MODELS AND METHODS IN MODERN SCIENCE
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A SPHERE AND ITS FRAGMENTS
Makhmudov Azam Kudratovich
Teacher of mathematics at the Terdu Academic Lyceum.
Mamaraimov Bekzod Qodirovich
Teacher of mathematics at the Terdu Academic Lyceum.
Musurmonov Maruf Akrom oglu
Teacher of mathematics at the Terdu Academic Lyceum.
https://doi.org/10.5281/zenodo.15293325
Abstract.
This article provides a simple and clear understanding of the
structure, main properties and parts of a sphere. It describes such shapes as a
hemisphere, sector, segment and quadrant, their application in everyday life and
their importance in developing spatial thinking of students.
Keywords:
Sphere, hemisphere, sector, segment, quadrant, spatial shape,
geometric imagination, sphere.
The history of geometry begins in the distant past of the ancient world, but
it undoubtedly arose in the countries of the East. The development of geometry
can be characterized by four periods, but its boundaries cannot be separated by
any specific years. The first period, the period of the emergence of geometry,
covers the period up to the 5th century BC and is closely related to the
development of land surveying in ancient Egypt, Babylonia, and Greece (the
word geometry itself is derived from the Greek words: geo - earth and metrio -
measure, and its lexical meaning is land measurement). According to the
information written by the Greek historian Herodotus (c. 465-425 BC), the first
knowledge of geometry began to be formed in Egypt. It is said that the kings
distributed rectangular plots of land to the Egyptians for farming and collected
taxes from the landowners accordingly. The plots that were destroyed as a result
of the flooding of the Nile River were re-measured and the amount of tax was re-
determined accordingly. A number of necessary needs, such as distributing land,
determining the amount of tax, measuring surfaces, and building irrigation
structures, were factors in the formation of geometry in Egypt. It is also worth
noting that Mathematics and geometry are one of the most important sciences
that develop human thinking, logical thinking, and intellectual potential. This
science teaches a person to think systematically and consistently, to analyze the
problem in depth, and to draw reasonable conclusions. Mathematics also
strengthens a person's internal discipline and develops positive qualities such as
determination, patience, thoroughness, and accuracy. That is why mathematics
is not only a system of scientific knowledge, but also a means of personal
development and awareness.
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Mathematics is the main tool for understanding the world. Every event
around us, the balance in nature, technological processes, and economic
activities are based on certain mathematical laws. Therefore, a person who has
mastered mathematics can succeed in many aspects of life. Mathematics is the
main basis of many disciplines, from physics to economics, from engineering to
information technology.
The practical goals of mathematics education are aimed at the following
main areas:
Combining theoretical knowledge with practice. It is necessary for students
to develop the skills to apply theoretical knowledge taught during the lesson to
problems encountered in real life. This develops independent thinking, a
creative approach, and mathematical literacy in them. Students consolidate their
knowledge by performing more practical exercises in working with numbers,
algebraic expressions, equations, and other mathematical concepts.
Effective use of technical and visual aids. Modern mathematics lessons are
taught more effectively with the help of visual materials, graphic devices,
interactive whiteboards, computer programs, mathematical modeling and
animations. These tools facilitate students' understanding of the subject,
increase interest in the lesson and help them to master concepts more deeply.
Formation of independent learning skills. Students are taught to study
independently using textbooks, popular science literature, and Internet
resources. This develops their skills in working on themselves, deeply analyzing
the topics studied, searching for and mastering new knowledge. In today's
digital age, this skill is of particular importance.
Development of creativity and analytical thinking. Logical problems,
mathematical puzzles and tasks requiring unconventional solutions given in
mathematics lessons activate students' creative thinking abilities. Through this,
students develop the skills to approach any problem in different ways and find
alternative solutions.
Preparation for life. Mathematics provides many knowledge and skills
necessary in everyday life. For example, mathematical knowledge is of great
importance in shopping, budgeting, time planning, and analyzing statistical data.
Also, mathematics teaching methodology is one of the important areas of
pedagogical science, which studies the theoretical foundations, methodological
principles, and effective teaching methods of teaching mathematics. This area
teaches teachers how to organize mathematics lessons, what didactic materials
to use, and what teaching methods to apply.
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One of the most important and widespread three-dimensional shapes in
geometry is the sphere. The sphere occupies an important place in both
theoretical and practical fields with its perfect symmetry and aesthetic
appearance.
A sphere is a spatial div consisting of a set of points located at the same
distance from its center. It is completely smooth and round, and looks equal
from all sides. The surface of the sphere is called the “sphere”. Its main elements
are the center, radius, and diameter. The sphere appears in life in shapes such as
a ball, globe, and watermelon.
The boundary of a sphere is called the surface of a sphere or sphere. Thus,
all points that are a distance equal to the radius from the center of the sphere
are points of the sphere. The line segment connecting the center of the sphere
with any point on the surface of the sphere is also called the radius. By
definition, the coordinates of an arbitrary point M (x; y; z) of a sphere with
center A (a; b; c) and radius R satisfy the equation (x-a)2+(y-b)2+(z-c)2 R.
We call it the equation of the sphere. If the center of the sphere is at point A
(0; 0; 0), its equation takes the form x2+y2+z2 R.
