Geometry Definition

Geometry is a branch of mathematics that is responsible for the study of figures and space, as well as their properties, their measurements and the relationships that exist between them. The word, rooted in the Greek γεωμετρία (geometry), refers to the “measurement of the Earth”.

I believe that there is no better way to introduce the importance of geometry in our lives. We live in a very complex world, full of shapes and figures, understanding them has been of vital importance in the development of our civilization, from the constructions of the first civilizations to the development of the Global Positioning Systems (GPS) that we use today. .

How was Geometry developed?

Geometry is one of the oldest disciplines that exist since there are records of its existence since Ancient Egypt. The true rise of geometry as a discipline occurred in Ancient Greece, where various philosophers used some of the ideas developed in Ancient Egypt. Some of the most recognized works were those of Pythagoras and Euclid, the latter is considered the father of geometry.

Most of us know Pythagoras from his famous theorem, in which he states that in a right triangle the square of the hypotenuse is equal to the sum of the squares of the legs. The Pythagorean Theorem was the basis for the development of trigonometry and continues to have applications in various branches of Physics and Engineering. In addition, Pythagoras studied how musical notes are produced with a string fixed at both ends and how dividing the string into different proportions gives rise to mutually consonant musical notes.

For his part, Euclid, who was born a couple of centuries after Pythagoras, laid the foundations of what we know today as Euclidean Geometry. Euclid’s Elements was his masterpiece and is considered the most successful book in the history of mathematics. Euclid’s work is based on five axioms that shaped the geometry developed later:

1) Given two points in space, you can always draw a line that joins them.

2) Any line segment joining two points can be extended in the same direction.

3) Given a point “O” and a segment that starts from it, a circle can be drawn whose center is the point “O” and whose radius is equal to the length of the segment.

4) All right angles are congruent to each other.

5) Given a straight line and a point external to it, a single line can be drawn that passes through that point and is parallel to the other line.

Of all these postulates, the last one has been the one that has been the most investigated over the years. Several mathematicians realized that the fifth postulate was independent of the other 4, that is to say, that there were cases in which the first four postulates were fulfilled, but the last one was not. These types of geometries were called Non-Euclidean Geometries and are the basis of modern GPS and the General Theory of Relativity.

René Descartes also made important contributions to the development of geometry. He proposed a system with which points, lines and figures could be located in space in order to describe them mathematically through algebraic expressions. This invention of Descartes would receive the name of Cartesian Coordinates and they are still used to this day.

The different branches of Geometry

As we can realize, geometry in turn is divided into different branches that study various cases and properties. Some of the most relevant are:

• Euclidean Geometry: It is the geometry developed from the five axioms of Euclid’s Elements and its field of application is flat spaces.

• Non-Euclidean Geometry: These are types of geometries in which the fifth postulate of Euclid’s Elements is not fulfilled, this happens in spaces that are not flat. Some types of non-Euclidean geometries are Elliptic Geometry, Hyperbolic Geometry, and Riemannian Geometry.

• Analytical Geometry: It is the one that arises from the union between Euclidean Geometry and Algebra. It was developed by René Descartes in conjunction with the Cartesian coordinate system.

• Differential Geometry: This type of geometry studies the properties of curves and surfaces that are subject to variations. Integrates Calculus and Algebra tools.

• Topology: This branch of geometry and mathematics studies the properties of geometric fields that are invariant under continuous transformations.

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