# Striking paradoxes of the Universe

Paradoxes can be found everywhere, from ecology to the geometry and the logic to chemistry. Even the computer on which you are reading an article, is full of paradoxes. Before you - ten explanations of curious paradoxes. Some of them are so strange, it is difficult to immediately understand what is the essence of ...

### 1. Banach-Tarski paradox

Imagine that you keep the ball in his hands. Now imagine that you started to tear the ball into pieces, with the pieces may be of any shape, what you like. After you put the pieces together so that you have got two balls instead of one. What will be the size of the balls compared to ball-original?

According to the theory of sets, the two resulting balloon will be the same size and shape as the balloon-original. Furthermore, given that in this case the balls are of different volume, any of the balls can be converted in accordance with another. This leads to the conclusion that can be divided into pea-sized balls with the sun.

The trick of the paradox lies in the fact that you can break the balls into pieces of any shape. In practice this is not possible - the material structure and ultimately atoms size impose some limitations.

To make it really possible to break the ball the way you like it, it must contain an infinite number of affordable zero-dimensional points. Then a ball of such points will be infinitely dense, and when you tear it forms lumps can get so complex that it will not have a certain volume. And you can collect these pieces, each of which contains an infinite number of points, a new ball of any size. A new ball will continue to consist of infinite points, and both balls will be equally infinitely dense.

If you try to translate the idea into practice, it will not work. But it turns out everything is fine when working with mathematical spheres - infinitely divisible number sets in three-dimensional space. Resolved paradox is called the Banach-Tarski and plays an important role in mathematical set theory.

### 2. Paradox Peto

It is obvious that the whales are much larger than us, it means that they have the bodies of more cells. And every cell in the body can theoretically become malignant. Consequently, the whales are much more likely to develop cancer than people, right?

Not this way. Peto paradox, named after the Oxford professor Richard Peto, argues that the correlation between the size of the animal and the cancer does not exist. In humans and whales chance of getting cancer is about the same, but some breeds of tiny mice are much more likely.

Some biologists believe that the lack of correlation in Peto paradox can be explained by the fact that larger animals are more resistant tumor: the mechanism works in such a way as to prevent the mutation of cells during division.

### 3. The problem of the present

That something might physically exist, it must be present in our world for some time. There can be no object length, width and height, and can not be an object without "duration" - "instant" object, that is, one that does not exist at least some amount of time, does not exist at all.

According to the universal nihilism, past and future do not take the time in the present. In addition, it is impossible to quantify the duration of which we call "real time": any amount of time, which you call "real time" can be divided into parts - past, present and future.

If this lasts, say, second, the second can be divided into three parts: the first part will be the last, the second - in this, the third - to the future. Third of a second, which we now call the present, can also be divided into three parts. Certainly the idea of you already understand - so you can go on forever. Thus, this actually does not exist, because it does not last over time. Universal nihilism uses this argument to prove that there is nothing at all.

### 4. The paradox Moravec

When solving problems that require thoughtful considerations people have difficulties. On the other hand, the main motor and sensory functions such as walking does not cause any trouble at all.

But if we talk about computers, the opposite is true: The computer is very easy to solve complex logical problems such as the development of chess strategy, but much more difficult to program a computer so that he could walk or reproduce human speech. This is the difference between natural and artificial intelligence known as the paradox of Moravec.

Hans Moravec, a robotics researcher at the University faculty of Carnegie Mellon University, explains this observation by the idea of reverse engineering our own brain. Reversible engineering the most difficult to carry out when the tasks that people perform unconsciously, for example, motor functions.

Because abstract thinking has become part of human behavior is less than 100 000 years ago, our ability to solve abstract problems is conscious. Thus it is much easier to create the technology for us that emulates this behavior. On the other hand, activities such as walking or talking, we do not comprehend, so make the AI do the same to us difficult.

### 5. Benford's Law

What is the chance that a random number starts with the number "1"? Or "3"? Or "7"? If you are somewhat familiar with the theory of probability, it can be assumed that the probability - one to nine, or about 11%. If you look at the actual numbers, you'll notice that the "9" is much rarer than in 11% of cases. Also, far fewer numbers than expected, starting with "8", but a whopping 30% of the numbers start with the digit "1". This paradoxical pattern manifests itself in all sorts of real cases, the number of people to share price and the length of the river.

Physicist Frank Benford first noted this phenomenon in 1938. He found that the frequency of occurrence of numbers as the first drops as the number increases from one to nine. That is, "1" appears as the first digit of about 30, 1% of "2" is about 17, 6% of the cases, "3" - approximately 12, 5%, and so on through "9" serving as the first digit only 4, 6% of the cases.

To understand this, imagine that you are consistently numeruete lottery tickets. When you tickets numbered from one to nine, any chance to become the first digit is 11, 1%. When you add ticket № 10, the chance of random numbers to begin with "1" is increased to 18 2%. You add tickets from number 11 to number 19, and the chance that the ticket number begins with "1" continues to grow, reaching a maximum of 58%. Now you add the ticket number 20 and continue numbered tickets. Chance that the number will begin with "2", is growing, and the likelihood that it will start with "1", falls slowly.

