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Honeycomb mathematics problem
Flying worker bees, worker bees are immature females, so there should be drones inside!

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Worker bees and queen bees are individuals with different division of labor developed from fertilized eggs. They are a group of sisters.

1 Wonderful honey world

Bees-insects with tight social organization

A robust bee colony consists of about 30,000 adult bees and 20,000 young bees. These individuals constitute the bee society. Adult bees shoulder social responsibilities and obligations and carry out strict social division of labor with each other. Young bees grow under the careful care of adult bees, and once they emerge into bees, they immediately join the ranks of division of labor. In the insect world, bees have such strict social organizers, only ants and termites. Because bees are easy to raise and observe and have important economic value, human beings are particularly interested in them. In the caves discovered in Spain 8000 years ago, murals of human hunting for honey appeared. The word "honey" has appeared in Oracle Bone Inscriptions unearthed from Yin Ruins in China, which shows that China people have a long history of using bee products. Aristotle, a great philosopher and naturalist 2300 years ago, also made a detailed observation and record of bees. These historical records show that humans and bees have been consistent along the way.

Maybe bees are too famous! Most people have been stung by bees, so the killer is a bee. In fact, there are many kinds of stinging bees, such as wasps (tiger bees), long-legged bees (wasps), bumblebees, flower bees, mud bees and so on. In fact, there are only four kinds of bees in the world, namely, Oriental bees, floria bees, Chinese bees and Italian bees, and there are only Oriental bees and Western bees in Taiwan Province Province. Generally speaking, it is bees that have stings in the stung parts. This is because the stings of bees have obvious barbs and their structures are easy to fall off. As we all know, bees died heroically soon after being stung because of internal organs damage.

The queen bee and the worker bee are both girls.

Many people may not know that bees are female society, let alone believe that hardworking worker bees are actually female bees. In recent years, the author has often served as a lecturer in the Nature Exploration Team of the National Taiwan Province Provincial Museum, leading students to explore the world of bees together. When I ask the sex of worker bees, about 80% of the students always think it is a boy. In fact, there is a strange phenomenon in the world of bees. The fertilized egg will develop into a female bee, while the unfertilized egg will become a male bee. Queen bee and worker bee are both female bees developed from fertilized eggs. A swarm of bees has only 65,438+0 queen bees. She specializes in laying eggs and can lay 1500-2000 eggs every day. The total weight of these eggs is far greater than that of the queen bee! The queen bee can live for 3-5 years; Most of the individuals in the bee colony are worker bees, they have no reproductive ability, and the ovipositor tube has been specialized into a stinging needle. Worker bees are responsible for all the work in the bee colony except laying eggs, and their life span is only 1-2 months.

Why is the queen bee also developed from fertilized eggs, but it has such "super power"? The key lies in the difference in nutrition. If the fertilized egg is laid in a special nest-Wangtai, the worker bees will feed her the mysterious food-royal jelly after the larvae hatch, and the female larvae who eat royal jelly will grow very fast, and will emerge as queen bees after 13 days. If fertilized eggs are laid in a general hexagonal nest, they will only eat worker bee jelly, a food with poor nutritional value, and will only develop into sterile worker bees in the future! In addition, in the adult stage, every time the queen bee lays 10 eggs, the bodyguard bees (also called worker bees) who are in charge of nursing will secrete royal jelly to supplement the nutrition for the queen bee, so that the queen bee can continue to lay eggs and live for 3-5 years. Adult worker bees eat pollen and honey, plus work fatigue, and their life span is only 1-2 months! Therefore, royal jelly has always been regarded as a sacred product of nutritional supplement by human beings.

A romantic and haunted drone.

