After reading this, you can brag about your tallness when you get together with friends. Pen ~ fun ~ pavilion www.biquge.info

The 2016 Nobel Prize in Physics was awarded to three scientists, David Solis, Duncan Haldane and Michael Kostellitz, for their discovery of the topological phase of matter and their theoretical contributions to topological phase transitions.

Zhang Shousheng, a professor of physics at Stanford University, said that topology is a geometric concept that describes the properties of geometric patterns or spaces that can remain unchanged after continuously changing shapes.

The topology is very high? Actually, it has the most down-to-earth theorem

[Theorem 1: You can never straighten out the hair on a coconut]

Imagine a sphere full of hairs, and you can comb all the hairs without leaving any pinches of hair like a cockscomb or a whirl like hair? Topology tells you that this is not possible.

This theorem is known as the "hairball theorem" and was first proved by Brouwer. In mathematical language, it is impossible to have a continuous unit vector field on the surface of a sphere. This theorem can be generalized to higher-dimensional spaces: for any even-dimensional sphere, there is no continuous unit vector field.

The hairball theorem has an interesting application in meteorology: since the wind speed and direction of the wind on the earth's surface are continuous, by the hairball theorem, there will always be a place on the earth with a wind speed of 0, that is, cyclones and eye of the storm are inevitable.

Another unexpected "application" of the hairball theorem is in video games! Many people will find a problem when playing first-person shooters: when you move the mouse up and make your character look up at the sky, a hand shake will find that your character turns 180 degrees in an instant, and the same phenomenon occurs in other games when looking down at the soles of your feet. This is the "spin" of the hairball that you encountered.

This happens because the game engine needs to solve a math problem: the data that the player enters with the mouse is just an axis of gaze, around which the game screen can theoretically rotate arbitrarily. So where should the actual picture be up and down? This requires each mouse data to correspond to a direction -- a vector field. Unfortunately, the hairball theorem states that there must be at least one discontinuous point in this field, so the slightest movement of the mouse near this point can cause the screen to flip dramatically.

This is not the case with VR devices, because it's not just the mouse position that determines the VR screen, it's the whole headset, so there's no spinning.

[Theorem 2: For any ham sandwich, a knife must be cut so that two slices of bread and a slice of ham are divided into two equal parts of the same size]

The word "any" is loose – the ingredients that make up a sandwich don't have to touch each other, and each ingredient doesn't have to be one slice of its own right, but can be many. It doesn't matter if you put your sandwich in a blender and make a sauce, or tear it up and feed it all to the duck – as long as your sandwich is divided into three parts, there must be a knife that can cut each part in half.

It can also be extended to the n-dimensional case: if there are n objects in n-dimensional space, then there is always an n-1-dimensional hyperplane that divides each object into two equal parts of "volume".

This theorem is called, as you might expect, the "ham sandwich theorem." It was first demonstrated by Stephen Barnach that it appeared in algebraic topology and has important uses in measure theory.

[Theorem 3: The International Date Line is Indispensable]

Time zones on Earth are connected by pairs, with East 8 followed by East 9, then East 10, and so on - with one exception: the International Date Line. It's a day apart on both sides.

Can we devise a time zone system that doesn't require an international date line? The answer is no, no matter how detailed and cumbersome it is. This is a corollary of the Bosuk-Ulam theorem in the one-dimensional case in topology, which was proposed by Ulam and proved by Bosuk in 1933.

In fact, the theorem itself is formulated as "Given any continuous function from an n-dimensional sphere to an n-dimensional space, two points symmetrical to the center of the sphere can always be found on the sphere, and their function values are the same." "When n = 1, it becomes the correspondence between the equator and time.

Another corollary of this theorem is that there are always two symmetrical points on Earth, and their values of temperature and atmospheric pressure are exactly the same.

Theorem 4: Hold a coffee cup full of coffee, and without letting go or spilling the coffee, you must rotate the cup twice to get your hands, arms, and cups back to their original shapes.

(Do not try this experiment with hot coffee.) ）

Method: Reach forward and backhand hold the coffee cup, then gradually rotate it towards your chest, passing under your armpit for the first lap. At this point, the coffee cup has finished spinning, but the arm has twisted into a strange shape. At this point, raise your arms and make a second turn from the top of your head to restore everything.

Hand remnants of the party's attention: You can use an empty cup, so as not to pour water into your neck.

In fact, the rotation of your hand and the coffee cup is called the rotation group so(3) in topology, and a complete return to the original state is equivalent to drawing a ring in so(3). In topology, the basic group of so(3) is "z/2" - this means that you have to revert the coffee cup twice to get your entire arm back once.

[Theorem 5: If you lay a local map on the ground, you can always find a point on the map, and the point on the ground below this point is exactly the location it represents on the map]

In other words, if you draw a map of the entire mall on the floor of the mall, you can always make a precise "you are here" mark on the map.

