a large line segment to the total length, then this ratio is equal to 0.6 18 ..., which is expressed by the formula Pb/AP = AP/AB = 0.665433. Interestingly, this number can be seen everywhere in nature and people's lives: the navel is the golden section of the total length of the human body, and the aspect ratio of doors and windows is also 0.6 18. On some plant stems, the included angle between two adjacent petioles just divides the circle into the included angle of two radii of 1: 0.6 18. And this angle has the best effect on plant ventilation and lighting. It is also found that the theme of some famous paintings, sculptures and photographs is mostly 0.6 18. ......

The number 0.6 18 is more concerned by mathematicians. Its appearance not only solves many mathematical problems, but also makes the optimization method possible. Optimization method is a method to solve the optimization problem. If it is necessary to add a chemical element to increase the strength of steel during steelmaking, it is assumed that the amount of a chemical element to be added per ton of steel is between1000-2000g. In order to find the most suitable dosage, experiments should be carried out in the range of 65,438+0,000 grams and 2,000 grams. Usually take the midpoint of the interval (i.e. 1500g) for testing. Then compare with the test results of 1000g and 2000g respectively, and proceed in turn until the best result is obtained. This experimental method is called dichotomy. But this is not the fastest experimental method. If the experimental point is 0.6 18 of the interval, the number of experiments will be greatly reduced. This is the one-dimensional optimization method, also known as the 0.6 18 method. Therefore, the great painter Leonardo da Vinci called 0.6 18 ... the golden number.

Mysterious Numbers in Plants

The "plum blossom" on playing cards is not a plum blossom or even a flower, but a clover. Clover is a very symbolic plant in western history. It is said that the first leaf represents hope, the second leaf represents confidence and the third leaf represents love. If you find a four-leaf clover, you will get lucky and find happiness. Looking for four-leaf clover in the wild is a game for western children, but it is difficult to find it. It is estimated that there is one four-leaf mutant for every ten thousand clovers.

In China, plum blossom has a similar symbolic meaning. According to folklore, the five petals of plum blossom represent Five Blessingg. In the Republic of China, plum blossom was designated as the national flower, claiming that the five petals of plum blossom symbolize the harmony of five ethnic groups, which has the significance of strengthening the five ethics, emphasizing the five permanents and applying the five religions. However, it is not unique that plum blossom has five petals. In fact, the most common number of petals is five. For example, other species belonging to Rosaceae, such as peach, plum, cherry blossom, apricot, apple and pear, all have five petals. The common number of petals is: 3, iris and lily (it seems to be 6, but it is actually two sets of 3); 8 pieces, delphinium; 13, chrysanthemum morifolium; The petals of sunflower are 2 1, and some are 34. Daisies have 34, 55 or 89 petals. However, there are few flowers with other petals. Why are the number of petals not randomly distributed? 3, 5, 8,13, 21,34, 55, 89, ... What's so special about these numbers?

Yes, they are Fibonacci series. Fibonacci (1 170- 1240) was a medieval Italian mathematician. Instead of counting the number of petals, he worked out this series when solving a problem about rabbit reproduction. Suppose you have a pair of newborn rabbits, a male and a female. They began to mate when they were one month old. At the end of the second month, the mother rabbit gave birth to another pair of rabbits. A month later, they also began to breed, and so on. Every female rabbit will give birth to a pair of rabbits every month when she begins to breed. Assuming no rabbits die, how many pairs of rabbits will there be in a year?

1 At the end of the month, the first pair of rabbits mated, but only 1 pair of rabbits; At the end of February, the mother rabbit gave birth to two pairs of rabbits. At the end of March, the oldest female rabbit gave birth to a second pair of rabbits, a total of three pairs of rabbits. At the end of April, the oldest female rabbit gave birth to a third pair of rabbits, and the female rabbit born two months ago gave birth to a pair of rabbits. There are five pairs of rabbits. ... class =' class1' > If calculated in this way, the logarithms of rabbits are: 1, 1, 2,3,5,8,1,34,55,89, 144, ... Starting from the third number, each number is the sum of the first two numbers.

