Some scientists and philosophers have expressed the whole truth universe can be found in the heart of the particle. Interestingly this statement has also been expressed in the speeches of mystics and Innocent Imams in Islam religion. For example, Hatef Esfahani says:
Open the eyes of soul / that What is invisible will be visible
When every particle splits / A sun will see in the middle
:We also have Attar’s Mantiq-Al-Tair
If the heart of a particle would open / It would see what life is inside it
In such a sea, which is the Great sea / the universe is a particle, and the particle is the universe
Although it is easy to say this elegant matter, it is not easy to accept. Many people find it possible to accept this principle only with a poetic and intuitive look. This short article will introduce an exciting idea that can open the door to a complete understanding of this fact.
To understand this article, every person should learn a simple mathematical concept called function and one-to-one correspondence introduced in junior high school mathematics. The only mathematical concept we describe in this article is the attractive concept of countless numbers, which is not very important to our reasoning. However, we try to explain this very simply to engagement. Finally, we present the main argument.
Definition of innumerable numbers
We are all familiar with finite numbers like 1, 2, 3, 100, 8794, Etc but there is an exciting division about infinite numbers:
A– Unlimited numbers that are countable (numeric numbers)
When we talk about an infinite number, which is countable, we are talking about a number that can be counted with a set of natural numbers and, in mathematical terms, correspond one by one to natural numbers. The set of natural numbers are the same numbers that we use to count objects in our daily lives and are: {1,2,3,4, … }
For example, the number of even numbers is unlimited and countable because 0 is the first even number, 2 is the second even number, 4 is the third even number, 6 is the fourth even number, 8 is the fifth even number, Etc. In this way, we can count even numbers. Another example of infinite numbers is odd numbers, multiples of 3, multiples of 10, and rational numbers. Rational numbers (or fractional numbers) are infinite, but they are countable because we can make a one-to-one correspondence between rational numbers and natural numbers.
In this diagram, all rational numbers are written in neat order, and all of them can be counted by moving over arrows. Of course, we do not consider the fractions that can be simplified because we have counted them before. Therefore, we can say that rational numbers correspond to natural numbers one by one.
B– Unlimited numbers that are not countable (innumerable numbers)
Innumerable numbers are not finite and cannot be counted. In other words, one cannot find a function that makes a one-to-one correspondence between an innumerable number and natural numbers. For example, the number of real numbers or R is in this category. Real numbers are the same numbers on an arbitrary line from negative to positive infinity.
Given that real numbers are made up of a set of rational and vague numbers, both of which are infinite, and that rational numbers in the previous section corresponded to natural numbers one by one, it is easy to prove that the number of real numbers is innumerable.
One-to-one correspondence between R and interval [-1,1]
The exciting point is that any arbitrary interval of real numbers, no matter how small, corresponds to the whole real numbers one by one, and in a sense, we can say that their number is equal. For example, the number of numbers in the interval [1,-1], which includes all real numbers between the two numbers 1 and -1 and 1, corresponds to the total number of real numbers one by one.
To prove this simple proposition, It suffices to find a function that creates a one-to-one correspondence between these two sets. If x belongs to the domain[-1,1], then the range of the following function goes from -∞ to +∞.
We should say a simple point before the final argument in this article. One of the properties of the corresponding sets is that if the three sets A, B, C correspond to the three sets E, F, D, respectively, then two sets A × B × C and E × F × D also correspond to each other one by one. Thus [-1,1] and R also correspond to each other.
Equivalency of a small particle and the world!
We are now preparing to make our final argument. Consider a spherical marble (instead of a hypothetical particle in the universe) of radius one with a spherical universe whose radius is infinite. If we assume the center of both spheres at the origin of the coordinate system, each point inside our small marble corresponds precisely to one and only one point inside the infinite sphere, or the universe.
Suppose that point (X, Y, Z) is an arbitrary point inside a small sphere to prove this. The numbers X, Y, Z are in [-1,1]. Therefore, for each of these points, one and only one point called X1, Y1, Z1 can be found on the axes of length, width, and height that correspond to them one by one. So the point (X, Y, Z) will correspond to one and only one point in the universe called (X1, Y1, Z1). In other words, according to the last point mentioned in the previous section, [-1,1] and R are in one-to-one correspondence with each other. So we could easily match the infinite world with very tiny marble and align them!
Concluding remarks
The exciting thing is that our marble can be as small as we want, and the only condition is that its radius is not zero or it is not lacking. It is enough to note that we did not consider a specific size or unit for the number 1 in the range [-1,1] in the previous section, and we can consider it as small as we want.
Another interesting result is that any sentence or fact about the points of the larger sphere is translatable into the language of the points inside the marble by substituting the corresponding points of the small sphere.
The above issue can point to the article beginning, which says: All the universe’s secrets at the heart of a particle.