Drones are the only males in the bee world. They are strong, but they have no ability to collect food, so they play a "soft role" in bee society. The only task of the drone is to mate with the virgin queen. In order to chase after his girlfriend, his sense of smell and vision are particularly developed. But there are very few drones that can really mate with the queen bee, because the queen bee only goes out to mate 1 time in her life. These 1 mating flights collected the sperm of many heroes (about 10 drones) in the sperm storage sac of the queen bee. The seminal vesicle is a sperm bank, which can store about 7 million sperm, and the activity of these sperm can last for more than 3 years. From this point of view, the drones lucky enough to have a kiss are really rare, rare and rare. Because the drone is not produced, once the food stored by the bees is insufficient, the drone will be swept out of the house and starve to death. However, if there is a chance to mate with the virgin queen, the drone will be miserable! After coitus, he died of exhaustion of physical strength and damage of internal organs, which confirmed the so-called "peony flowers die, and ghosts are also romantic."

Worker bees with division of labor and cooperation

Worker bees are the backbone of bee society. Their number is about 30 thousand, which is responsible for most of the work of bee colonies. Therefore, it is necessary to adopt the mode of division of labor and cooperation to maintain the operation of bee society. The division of labor of worker bees is based on "day age", and worker bees aged 1-20 are office bees, which are responsible for all the work in the nest; After 2 1 day, she is a bee, responsible for collecting water, nectar, pollen and propolis outside the nest.

The work of office bees is very complicated, and the general work is broken down by age. The new worker bees began to clean up their nests. After about 1 week, their food secretion glands developed completely, and they began to be responsible for raising young bees (called nurse bees) and queen bees (bodyguard bees). Then, their familiarity with the hive increased and they began to play the role of "porters"; They remove the debris from the nest and receive the nectar and pollen collected from the field to the warehouse for storage. Subsequently, the wax glands of worker bees mature, and they also undertake the work of nesting. If the weather is too hot, they have to use themselves as electric fans-to adjust the temperature in their nests. Finally, the venom content of 19-20-day-old worker bees reached a peak, and they were responsible for guarding the nest door. These guard bees are particularly fierce, so it is best not to provoke the guard bees at the door of the nest.

Guarding is the last task of domestic bees, and then they will join the ranks of farm bees.

The language of the bee world

Social animals must develop a set of ways to communicate with each other, otherwise the whole social organization will not function normally because of poor communication. Bees can't talk. They use body movements and chemicals as the language of communication.

Wild bees specialize in collecting food, and the collected food must meet the needs of the population. This task is very arduous, so effective collection strategies must be formulated. Early in the morning, the reconnaissance bees in the field went out to look for food. After collecting food and returning to the nest, they tell their companions the direction and distance of food source by "dancing" to facilitate their companions to collect it. After these companions collected it and returned to the nest, they told other companions in the same way. In this way, wild bees can go directly to the food source to collect food without wasting their aimless groping.

In addition to body language, bees also have a variety of exocrine glands, which secrete a variety of chemicals to transmit information. This chemical used for communication between the same kind is called pheromone. Queen bees, worker bees and drones all secrete different pheromones, and even larvae secrete pheromones to communicate with adult bees. Queen bees use pheromones to attract males to mate, and the secretion of maxillary glands inhibits the development of worker bees' ovaries. When worker bees sting people, they will also secrete warning pheromones, causing other worker bees to sting people.

The mystery of honeycomb structure

Darwin (1809- 1882) said: "It is a fool to observe the structure of a beehive without praising it. What's the secret of the hive?

Worker bees secrete beeswax from the wax glands in the abdomen to build a hive, which is used as a storage room for the queen bee to lay eggs, brood and store honey and pollen. It is estimated that worker bees need to consume 16 kg of honey to secrete 1 kg of beeswax, while collecting 1 kg of honey requires field bees to fly 320,000 kilometers to complete, which is equivalent to the distance of eight circles around the earth! So beeswax is very precious to bees. Instinctively, bees use the "most economical principle" to build hives, that is, using the least material-beeswax to create the largest space-the nest room.

Seen from the front, the hive consists of many regular hexagonal hollow columnar storage rooms; From the perspective of three-dimensional section, it has storage rooms on the left and right sides, and the bottom is composed of three congruent rhombic surfaces with internal angles of 70 32' and109 28' respectively.

Scientists were surprised by the observation of honeycomb structure, and then put forward two mathematical problems:

(1) Why is the front of the hive a regular hexagon?

(2) Why is the bottom edge composed of three congruent rhombic faces? Why are its internal angles 70 32' and109 28'?