Plants seem to be fascinated by Fibonacci. Fibonacci number exists not only in flowers, but also in morphological characteristics such as leaves, branches, fruits and seeds. Leaf order refers to the arrangement of leaves on the stem, and the most common one is alternating leaf order, that is, only 1 leaf is born on each node, which is interactive. Starting from any leaf, connect the falling point of each leaf with an upward line. It can be found that this is a spiral, circling upwards until the falling point of another leaf above coincides with the falling point of the initial leaf, which is the end point. The number of spirals around the stem from the first leaf to the last leaf is called leaf order. The leaf order of different plants may be different and the number of leaves may be different. For example, the leaf order of elm is 1 (that is, 1 around the stem) and there are 2 leaves; Mulberry with leaf order 1 and 3 leaves; Peach, leaf order is 2 leaves and 5 leaves; Pear, the leaf order is 3 leaves and 8 leaves; Apricot, leaf order 5, with 13 leaves; The leaf order of pine is 8, with 2 1 leaf ... expressed by the formula (the number of leaves around the stem is the numerator and the number of leaves is the denominator), which are 1/2, 1/3, 2/5, 3/8, 5/ 13, respectively.

If you look at the sunflower disk, you will find that its seeds are arranged to form two sets of embedded spirals, one is clockwise and the other is counterclockwise. Counting the number of these spirals, although different varieties of sunflower will be different, the number of these two groups of spirals is generally 34 and 55, 55 and 89 or 89 and 144, in which the former group of numbers is clockwise and the latter group is counterclockwise, and each group of numbers is two adjacent numbers in Fibonacci series. Look at the arrangement of scales on pineapples and pinecones. Although it is not as complicated as a sunflower disk, there are two groups of similar spirals, the number is usually 8 and 13. Sometimes this spiral is not so obvious and needs careful observation, such as cauliflower. If you study a cauliflower carefully, you will find that the arrangement of small flowers on the cauliflower also forms two sets of spirals. If you count the number of spirals, are they also two adjacent Fibonacci numbers, such as 5 clockwise and 8 counterclockwise? Fold a small flower and observe it carefully. It is actually composed of smaller flowers arranged in two spirals, and its number is also the number of two adjacent Fibonacci numbers.

Why do plants prefer Fibonacci numbers so much? This is related to another "mysterious" number, which was noticed and even worshipped by people as early as ancient Greece. Suppose there is a number φ, which has the following interesting mathematical relationship:

φ^2 - φ^ 1 -φ^0 =0

Namely:

φ^2 -φ - 1 =0

This equation has two solutions:

( 1 + √5) / 2 = 1.6 180339887...( 1 - √5) / 2 = - 0.6 180339887 ...

Note that the decimal parts of these two numbers are exactly the same. The positive solution (1.6180333865 ...) is called golden number or golden ratio, which is usually expressed by φ. This is an irrational number (decimals are infinitely cyclic and cannot be expressed by fractions), which is the most unreasonable irrational number. It is also an irrational number. π π can be accurately approximated by 22/7, natural constant e can be accurately approximated by 19/7, √ 2 can be accurately approximated by 7/5, and φ can't be accurately approximated by fractions with single denominator.

The golden number has some wonderful mathematical characteristics. Its reciprocal is exactly equal to its fractional part, namely 1/φ = φ- 1. Sometimes this reciprocal is also called the golden number and the golden ratio. If a straight line AB is divided by point C and AB/AC = AC/CB, then this ratio is equal to the golden section number, and point C is called the golden section point. If the top angle of an isosceles triangle is 36 degrees, then the ratio of its height to the bottom line is equal to the golden number. Such a triangle is called the Golden Triangle. If the aspect ratio of a rectangle is the golden number, then a square with its width is cut from the rectangle, and the aspect ratio of the remaining small rectangles is still the golden number. Such a rectangle is called a golden rectangle, which can be cut infinitely by the above method to get smaller and smaller golden rectangles, and if the diagonal corners of these golden rectangles are connected by arcs, a logarithmic curve is formed. Common newspapers, magazines, books, paper, ID cards and credit cards are all close to the golden rectangle, which is said to make people look comfortable. Indeed, in our life, golden numbers are everywhere, and buildings, works of art and daily necessities all like to use them in design, because it makes us feel beautiful and harmonious.

So what is the relationship between the golden number and Fibonacci number? According to the above equation: φ 2-φ- 1 = 0, we can get:

φ = 1 + 1/φ = 1 + 1/ ( 1 + 1/φ) = ...= 1 + 1/( 1 + 1/( 1 + 1/( 1 +...)))