Benford's law does not apply to all cases of the distribution of numbers. For example, sets of numbers, the range of which is limited (human growth or weight) does not fall under the law. It also does not work with sets that have only one or two orders of magnitude.

However, the law applies to many types of data. As a result, power can use the law to detect fraud when the information provided does not follow Benford's law, the authorities may conclude that someone fabricated the data.

### 6. C-paradox

Genes contain all the information necessary for the creation and survival of the organism. It goes without saying that complex organisms must have the most complex genomes, but this is not true.

Celled amoeba genomes have 100 times more than a man, in fact, they have almost the largest known genomes. And very similar to each other species genome can vary dramatically. This oddity known as the C-paradox.

Interesting output from the C-paradox - gene may be larger than necessary. If all genomes in human DNA are used, the number of mutations per generation is extremely high.

The genomes of many complex animals like humans and primates includes DNA that does not code for anything. This is a huge number of unused DNA varies significantly from the spirit to the merits, it seems, neither of which does not depend on what makes the C-paradox.

### 7. Immortal ant on a rope

Imagine an ant crawling on a rubber rope length of one meter at a rate of one centimeter per second. Also imagine that every second rope stretched one kilometer. Does ant will reach sometime before the end?

It seems logical that a normal ant is not able to, because its speed is much lower than the speed at which the rope is stretched. However, eventually the ant gets to the opposite end.

When an ant not even started moving, before it is 100% of the rope. A moment later a rope has become much more, but also an ant walked some distance, and if we consider the percentage, the distance he has to go, decreased - it has less than 100%, even if slightly. Although constantly stretched rope, a small distance traveled ant becomes greater, too. And, although in general the rope is extended at a constant rate, the way ants every second becomes a little less. Ant, too, all the time continues to move forward at a constant speed. Thus, every second the distance that he has already passed, increases, and then he has to go - is reduced. As a percentage, of course.

There is one condition, that the problem could have a solution: the ant should be immortal. So the ant comes to end after 2, 8 * 1043.429 seconds, which is a little longer than the universe exists.

### 8. The paradox of the ecological balance of

model "predator-prey" - this is an equation describing the actual environmental situation. For example, the model can determine how to change the number of foxes and rabbits in the forest. Assume that the grass, which feed on the rabbits in the forest becomes more and more. We can assume that such an outcome is for rabbits is favorable because an abundance of grass they will be well to reproduce and increase in numbers.

The paradox of the ecological balance of claims that is not the case: first, the number of rabbits really grow, but the growth of the population of rabbits in a closed environment (forest) will lead to an increase in the population of the foxes. Then the number of predators will increase so much that they will destroy all the prey first, and then die out themselves.

In practice, this paradox does not apply to most species - if only because they do not live in a closed environment, so the animal populations are stable. In addition, the animals are able to evolve: for example, in the new conditions, new safeguards will be mining.

### 9. The paradox of Triton

Gather a group of friends and watch together this video. When you're done, let everyone express their opinion, increases or decreases the sound during all four colors. You will be surprised how different are the answers.

To understand this paradox, you need to know something about the musical notes. Each note has a certain height, which determines the high or low sound we hear. Note the next higher octave sounds in the two times higher than the previous note octave. And every octave can be divided into two equal tritone interval.

In the video Triton separates each pair sounds. In each pair, one sound is a mixture of the same notes of different octaves - e.g., a combination of two notes to where one above the other sounds. When sound Triton passes from one note to another (e.g., G-sharp between before) can be rightly interpreted as a note higher or lower than the previous one.

Another paradoxical feature newts - a feeling that the sound is constantly becoming lower, although the pitch remains the same. On this video you can see the effect for a full ten minutes.

### 10. The effect Mpemba

Before you two glasses of water, absolutely identical in all except one: the water temperature in the left glass is higher than the right. Place both cups in the freezer. In a glass of water will freeze faster? You can decide that the law, in which the water was originally colder, but the hot water freezes faster than water at room temperature.

This strange effect is named for a student from Tanzania, who observed it in 1986, when to freeze the milk to make ice cream. Some of the greatest thinkers - Aristotle, Francis Bacon and René Descartes - and the previously noted this phenomenon, but have not been able to explain it. Aristotle, e.g., hypothesized that any quality is enhanced in a medium opposing to this quality. Mpemba effect is possible due to several factors. Of water in a beaker with hot water can be smaller, since some of it will evaporate and the resulting freeze should minimal amount of water. Also hot water contains less gas, and therefore, in such water is easier to occur convection currents, therefore, it will be easier to freeze.

Another theory is based on the fact that weakens the chemical bonds that hold water molecules together. A water molecule consists of two hydrogen atoms bound to one oxygen atom. When water is heated, the molecules are moved slightly apart, communications between them diminishes, and the molecules lose little energy - this allows hot water to cool faster than cold.