Also, by definition, the coordinates of the points M ( x; y; z) of a sphere with
center A at point (a; b; c) and radius R satisfy the inequality (x-a)2+(y-b)2+(z-
c)2 R.
If we divide a sphere into two equal halves by a plane from the center, two
equal parts are formed. Each of these parts is called a "hemisphere".
Hemispheres are widely used in architecture (domes), containers, and other
fields.
When a sphere is cut by an arbitrary plane, a circle is formed in the section,
and the center of this circle is the perpendicular base dropped from the center of
the sphere to the cutting plane. The plane passing through the center of the
sphere is called the diameter plane. The intersection of the diameter plane with
the sphere is called a great circle, and the intersection with the sphere is called a
great circle. The arbitrary diameter plane of the sphere is its plane of symmetry.
The center of the sphere is its center of symmetry.
A sector of a sphere is a cone-shaped section extending outward from the
center. It forms a certain section inside the sphere. Such shapes play an
important role in technical and scientific modeling.
A segment is a piece of a sphere cut from the top or bottom, resembling a
slice of cake or watermelon. This shape is often found in the fields of
construction and design.
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If a sphere is cut through three mutually perpendicular planes, it is divided
into four equal parts. Each of these parts is called a quadrant. Quadrants are
used to build spatial models in the fields of physics and medicine.
The sphere and its parts are widely used not only theoretically, but also in
everyday life.
• Astronomy: The shape of the Earth and other planets is spherical.
• Medicine: Used to create models of internal organs and the brain.
• Technology and engineering: Mechanisms and details are designed.
• Education: Explained to students to develop spatial thinking.
The shape of a sphere plays a very important role in physics. The following
physical processes are associated with the spherical shape:
• Gravity: Planets, stars, and other celestial bodies assume a spherical shape
under the influence of gravity, because gravity acts uniformly at all points.
• In particle models: The spherical shape is used to model particles under
the influence of the forces between the atomic nucleus and electrons.
• Gases and liquids: Droplets of gas or liquid assume a spherical shape,
because this shape provides the minimum energy of the droplet.
Uses of the spherical shape in architecture and engineering:
Domes: The spherical shape is one of the most efficient structures in
architecture, evenly distributing large loads and providing structural strength.
For example, domes are used in mosques and other religious buildings, as well
as in modern engineering projects.
Bearings and mechanical systems: Spherical bearings and other mechanical
systems increase energy efficiency. For example, ball bearings are used in
machines and cars to increase speed and improve work efficiency.
Technologies based on the sphere and its parts:
GPS and navigation systems: Space networks and satellites are spherical in
shape, allowing for accurate and efficient measurement of the Earth's
circumference. These technologies are widely used in GPS systems and
navigation services.
In spherical cameras: Spherical cameras are designed to capture 360°
videos based on the shape of a sphere. This technology is used in virtual reality
(VR) and the film industry.
That is, the study of the shape of the sphere is of great importance in our
daily lives, in scientific research, technological achievements and in
understanding natural phenomena. This not only helps to understand
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mathematical and physical processes, but also ensures the effective functioning
of many technological and biological systems in life.
There are specific laws and rules in the creation of geometric shapes,
whatever is created, these did not happen by itself. Everything is discovered in a
state of inextricably linked with each other. The unique architectural
construction projects of cultural heritage monuments left by our ancestors from
the past history, the geometric shapes that reflect their charm, Islamic patterns,
interior and exterior designs have acquired a special significance and have been
spreading their name to the world from ancient history to the present. The
works of our great scholars and pioneering scientists, including mathematics,
astronomy, geography, philosophy, and medical laws, are among them. It is
impossible to imagine the whole world, the universe without geometry. The
intersection of horizontal and vertical axes at a point under 90 degrees forms
the (x – y) axis. This is the creator of geometric shapes, formed by the
intersection of circles or curves drawn on it using a compass at one point.
Conclusion, The concepts of the sphere and its parts help students to
master the science of geometry more deeply and form spatial thinking. With
their help, the student learns not only to see shapes, but also to imagine and
apply them in practice. This topic occupies an important place in mathematics.
References:
1.
To‘xtayev N. Geometriya asoslari. – Toshkent: O‘qituvchi, 2019.
2.
Karimov A.A. Maktab geometriyasidan masalalar. – Toshkent: Fan va
texnologiya, 2020.
3.
Axmedov M. Fazo jismlari va ularni tasavvur qilish. – Samarqand: Ilm Ziyo,
2018.
4.
Boboyev B.X. Geometrik shakllar va ularning modellashtirilishi. –
Toshkent: Yangi asr avlodi, 2021.
5.
Vilenkin N.Ya. Matematika va mantiq asoslari. – Moskva: Prosveshcheniye,
2015.
6.
Khan Academy – Geometry Course Materials, www.khanacademy.org
(https://www.khanacademy.org/)
7.
OpenStax – Geometry Textbook, https://openstax.org
8.
Sultonov I., Yuldashev R. Matematika o‘qitish metodikasi. – Toshkent:
TDPU nashriyoti, 2022.
9.
Djamalov R.M. Matematika kursi bo‘yicha amaliy mashg‘ulotlar. –
Toshkent: Universitet, 2016.
10.
International Bureau of Education (IBE). Mathematics Teaching for the
21st Century. – UNESCO Publishing, 2017.