According to the above formula, you can use a calculator to calculate φ: enter 1, take the reciprocal, add 1, take the reciprocal, add 1, take the reciprocal, ... and you will find that the sum is getting closer and closer to φ. Let's use fractions and decimals to represent the above approximate steps:

φ ≈ 1 φ ≈ 1 + 1/ 1 = 2/ 1 = 2φ ≈ 1 + 1/( 1+ 1/ 1) = 3/2 = 1.5φ ≈ / kloc-0/ + 1/ ( 1+ 1/( 1+ 1)) = 5/3 = 1.666667φ ≈ 1 + 1/( 1+ 1/( 1+( / kloc-0/+ 1))) = 8/ 5 = 1.6φ ≈ 1 + 1/( 1+ 1/( 1+( 1+( 1+ 1)))) = 13/ 8 = 1.625φ ≈ 1 + 1/ ( 1+ 1/( 1+( 1+( 1+( 1+ 1))))) = 2 1/ 1 3 = 1.6 15385φ ≈ 1 + 1/( 1+ 1/ ( 1+( 1+( 1+( 1+( 1+ 1)))))) = 34/2 1 = 1.6 19048φ ≈ 1 + 1/( 1+ 1/ ( 1+( 1+( 1+( 1+( 1+( / kloc-0/+ 1))))))) = 55/34 = 1.6 17647φ ≈ 1 + 1/( 1+ 1/ ( 1+( 1+( 1+( 1 +( 1+( 1+( 1+ 1)))))))) = 89/55 = 1.6 18 182 ...

Did you find it? The numerator and denominator of the above scores are adjacent Fibonacci numbers. It turns out that the ratio of two adjacent Fibonacci numbers is about equal to φ. The bigger the number, the closer it is. When it is infinite, the ratio is equal to φ. Fibonacci number is closely related to gold number. Plants like Fibonacci numbers, but in fact they like golden numbers. Why is this? Is there some arrangement that God wants to make the world full of beauty and harmony?

The branches, leaves and petals of plants are homologous, and they all germinate and differentiate from the meristem of stem tip in turn. The new bud grows in a different direction from the previous bud and rotates at a fixed angle. If we want to make full use of the growth space, the growth direction of new buds should be as far away from the old buds as possible. So what is the best angle? We can write this angle as 360 × n, where 0 < n < 1. Because the left and right angles are the same (only the rotation directions are different), such as n=0.4 and n=0.6, the results are actually the same, so we only need to consider the case of 0.5 ≤ n < 1. If the new bud wants to be as far away from the previous old bud as possible, it should grow in the opposite direction, that is, n = 0.5 = 1/2, but in this case, the second new bud is in the same direction as the old bud, and the third new bud is in the same direction as 1 new bud ..., that is, it only overlaps around 1 week. If There will be overlap after three rounds, and there are only five directions in total. In fact, if n is a true fraction p/q, it means that there is overlap around p, and * * * has q growth directions.

Obviously, if n is an irrational number that cannot be expressed by a fraction, it is much more "reasonable". What kind of irrational numbers do you choose? Pi, natural constant e and √2 are not good choices, because their fractional parts are very close to 1/7, 5/7 and 2/5 respectively, that is, they overlap around 1, 5 and 2 weeks respectively, with only 7, 7 and 5 directions in total. So the conclusion is that the more irrational the irrational number, the better and more "rational". As mentioned earlier, the most unreasonable irrational number is the golden number φ ≈ 1.438+08. That is, the optimal value of n is ≈0.6 18, that is, the optimal rotation angle of new shoots is about 360× 0.6 18 ≈ 222.5 or 137.5.

As mentioned above, the most common leaf orders are 1/2, 1/3, 2/5, 3/8, 5/ 13 and 8/2 1, indicating how many leaves there are each week. If we want to convert them into n (indicating how many times each leaf turns), we will find that. They are the ratio of two adjacent Fibonacci numbers, which are close to 1/φ in different degrees. In this case, the buds of plants can have the most growth directions and occupy as much space as possible. For leaves, it means getting as much sunlight as possible for photosynthesis, or taking as much rainwater as possible to irrigate the roots; For flowers, it means showing yourself as much as possible to attract insects to pollinate; For seeds, it means arranging them as closely as possible. All these are of great benefit to the growth and reproduction of plants. It can be seen that the reason why plants prefer Fibonacci number is the result of evolution under the natural selection of survival of the fittest, which is not mysterious.