THE Geometry it is one of the three major areas of Mathematics, alongside calculus and algebra. The word “geometry” has a Greek origin and its literal translation is: “to measure the earth”. This information gives us clues as to how it was born and why it developed over the centuries.
THE Geometry it is the study of the forms of objects present in nature, the positions occupied by these objects, the relationships and properties relating to these forms.
How is geometry constructed?
THE geometry is built on primitive objects: point, line, plane, space, among others. These objects have no definition, but they have characteristics that make their identification possible.
Using these primitive objects is that the first geometric shapes of the plane: line segments, polygons and angles. From them, the definition of the distance between two points is made, on which the definition of a circle depends. All of this serves as the basis for building the spatial geometry.
THE geometry is also responsible for properties of the
geometric figures. These properties are nothing more than results of relationships analyzed in objects and geometric figures. A property of circles, for example, is the following: the result of dividing the perimeter of a circle and its diameter will always be equal to π (approximately 3.14).Thus, the geometry it is built by relating basic objects in order to obtain more elaborate objects. These are related to each other to arrive at even more elaborate objects and so on.
Geometry divisions
Currently, geometry is divided into two sets: Euclidean Geometry and Non-Euclidean Geometries.
Non-Euclidean Geometries
Euclides, great mathematician and writer, probably lived in the III century; Ç. and is called the father of geometry. He was the first to bring together all the geometry in a single work, called “The Elements”. This mathematician based the plane geometry on five postulates.
The fifth of these postulates is much more sophisticated than the other four. This raised doubts among mathematicians from his time until the mid-19th century, when Lobachevsky, a Russian mathematician, decided to reconstruct the geometry, but using the negation of Euclid's fifth postulate.
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This postulate stated: Through a point outside a line passes a single line parallel to the given line. Lobachevsky considered the opposite: Through a point off a straight line passes more of a line parallel to the given line.
Geometric objects and figures are defined in the same way as in plane geometry, the only difference being actually the fifth postulate.
The results obtained by Lobachevsky are divided as follows: those that do not depend on Euclid's fifth axiom are identical to traditional geometry. Those that depend are different. For example, the sum of the inner angles of a triangle, in geometries constructed after Lobachevsky, is not equal to 180°.
Lobachevsky's studies gave rise to Rhiemannian geometry and opened a door for the construction of other geometries completely different from the plane and spatial geometry we know. The most interesting fact is that his results have many applications in everyday life.
Euclidean Geometry
It is the geometry discussed in elementary and high school and the only geometry known to man until the mid-19th century. Euclidean geometry is divided into the following subareas:
plane geometry: All figures, shapes and definitions are made for objects belonging to the plane, that is, they have only width and length, but no depth.
The concepts discussed by plane geometry are point, line, plane, relative positions, distance between two points, angles, polygons, areas and trigonometry, among others.
Spatial Geometry: Objects belong to three-dimensional space, that is, now there is the possibility to consider their depth.
The concepts discussed in spatial geometry are: all those of plane geometry, in addition to planes, polyhedra and round bodies.
Analytical Geometry: Subarea that relates geometry to algebra and uses one to solve problems arising from the other.
The concepts discussed in analytic geometry are: all concepts and definitions of plane geometry and from an algebraic point of view, coordinates, vectors, matrices, quadrics and solids of revolution, among others.
By Luiz Paulo Moreira
Graduated in Mathematics
Would you like to reference this text in a school or academic work? Look:
SILVA, Luiz Paulo Moreira. "What is geometry?"; Brazil School. Available in: https://brasilescola.uol.com.br/o-que-e/matematica/o-que-e-geometria.htm. Accessed on June 27, 2021.
