what is Geometry ?

what is Geometry ?

The part of mathematics deals with the shape of individual objects and the properties of encompassing space. Also we can say Geometry ,the branch of mathematics is primarily concerned with the characteristics of figures as well as shapes. practically, geometry plays a great role in determining the areas, volume, and lengths. Geometry is a word coming from ancient Greek. Geo-“earth”, metron-“measurement “.
Euclid is considers to be the father of geometry.
 what is Geometry ?

History of Geometry:-

The construction of Yajnavedis in India attracted the attention of mathematicians to the study of geometry, their study was more akin to field committee. Historians believe that Indians knew such relationships as 32 + 42 = 52 (32 + 42 = 52) 1,000 years before Christ, but they did not study such relationships in any other way except for a few simple equations. About 600 years before Christ, the Roman mathematician Pithgauras studied this relationship in a logical manner and stated that the square on the hypotenuse in a right triangle is equal to the sum of the squares above the other arms.
Although the study of geometry started almost simultaneously in all the old civilized countries, such as Egypt, Babylonia, China, India and Greece, but no other country has progressed as much in this science as Greece. About 300 years before Christ, Euclid, a mathematician from Greece, sorted all the facts that were known by that time in a very logical manner. He tried to prove other facts on the basis of known facts. Thus, on sorting the facts, he reached some preliminary facts which are difficult to prove. Well they seem very clear. These facts are so simple that Euclid accepted them as axiomatic and called them facts themselves. The proof of theorems of geometry is dependent on these facts. Those facts are the following:
1. Those things, which are equal to the same thing, are also equal among themselves.
2. If equal items are added to equal items, then the sum is equal.
3. If equal items are subtracted from equal items then the remainder are equal.
4. Equals equal to equal objects.
5. If the two lines are cut by the third line and the sum of the inner angles of one side is less than two right angles, then where the joint is less, the two lines will meet at a point when extended.
4. Similarly, in creation, you can also create from one composition to another, but in the end, we reach some such compositions, whose use does not depend on other experiments. These compositions can also be taken forward only after considering themselves. They
(1) A line can be drawn from any point.
(2) Limited lines can be extended on both sides.
(3) Taking a point as center, we can draw a circle of radius.
Apart from this, they do not accept any other facts without proving them. Four of the above five facts themselves are so simple and clear that proving them is equivalent to proving your hand, but the fifth does not seem to be axiomatic. Mathematicians objected to this fact as axiomatic and tried hard to prove it. These efforts resulted in great inventions. Similarly, new geographical terms are mentioned in geometry. The definition of one word depends on the definition of other words. Finally, let us see that these definitions are based on the definitions of point, line and plane. According to Euclid, the plane is one in which the length is the width, but the thickness is not. Many people are skeptical of this definition as well, but a little reflection will make it clear that the definition is correct. For example, if two liquids are filled in a glass vessel that do not meet each other, then when they become stable then we will see that one plane separates the two substances. It does not have thickness. If it were, there would be a space between the two fluids that neither contained the substance below nor the top, but it is impossible. This example would have made it clear that the floor does not have thickness. It only has length and width. Similarly, looking at the shade of a flat wall in the sun, we can say that the line does not have width. The line lies in the plane, so the thickness of the floor is the thickness of the line. Therefore the line has neither thickness nor width, only length. If the lines meet at a point, then the width of the line is the length of the point, that is, there is neither length, nor width, thickness in the point. There is only space.
Everyone will be aware that in geometry, the properties of triangles, squares, circles, cones, cylinders, etc. are studied. 1 In the old times, some questions puzzled mathematicians. The solutions to those questions gave a lot of thought, there is no doubt in it, like making a cube whose cube is twice the given cube. At that time, the meaning of composition was understood to be composed only with the help of tracks and compasses. The second question was to make a square whose area is equal to the area of ​​the given circle. The third question was to divide a given angle into three equal parts. This work is impossible with track and compass, but can be done by other means. These questions kept mathematicians busy for centuries. Mathematics was greatly benefited by his deliberations, no doubt.
The Greeks also studied the properties of the ellipse, parabola, and hyperbola curve formed by cutting a cone from a plane.

