The world around us, despite the diversityobjects and phenomena occurring with them, is full of harmony due to the precise action of the laws of nature. Behind the seeming freedom with which nature draws the outlines and creates the forms of things, lurking clear rules and laws involuntarily suggesting the presence of some kind of higher power in the process of creation. On the verge of pragmatic science, which describes the occurring phenomena from the position 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 happening to them.
The ball as a geometric shape is the mostoften found in nature form for physical bodies. Most of the bodies of the macrocosm and the microworld are in the shape of a ball or they tend to come closer to that. In essence, the ball is an example of an ideal shape. The generally accepted definition for a ball is considered to be the following: it is a geometric body, a set (set) of all points of space that are from the center at a distance not exceeding the specified one. 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, all points located at a distance from the center not exceeding the radius length are enclosed in the volume of the ball.
Шар еще рассматривают как результат вращения semicircle around its diameter, which at the same time remains fixed. 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 deal with a closed ball, then it includes this sphere, if with an open one, then it excludes it.
Considering additionally related to the balldefinitions should be said about cutting planes. A cutting plane passing through the center of the ball is usually 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 are faced withquite fascinating patterns and features. It turned out that this quantity either repeats completely or is very close in the method of determination 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 smaller than the volume of this cylinder.
Привлекательно и оригинально выглядит способ the derivation of the formula for the volume of the ball using the principle of Cavalieri. It consists in finding the volume of any figure by adding the areas obtained by its section with an infinite number of parallel planes. For output, take a hemisphere with radius R and a cylinder having a height R with a base circle with a radius R (the bases of the hemisphere and the cylinder are located in the same plane). In this cylinder we enter the cone with the apex in the center of its lower base. Proving that the volume of the hemisphere and the parts of the cylinder that are outside the cone are equal, we can 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). It is easy to prove, having spent the general cutting plane through a hemisphere and the cylinder. The areas of the small circle and the ring, bounded on the outside by the sides of the cylinder and the cone, are equal. And, using the Cavalieri principle, it is not difficult to come to the proof of the basic formula, with the help of which we determine the volume of the ball.
But not only with the problem of studying natural bodies.connected finding ways to determine their various characteristics and properties. Such a figure of stereometry as a ball is very widely used in practical human activity. The mass of technical devices has in its designs details not only of spherical shape, but also composed of ball elements. It is copying of ideal natural solutions in the process of human activity that gives the highest quality results.