In 1912, the Dutch mathematician Brouwer proved a theorem: assuming d d is the set of points in a disk, and f is a continuous function from d to itself, then there must be a point x, such that f(x)=x. In other words, if all the points in a disc move continuously, there will always be one point that can return exactly to the position before the motion. This theorem is called Brouwer's fixed-point theorem.

In addition to the "map theorem" above, there are many other fantastic corollaries of Brouwer's fixed point theorem. If you take two sheets of paper of the same size, crumple one of them and place it on the other, according to Brouwer's fixed point theorem, there must be a point on the ball of paper that is just above the same point on the paper below.

This theorem can also be extended to three-dimensional space: when you have finished stirring your coffee, you will always find a point in the coffee that is in the same position before and after stirring (although this point may have been somewhere else during the stirring).

There is also a headphone cable......

Why do the headphone cables always wrap up in a ball?

Every time I take out my headphones from my bag and try to listen to music, I will find:

No matter how neatly the cable is wrapped beforehand, it will always twist into a mess in the bag.

In recent years, physicists and mathematicians have been pondering why wires are so disobedient. Through experimentation, they found that there are a number of interesting solutions to wire knotting. In 2007, researchers from the University of California, San Diego, put several pieces of rope in a box and turned them upside down in an attempt to explain why the cord gets knotted when it dangles in your backpack.

The results of the study show that random motion always seems to end up leading to knots.

A long, soft rope spontaneously forms many different configurations in its natural state: perhaps in a neat straight line, or perhaps one end of the rope bends and crosses the middle. In practice, the latter is the majority: the rope always tends to wrap itself around itself and eventually clump together. In these random configurations, there are basically no lumps, so these ropes will basically end up in a mess. Once knotted, it is less likely to untie it automatically. As a result, the knots of the rope will only increase.

Knotting rope is not a simple problem, and mathematicians have created a sub-discipline of topology called knottheory, which is used to study the mathematical properties of knots.

The mathematical definition of a kink is any simple closed curve in three-dimensional space. Using this definition, mathematicians have divided knots into several categories: for example, in the simplest trilobal knot, where the rope crosses itself only three times, and similarly, there is a knot formed by the rope crossing itself four times, known as the figure-eight knot. Mathematicians have found a set of numerical formulas called Jones polynomials to define each type of kink. However, for a long time, knot theory was considered a somewhat inscrutable branch of mathematics.

In 2007, physicist Douglassmith and his understudent at the time, Dorianraymer, decided to test the knot theory with real rope. In the experiment, they put a piece of string into the box and then flip the box for 10 seconds. Subsequently, Reimer changed the length, hardness, box size, flipping speed and other parameters of the rope and performed about 3,000 replicates.

The results showed that there was a probability of about 50% that the rope would tie a knot. One of the main factors influencing this outcome is the length of the rope: ropes less than 1.5 feet (about 46 centimeters) in length are less likely to get knotted, and as the length increases, so does the chance of knotting. However, there is also an upper limit to this, and when the length of the rope reaches 5 feet (about 152 cm), it will fill the entire box, and in more than 50% of cases it will not be knotted.

Reimer and Smith also used Jones polynomials, invented by mathematicians, to classify the knots they observed. After each flip, they take a picture of the rope and feed the image data into a computer algorithm to classify the knots. According to kink theory, there are 14 basic knots, all of which contain no more than 7 crosses. Reimer and Smith observed all 14 knots during their experiments, and also found more complex knots, some with as many as 11 crossings.

The researchers eventually built a model to explain their observations. In general, in order to put the rope in the box, the rope must be coiled. At this point, the end of the rope will be parallel to some of the segments of the rope. When the box is flipped over, there is a chance that the end of the rope will fall into the middle of the parallel segments and form a cross. After multiple crossings, the end of the rope is essentially wrapped around a certain segment of the rope, forming a different knot.

In fact, after watching for a long time, what we want to know most is whether there is a way to make the wire not knotted? Researchers have observed in experiments that if a stiffer rope is used, the probability of knotting will be reduced. Probably that's why Apple has chosen stiffer materials for the power cords of recent generations of laptops. This explains why the long, thin Christmas tree lights are always a mess, while the short, thick wiring boards are always flat.

In addition, the smaller container prevents knots. Experiments have found that when a longer rope is in a smaller box, it will stick to the inner wall of the box because the rope has a tendency to unfold, so that the end of the rope will not fall into the middle of the rope and wrap around when the box is flipped. This is the reason why the probability of umbilical cord knotting is low (about 1%): the space in the uterus is not enough for the umbilical cord to knot.

Finally, the higher speed at which the box is flipped reduces the chance of the rope getting knotted. Because of the centrifugal force, the rope will stick to the inner wall of the box, and there is no possibility of knotting at all. However, this method doesn't seem to be able to solve the problem of the cable getting knotted in the bag. Maybe you can do somersaults or buy some clothes with small pockets - I think it's more realistic to do the latter. (To be continued.) )