Euclidean Geometry

Euclidean geometry is a branch of mathematics that explores the properties and relationships of points, lines, surfaces, and shapes in three-dimensional space.

It is founded on a set of basic geometric concepts, along with postulates and axioms, known as primitive elements. These form the basis for a consistent framework of logical reasoning and deductions.

This framework was introduced by the Greek mathematician Euclid in his treatise "Elements," written around 300 BCE. As a result, this field of study is known as "Euclidean geometry."

Euclid’s geometry is built upon five fundamental axioms, or postulates, which are as follows:

1. The Line Postulate: A straight line can be drawn connecting any two distinct points, and there is only one such line.
2. The Extension Postulate: A line segment can be extended indefinitely in either direction.
3. The Circle Postulate: Given a point and a distance, a circle can be drawn with that point as its center and the distance as its radius.
4. The Right Angle Postulate: All right angles are equal to one another.
5. The Parallel Postulate: Given a line and a point not on that line, there is exactly one line parallel to the first that passes through the point.

Beyond these axioms, Euclidean geometry encompasses a wealth of theorems derived from them. Among the most famous are the Pythagorean theorem and Thales' theorem.

Often referred to as "plane geometry," Euclidean geometry serves as the foundation of school-level geometry and provides the building blocks for studying more advanced and abstract geometric systems.

Euclidean geometry is applicable in "flat" spaces, whether two-dimensional or three-dimensional, but it does not hold for curved surfaces.

For example, on the surface of a sphere, lines that appear to be parallel eventually intersect, violating Euclid’s fifth postulate.

This realization paved the way for the development of non-Euclidean geometries, such as spherical, elliptic, and hyperbolic geometry.

In summary, despite its limitations, Euclidean geometry remains a fundamental part of classical mathematics, providing essential tools for understanding and describing the space around us.