Paradoxes can be found everywhere, from ecology to geometry and from logic to chemistry. Even the computer on which you are reading the article is full of paradoxes. Before you — ten explanations quite fascinating paradoxes. Some of them are so strange that we just cannot fully understand what is the point.

1. The paradox of Banach-Tarski Imagine that you are holding a ball. Now imagine that you began to tear the ball to pieces, and the pieces can be any shape you like. After you fold the pieces together so that you get two balls instead of one. What will be the size of these balls compared to ball-original?

According to the theory of sets, the two resulting ball will be the same size and shape as the ball-the original. In addition, given that the balls have a different amount, then any of the balls may be converted in accordance with the other. This allows us to conclude that a pea can be divided into balls the size of the Sun.

The trick of the paradox is that you can break the balloons into pieces of any shape. In practice it is impossible — the structure of the material and ultimately the size of atoms impose some limitations.

To make it really possible to break the ball the way you like it, it must contain an infinite number of available zero-dimensional points. Then the Orb of these points will be infinitely dense, and when you break it, shapes of pieces can be so complex that will 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, then nothing happens. But all good when dealing with mathematical fields — infinitely divisible numeric sets in three-dimensional space. Resolved paradox is called the theorem of Banach-Tarski and plays huge role in mathematical set theory.

2. The Peto paradox is Obvious that whales are much larger than us, which means that their bodies are much more cells. And every cell in the body can theoretically become malignant. Therefore, whales are much more likely to get cancer than humans, right?

Not so. The Peto paradox, named after Oxford Professor Richard Peto, States that the correlation between animal size and cancer does not exist. In humans and whales the chance of getting cancer is about the same, but some breeds of tiny mice have a much greater chance.

Some biologists believe that the absence of correlation in the paradox of pet can be explained by the fact that larger animals are able to better resist tumor: the mechanism works in such a way to prevent the mutation of cells in the fission process.

3. The problem now So that it could physically exist, it must exist in our world for some time. Can be no object without length, width and height, and also cannot be the object without a "duration" — "instant" object, i.e. one that does not have at least some amount of time, does not exist.

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

If the present lasts for, say, second, that moment can be divided into three parts: the first part is past, second present, third future. A third of a second, which we now call real, can also be divided into three parts. Certainly the idea you already knew — so can continue indefinitely.

Thus, the present actually exists, because it does not continue in time. Universal nihilism uses this argument to prove that nothing exists.

4. The paradox Moravek in solving problems that require thoughtful reasoning, people happen difficulties. On the other hand, basic motor and sensory functions like walk does not cause any difficulties at all.

But if we talk about computers, the opposite is true: computers are very easy to solve complex logical problems like the development of chess strategy, but much harder to program a computer to be able to walk or reproduce human speech. This distinction between natural and artificial intelligence is known as the paradox of Moravek. Hans Moravec, a researcher from the faculty of robotics at Carnegie Mellon, explains this observation through the idea of reverse engineering our own brains. Reverse engineering of the hardest to hold when the tasks that people perform unconsciously, for example, motor functions.

Because abstract thinking became part of human behavior less than 100,000 years ago, our ability to solve abstract problems is a conscious. Thus, for us it is much easier to create technology that emulates such behavior. On the other hand, such actions like walking or talking, we don't comprehend, so to get artificial intelligence to do the same difficult for us.

5. Benford's law What is the chance that a random number begins with the digit "1"? Or with the digit "3"? Or "7"? If you are somewhat familiar with probability theory, we can assume that the probability is one in nine, or about 11%.

If you look at the actual numbers, you'll notice that "9" is much rarer than in 11% of cases. Also much smaller numbers than expected, started with 8, but a whopping 30% of numbers start with the digit "1". This paradoxical pattern is manifested in various real cases, the number of the population to stock prices, and lengths of rivers.

Physicist Frank Benford first noted this phenomenon in 1938 year. He found that the frequency of appearance of digits as the first drops as the number increases from one to nine. That is, a "1" appears as the first digit approximately 30.1% of cases a "2" appears about 17.6% of cases, 3 is about 12.5%, and so on up to "9", serving as the first digit only 4.6% of cases.

