What are the mathematical theorems of terror? 1. Drunk bird.
Theorem: A drunken man can always find his way home, but a drunken bird may never get home.
Suppose there is a horizontal straight line, starting from a certain position, there is a 50% probability of going left 1 m and a 50% probability of going right 1 m. What is the probability that you will eventually return to the starting point if you wander indefinitely like this? The answer is 100%. In the process of one-dimensional random walk, as long as the time is long enough, you can always return to the starting point.
Now consider a drunk walking randomly in the street. Assuming that the streets of the whole city are distributed in a grid, every time an alcoholic comes to a crossroads, he will choose a road (including the one he came from) with equal probability to continue walking. So what are the chances that he can finally return to the starting point? The answer is still 100%. At first, the drunk may go further and further, but in the end he can always find his way home.
However, drunken birds are not so lucky. If a bird chooses a direction equally from up, down, left, right, front and back every time it flies, it will probably never return to its starting point. In fact, the probability of walking randomly in a three-dimensional grid and finally returning to the starting point is only about 34%.
This theorem was proved by the famous mathematician Paulia in 192 1. With the increase of dimensions, the probability of returning to the starting point will be lower and lower. In the four-dimensional grid, the probability of returning to the starting point is 19.3%, while in the eight-dimensional space, the probability is only 7.3%.
2. Irretrievable hairball
Theorem: You can never straighten the hair on a coconut.
Imagine a sphere with hair on its surface. Can you comb all your hair flat, without leaving a lock of hair like a comb or a lock of curly hair like hair? Topology tells you that this is impossible. This is called the hairball theorem, which was first proved by Brouwer. In mathematical language, it is impossible to have a continuous unit vector field on the sphere. This theorem can be extended to higher dimensional space: there is no continuous unit vector field for any even dimensional sphere.
Hairball theorem has an interesting application in meteorology: because the wind speed and direction on the earth's surface are continuous, there will always be a place where the wind speed is zero, which means that cyclones and eye holes are inevitable.
Divide the ham sandwich equally.
Theorem: Given any ham sandwich, there is always a knife to cut it, so that ham, cheese and bread are just divided into two equal parts.
More interestingly, the name of this theorem is really called "Ham Sandwich Theorem". It was written by mathematician Arthur? Si Tong and John? Proved by John Tukey in 1942, it is of great significance in measure theory.
The ham sandwich theorem can be extended to the N-dimensional case: if there are N objects in the N-dimensional space, there is always an n- 1 dimensional hyperplane, and each object can be divided into two equal "volumes". These objects can be any shape, disconnected (such as slices of bread), or even grotesque point sets, as long as the point sets are measurable.