The first problem involves the ancient isoperimetric problem, that is, on the plane, a closed area should be surrounded by fixed-length line segments to maximize its area. How to surround it?

For this isoperimetric problem, the ancient Greek mathematician Zhi Nuo Dorothy (BC 180) has proved the following results:

Of all the N-polygons, the regular N-polygon has the largest area, and the more sides, the larger the area. ※.

On the other hand, the ancient Egyptians knew that there were only three options for paving the floor with regular polygon tiles of the same shape and size: regular triangle, square and regular hexagon. From the above mathematical theory, it is found that choosing a regular hexagon on the front of the bee's hive conforms to the principle of mathematical extreme value, and creates the largest nest space with limited beeswax raw materials.

The second question is more difficult. Why is the bottom of the nest room composed of three congruent rhombic faces, rather than a flat structure? In fact, the reason is also related to the extreme value, that is to say, three congruent rhombic surfaces create the largest space and save the most materials.

17 12 years, the astronomer G. F. Maraldi of Paris Observatory measured the angle of the diamond-shaped surface of the honeycomb, and the results were 70 32' and109 28'. This result aroused the interest of the famous French naturalist Leo Moore. He thinks there must be a reason for this angle, which may be related to bees building the largest nest with the least beeswax. Rêaumur went to consult the Swiss mathematician Samuel K. Nigel, the following questions:

* Given a regular hexagonal cylinder with three congruent diamonds at the bottom, what can be done to save the most materials?

Rêaumur didn't tell k? Nigger, this is about the honeycomb structure.

k? Nig solved this extreme value problem by differential method, and the calculated results are 70 34' and 65 438+009 26', which are only 2' different from the observed values of honeycomb. They were surprised that bees used the principles of advanced mathematics to build their nests. About the difference of two points, later other mathematicians recalculated and found that the bee was right, K? Nigel was wrong. He made a small mistake in the calculation, so the result was wrong.

Asked Burt's Bee University

Bee has wonderful biological characteristics and the importance of economic production, so it is the most studied insect by scientists in the insect field, with a total of about 6.5438+0 million species. Although Aristotle began to study and observe bees as early as 2300 years ago, today, the world of bees still leaves many unknown mysteries, waiting for you and me to explore together!

2 "worker bee escape"

As we all know, worker bees and queen bees are individuals with different division of labor developed from fertilized eggs. They used to be a group of close sisters, just because the queen bee lived in the queen bee platform without worry and ate magical royal jelly all her life, and then she became the leader of the whole group, while the worker bees, who mainly ate pollen and nectar, later developed into incomplete female bees and became all beings in the bee kingdom. Worker bees, who have worked hard all their lives, are the embodiment of diligence, courage and selflessness. Shortly after birth, they began to engage in internal work such as cleaning nests, raising children, making honey and building nests. Bravely provoke the heavy responsibility of field work after 2-3 weeks. They fly out of the nest door more than ten times a day, braving the scorching sun and wind and rain, looking for honey sources and collecting honey powder. During the honey flow period in summer, worker bees leave early and return late, covered with gold powder, with nectar and wrapped in their bellies. At night, worker bees flutter their wings, work overtime to collect honey, remove pollen wax cover, spit out pulp to raise insects and build nests, and work day and night. Their lives shine with hard work and selfless dedication. Unfortunately, life is too short, too short. 30-60 days after the birth of worker bees, their lives are exhausted in hard work. If they combine work and rest and cherish life, it will create more wealth for mankind! How nice it would be!

However, during the overwintering period of bees, worker bees generally move in the hive except for excretion in sunny days, leading a semi-secluded life, saving their physical strength and preparing for the busy next spring. Their life span can generally reach several months, from October of last year to March and April of the following year. The formula of life span of organisms discovered by humans = the life span of organisms is 5-7 times of the date of development of organisms, and the development cycle of worker bees is 2 1 day, so their time limit for seeking help should be about 2 1×7= 147 days. It can be seen that in winter, the help of worker bees will generally reach the limit of their lives. The fact that worker bees live for a long time tells us that the life span of living things is a variable that can be long or short. Life needs both exercise and rest. If you go to extremes, it will be counterproductive.