Application of Geometry:-

Since birth ,humans are attracted to diverse shapes ,designs ,and colours. The aforesaid can be reinforced by the fact that while buying things in the market, humans are allured by the fabrics with fascinating patterns, books with eye-catching covers, sunglasses of one-of-it kind shapes, jewelry with captivating patterns, tea migs with beautiful forms ,and what not! Geometry can be referred to as being “omnipresent”. Major applications are in the field of constructions, roads, dams, bridge etc. and also in surveying, mapping, navigation. In software industry geometry is  used in animation, gaming, graphics design etc.

Branches of Geometry:-

Some major branches of Geometry are-analytical geometry, Euclidean Geometry, projective Geometry, differential geometry, non Euclidean Geometry, topology.

Analytical Geometry:-

Analytical geometry is also called coordinate geometry ,number geometry,  or Cartesian geometry, which constitutes the examination of geometry utilizing the coordinate system and these rules of algebra and reasoning. This contrasts with the general approach of Euclidean geometry, which holds a number of geometric concepts as primitives, and use deductive reasoning based on axioms and theorems get the facts. Analytical geometry is the foundation of most modern areas of geometry, including algebraic geometry, differential geometry and discrete geometry and calculations, and are widely used in physics and engineering.

Euclidean Geometry:-

Euclidean geometry, is also known as parabolic geometry. It follows some propositions that are based on five postulates of Euclid.
1:There is unique line joining two distinct points.
2:-If you have  a straight lines ,it is possible to extend in any direction to infinity.
3:It is possible to draw a circle given any center and a radius.
4:All right angles are equal.
5:If you have two straight lines ,and a third line crossing them, and the sum of the interior angle measures of the two lines is less than, two right angle.
Euclidean Geometry is of two types.
Plane geometry, I.e. two dimensional geometry and it is the study of line and shapes of flat surfaces.
Solid geometry, the three dimensional.

Non-Euclidian Geometry:-

How it came into existence :-Euclid’s fifth postulate that is also known as parallel postulate. Geometers were faced by the complexity of the parallel postulate and considered it could be proved as a theorem considering other four postulates .There appeared a problem with the parallel postulate whether it really was a postulate or it could be deduced from other axioms, propositions. The history of these attempts gave  rise to the establishment  Non-Euclidian Geometry and the Independence of the parallel postulate.
Non-Euclidian, its name tells geometry which is different from Euclidean geometry. It is consistent system of assumptions, definitions, and proofs that depict such objects as lines, points, planes .Spherical geometry and hyperbolic geometry are two most common Non-Euclidian Geometry. It is different from Euclidean geometry on the basis of the nature of parallel lines.

Projective Geometry:-

Projective Geometry ,it’s name satisfying it’s meaning. It all about the properties of geometric figures projection. Means projective Geometry is to deal with those properties of geometric figures that are not altered by projecting their image onto another surface. It’s development made in the 19th century which is a result of the movement away from the Euclidean geometry.

Differential geometry:-

It’s name define it’s meaning. It is the study of geometric properties with help of integral calculus and differential. The German mathematician Carl Friedrich Gauss (1777-1855),who is known as the father of differential geometry. His work on practical problems of surveying and geodesy ,developed the field of differential geometry. He characterized the intrinsic properties of curves ,surfaces, and manifolds using differential calculus.

Topology:-

Topology or topology is a major area of ​​mathematics. This is seen as an extension of geometry. It studies the properties that persist in objects upon continuous deformation. For example, distortions that occur when stretching something without tearing or cutting it. Topology has evolved from geometry and set theory.
The term ‘topology’ implies two things:
(1) Mathematics (topology), and
(2) For the union of sets used to define the basic concepts of this field
Topology is a broad field subject. It has several subfields. Some of its major areas are: Who used the principles of topology, a branch of mathematics