Tisch
Devlin, a Canadian science journalist, wrote in the Globe and Mail that after 1600 years of efforts, mathematicians finally proved that bees are the builders of work efficiency in the world.
Pepos, an ancient Greek mathematician in the 4th century, pointed out that the beautiful shape of beehives is the representative of the most efficient labor in nature. He guessed that the honeycomb with hexagonal cross section was built by bees with the least amount of beeswax. His conjecture is called the "hive conjecture", but no one can prove it.
Hale, a mathematician at the University of Michigan, announced that he had cracked the conjecture. Honeycomb is a very sophisticated construction project. When bees build nests, young and middle-aged worker bees are responsible for secreting flaky fresh beeswax, each with only one needle, and other worker bees are responsible for carefully placing these beeswax in a certain position to form a vertical hexagonal column. The thickness and error of each beeswax partition wall are very small. The width of the six partition walls is exactly the same, and the included angle between walls is exactly 120 degrees, forming a perfect geometric figure. People have always wondered why bees don't make their nests into triangles, squares or other shapes. Why is the partition flat, not curved? Although the beehive is a three-dimensional building, each beehive is a hexagonal cylinder, and the total area of beeswax wall is only related to the cross section of the beehive. This leads to a mathematical problem, that is, to find the plane figure with the smallest area and perimeter.
1943, the Hungarian mathematician Taos skillfully proved that the perimeter of a regular polygon is the smallest of all regular polygons connected end to end. But what if the edges of a polygon are curves? Taos thinks that a regular hexagon has the smallest circumference compared with other figures, but he can't prove it. However, when Hale thinks that the periphery is a curve, it is proved that the circumference of a figure composed of many regular hexagons is the smallest, no matter whether the curve protrudes outward or sinks inward. He has put the proof process of 19 page online. Many experts saw this proof and thought that Hale's proof was correct.
extreme
The hen, the triangle and the bad fox hatched four chicks. She was happy and worried. Happily, all four chicks are jumping, which is really pleasing; The worry is that the bad fox will steal the chicken.
In order to prevent the bad fox from stealing chickens, the hen found many boards and sticks and built a flat-topped cabin. Mother hen thinks that she is not afraid of bad foxes when she has a house.
Late at night, the fields were quiet. In the moonlight, a black shadow quickly ran near the cabin.
"Bang, bang!" The knock at the door woke the hen. "who?" Asked the hen.
"It's me, it's my husband's chicken. Open the door quickly. " A very ugly voice is answering.
The hen thinks: No! Husband's chicken is out of town, and it will take many days to come back. Besides, this ugly voice is not the voice of the husband chicken at all. The hen said loudly, "You are not an old hen, you are a bad fox. Go away quickly!" " "
When a bad fox can't be fooled, he shows his ferocious face. He snapped, "Give me the chicken! Or I'll tear down your house and eat you all! "
Although the hen was afraid, he said, "No, no, no! My chicks can't be eaten for you. "
The bad fox was furious and shook the flat-topped wooden house hard. Four chickens were so scared that they hid under the hen's wings. After shaking for a while, the frame of the house tilted. There was a big crack between the roof and the wall. A big fox's paw came in and caught a chick and ran away.
At dawn, birds fly around looking for food. A cry alarmed them.
Little yellowbird asks, "Hen, why are you crying?"
The hen cried and said, "I built a flat-topped wooden house to prevent bad foxes from stealing chickens." Who knows that the flat-topped wooden house is not strong, so the bad fox pushed it three times and two times. The bad fox caught a chick, blare ... "
The woodpecker said, "Little magpie can build a house. Please ask him to help you build a strong house!" " "
After a while, the woodpecker invited the magpie. The magpie said, "I can only build nests. Where can I build a house?"
"What shall we do?" Everyone is worried.
The magpie said, "Once I was in a big tree, I heard several builders say that the triangular roof is the strongest."
The woodpecker said anxiously, "Who has seen what a triangle looks like?"
The magpie made a triangle from three branches.
Everyone said, "Let's build it like this."
Some birds are holding branches, some are holding mud, woodpeckers are pecking holes in the wood, and magpies are tying up all the wood with twigs. When the sun went down, a new house with a triangular roof was built.
In the evening, the bad fox came again. This time, he said nothing and shook the wooden house desperately. Strange, why can't this wooden house shake tonight? ! The bad fox shook it again with all his strength, but it still did not move.
It's almost dawn, and the bad fox said bitterly, "Even if I spare you now, I'll come back tomorrow." If you dare to come out, I will eat you! " "
Early in the morning, the bird saw the hen worried about guarding the cabin again.
Little Shan Ying asked, "Mother hen, your cabin is very nice. What are you worried about? "
The hen said, "the triangular roof is safer, but we can't always stay in the house!" The bad fox said he would catch the chicks as soon as we came out. "
The lark said, "I have a good idea. We helped mother hen fence the house and install a wooden fence door to get in and out. Isn't this a good way to prevent bad foxes? "
Everyone said it was a good idea, so they started to build a wooden fence together. They also sharpened the top to prevent bad foxes from jumping in. Finally, install a rectangular wooden fence door.
At night, the bad fox really came again. He saw the baby chicken jumping around in the fence, drooling. The bad fox turned around the wooden fence twice and found that it was the easiest to break the fence door. He shook the wooden fence door with two claws. As a result, the rectangular door became a parallelogram, revealing a gap. The bad fox jumped in. If the hen doesn't bring the chick to the house quickly, I'm afraid it will be a disaster.
The bad fox is gone. The little magpie flew in and said, "Rectangular doors are easy to deform. Nail a board diagonally and turn it into two triangles, which will be much stronger. "
The lark said, "We can't always guard against bad foxes. We should do this ... like this. " Everyone was very happy and busy for a while before leaving.
The bad fox came quietly again without eating the chicks. He went straight to the wooden fence door and shook it vigorously. Hey, why can't you shake it this time? The fox shook it with all his strength and fell into the trap with a splash. The bottom of the trap was covered with triangular spikes, and the sly fox died.
The hen said happily, "triangles are really useful!" "
Tisso
Sisyphus is a string in ancient Greek mythology. Sisyphus, king of Corinth, was punished for pushing a boulder up a mountain, but no matter how hard he tried, the boulder inevitably rolled down before reaching the top of the mountain, so he had to push it again, endlessly. The String of Sisyphus is named after this story.
What is a Sisyphus string? That is to say, take any number, such as 35962, and count the even number, odd number and all numbers in this number, and you can get 2(2 even numbers), 3(3 odd numbers) and 5 (five digits in total), and use these three numbers to form the next number string 235. Repeat the above process for 235 to get 1, 2, 3. Repeating the sequence of 123 will still get 123. 123 is a digital black hole for this "universe" composed of programs and numbers.
Can each number finally get 123? Try with a large number. For example: 88883377499222, in this number, the number of even numbers, odd numbers and all numbers are 1 1, 9 and 20 respectively. Add these three numbers to get 1 1920, which is 1 1920.
This is the mathematical black hole "Sisyphus string".