Health and longevity is an eternal topic that human beings dream of. The famous saying that "life lies in exercise", which advocates people's diligence and housekeeping, has become the common sense that women and children know health and longevity.

Life has reached its glory in moderate exercise. Scientific research shows that exercise can promote blood circulation in the body, improve the functions of various tissues and organs, enhance disease resistance, accelerate the excretion of metabolites, and significantly increase the amount of anti-arteriosclerosis (such as high-density lipoprotein lipase) and anti-aging substances (such as superoxide dismutase) in the body, which is undoubtedly beneficial to people's longevity and quality of life. However, strenuous and large-scale exercise, especially long-term and large-scale exercise, will only lead to damage to tissues and organs, accelerate aging and promote death. The short life of worker bees who worked hard day and night and collected diligently is a good proof.

Modern medicine shows that people's life activities are a contradictory process: exercise not only has the above benefits for people, but also sharply increases the consumption of oxygen temperature in the body due to exercise, producing a large number of active oxygen, which is an important substance to promote human aging. Inactive animals, such as reptiles, turtles and soft-shelled turtles, have become the king of animal longevity because of their low oxygen consumption. Think about it, the life span of worker bees in the wintering period is several times that of high-intensity exercise in summer, which once again shows that life also lies in rest.

It seems to be a contradictory conclusion that life lies in exercise and rest. In fact, it is the unity of a pair of contradictions. In moderate exercise, life reduces the decline of organs, gets normal metabolism and enhances immunity. Life is adjusted and repaired in proper rest to remove active oxygen, resist aging and prolong life. As long as people master the rhythm of exercise and rest, health and longevity will no longer be a dream.

3) Worker bees control the number of worker bees through pheromones.

Bees have always been known for their high degree of organization. In the "society" composed of bees, each bee plays a carefully assigned role. However, how can they adjust their total labor force according to their own needs? A new study shows that worker bees control the development of bee larvae through a pheromone, thus controlling the number of "battle bees" in the bee colony.

When worker bees were born, they took on the job of cleaning the hive, and then gradually expanded their responsibilities to look after the bees and store food. At the age of two or three weeks, they finally "advanced" to their ultimate occupations-"scouts foraging around" and "soldiers" defending their territory. Among them, the evolution to "fighting bees" is related to the improvement of hormone levels in bees, the changes of brain structure and the expression of militant genes (sciencenow, 10, 2003), but the reasons for these changes are still unclear. Scientists speculate that adult bees control the development of young bees through some chemicals, and experience shows that this can only be achieved when they can contact each other.

Isabelle Leoncini, a graduate student at avignon University, and Gene Robinson, an entomologist at the University of Illinois at Urbana-Champaign, began to look for this mysterious substance. They found that compared with "wet nurse bees", the concentration of ethyl oleate in the honey stomach of battle bees is about 30 times higher-this fatty acid mixture, as a component of pheromones in bee colonies, controls the evolution of bee behavior. Researchers reported in this week's online edition of Proceedings of the National Academy of Sciences that young worker bees ate a mixture of ethyl oleate and honey and became impulsive and aggressive after two or three days. Robinson said that the life span of bees is only five to six weeks, but this time delay does exist. They believe that the battle bees probably passed this pheromone to the small worker bees when exchanging food.

Francis Ratnieks, an insect behaviorist at the University of Sheffield in the United Kingdom, said that this discovery revealed an interesting negative feedback mechanism. He explained that the "honey-collecting bees" who are responsible for collecting nectar are most likely to receive pheromones handed down by the fighting bees, and they happen to be the next batch of bees to become "fighting bees". When there are a large number of battle bees in the beehive, such as rainy days, the content of ethyl oleate in the beehive will increase, and the process of transforming bee picking into battle bees will be delayed. On the contrary, after the number of fighting bees decreased, the level of ethyl oleate decreased and the growth rate of bee picking increased. He said that through this method, the hive can optimize the allocation of "bee power resources" according to its own needs and environmental conditions without any "centralized rule".