The world around us, despite the variety of objects and the phenomena occurring with them, is full of harmony thanks to the clear action of the laws of nature. Behind the apparent freedom with which nature draws outlines and creates the forms of things, there are clear rules and laws that involuntarily suggest the presence of some higher power in the creation process. On the verge of pragmatic science, which describes the phenomena that occur from the perspective of mathematical formulas and theosophical worldviews, there is a world that gives us a whole bunch of emotions and impressions from the things that fill it and the events that happen to them.
The ball as a geometric figure is the most common form in nature for physical bodies. Most of the bodies of the macrocosm and the microworld have the shape of a ball or tend to approach one. In fact, the ball is an example of a perfect shape. The generally accepted definition for a ball is considered to be the following: it is a geometric body, the set (set) of all points in space that are located from the center at a distance not exceeding a given. In geometry, this distance is called the radius, and in relation to this figure it is called the radius of the ball. In other words, in the volume of the ball are all points located at a distance from the center not exceeding the length of the radius.
The ball is also considered as the result of the rotation of a semicircle around its diameter, which at the same time remains stationary. At the same time, the axis of the ball (fixed diameter) is added to such elements and characteristics as the radius and volume of the ball, and its ends are called the poles of the ball. The surface of the ball is called a sphere. If we are dealing with a closed ball, then it includes this sphere, if it is open, then it excludes it.
Considering the definitions additionally related to the ball, it should be said about secant planes. The secant plane passing through the center of the ball is called a large circle. For other flat sections of the ball, it is customary to use the name "small circles". When calculating the areas of these sections, the formula πR² is used.
Calculating the volume of the ball, mathematicians were faced with quite fascinating patterns and features. It turned out that this value either completely repeats, or is very close in the way of determining to the volume of the pyramid or the cylinder described around the ball. It turns out that the volume of the ball is equal to the volume of the pyramid, if its base has the same area as the surface of the ball, and the height is equal to the radius of the ball. If we consider the cylinder described around the ball, then we can calculate the pattern according to which the volume of the ball is one and a half times less than the volume of this cylinder.
The way to derive the formula for the volume of a ball using the Cavalieri principle looks attractive and original. It consists in finding the volume of any figure by adding up the areas obtained by its section by an infinite number of parallel planes. For output, we take a hemisphere of radius R and a cylinder having a height R with a base-circle of radius R (the bases of the hemisphere and cylinder are located in the same plane). In this cylinder we enter a cone with a vertex in the center of its lower base. Having proved that the volume of the hemisphere and the parts of the cylinder that are outside the cone are equal, we easily calculate the volume of the ball. Its formula takes the following form: four third products of a cube of radius on π (V = 4 / 3R ^ 3 × π). This can easily be proved by drawing a common secant plane through the hemisphere and cylinder. The areas of the small circle and the ring bounded on the outside by the sides of the cylinder and cone are equal. And, using the Cavalieri principle, it is easy to come to the proof of the basic formula, with the help of which we determine the volume of the ball.
But not only the problem of studying natural bodies is associated with finding ways to determine their various characteristics and properties. Such a stereometry figure as a ball is very widely used in the practical activities of man. The mass of technical devices has in their designs parts not only of a spherical shape, but also composed of ball elements. It is the copying of ideal natural solutions in the process of human activity that gives the highest quality results.