To understand this, imagine that you are consecutively numbered raffle tickets. When you have tickets numbered from one to nine, the chance of any digit to be the first is 11.1%. When you add the ticket number 10, the chance of random numbers begin with "1" increases to 18.2%. You add the tickets from No. 11 to No. 19, and the chance that the ticket number will begin with "1" continues to rise, peaking at 58%. Now you add the ticket number 20 and continue to number the tickets. The chance that the number will begin with "2", increases and the probability that it starts with 1, slowly decreases.

Benford's law does not apply to all cases of distribution of numbers. For example, sets of numbers, the range of which is limited (human height or weight) under the law do not fall. 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. Ultimately, the government 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-the paradox of the Genes contain all the information necessary for the creation and survival of the organism. Needless to say that complex organisms must have the most complex genomes, but this is not true.

Single-celled amoeba have genomes up to 100 times more than humans, actually they have almost the largest known genomes. And very similar between species genome may be vastly different. This oddity is known as a paradox. An interesting conclusion from the C-paradox — the genome can be more than necessary. If all genomes in human DNA are used, the number of mutations per generation will be incredibly high.

The genomes of many complex animals like humans and primates include DNA that does not encode. It's a huge amount of unused DNA, greatly varies from creature to creature, it seems, does not depend on anything, which creates the C-paradox.

7. Immortal ant on a rope Imagine an ant crawling on a rubber rope one meter with a velocity of one centimeter per second. Also imagine that the rope every second stretched for one kilometer. Get whether the ant ever to end?

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

When the ant has not even started moving, before it is 100% rope. A second later, the rope was much longer, but the ant also went some distance, and if you count in percent, the distance that he must go, reduced — it's less than 100%, albeit not by much.

Although the rope constantly stretched, the small distance traveled by the ant also becomes larger. And, although in General, the rope elongates with a constant velocity, the path of the ant 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 passed, increased, and that he must pass decreases. In the interest, of course.

There is one condition that the task might have the solution: the ant must be immortal. So the ant will reach the end through a 2.8×1043.429 seconds, which is slightly longer than the universe exists.

8. The paradox of ecological balance model of the "predator-prey" is the equation describing the real environment. For example, the model can determine how to change the number of foxes and rabbits in the woods. Suppose the grass, which feed on rabbits, in the forest becomes more and more. We can assume that for rabbits the outcome is favorable, because the abundance of grass will multiply and increase.

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

In practice, this paradox is not valid for most types of animals — at least because they do not live in a closed environment, so the animal population stable. In addition, animals are able to evolve: for example, new conditions of production will be a new defense mechanisms.

9. The paradox of newt Gather a group of friends and all together this video. When finished, let everyone Express their opinions, the sound increases or decreases during all four colours. You will be surprised how different the answers will be.

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

In the video, the Triton shared by each pair of sounds. In each pair, one sound is a mixture of the same notes from different octaves — for example, the combination of the two notes up to where one sounds higher than the other. When the sound in the Triton goes from one note to another (for example, g-sharp between the two before), we can rightly interpret the note as higher or lower than the previous one.

Another paradoxical property of the tritons is the sensation that the sound is constantly becoming lower, although the pitch of the sound does not change. In our video you can see the effect within ten minutes.

10. Mpemba effect Before you two glasses of water, identical in every way but one: the water temperature in the left beaker is higher than in the right. Place both cups in the freezer. Which Cup of water freezes faster? You can decide what is right, where the water was initially colder, but hot water will freeze faster than water at room temperature.

This strange effect is named in honor of a student from Tanzania, who observed it in 1986-m to year when I froze the milk to make ice cream. Some of the greatest thinkers — Aristotle, Francis bacon and Rene Descartes — had previously noted this phenomenon, but have not been able to explain it. Aristotle, for example, hypothesized that any quality enhanced in an environment opposite to this quality.

Mpemba effect is possible due to several factors. Water in a glass of hot water can be less, since part of it will evaporate, and the result should freeze less water. Also hot water contains less gas, and thus the water is easier to arise convection currents, thus freezing it will be easier.

Another theory is based on the fact that weakens the chemical bonds that hold water molecules together. The water molecule consists of two atoms of hydrogen linked to one atom of oxygen. When water is heated, the molecules are slightly moved away from each other, the connection between them is weakened, and the molecules lose a bit of energy — this allows hot water to cool faster than cold.

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