Geometry is a mathematical branch that studies shape, size and nature of space. For centuries, Euclidean geometry—based on a posture filed by Euclid—became the main reference to understanding space. However, in the 19th century, there was a revolution in mathematical thinking by introducing non-Euclidean geometry, which expanded the concept of space and dismantled traditional geometry.

What is NonEuclidean Geometry?

Non-Euclidean geometry is one type of geometry that doesn't obey one of Euclid's five postulates, particularly parallel postulates that say that "through an outer point of a line, there is only one line that is parallel to that line." In nonEuclidean geometry, there are two major forms that are often discussed:

1. Hyperbolic geometryIn this geometry, there are many lines that are parallel to the same line through the outer point. Hyperbolic space is often visualized as a curved surface into, like a clam shell. In this model, triangles have less than 180 degrees of angle, and the distance between points acts differently than in Euclidean's geometry.
2. Elliptic geometryIn this geometry, there are no parallel lines. All lines meet at two points at the end of space. The common model for elliptical geometry is the surface of the sphere, where triangles are formed by three points on the surface of the sphere have a greater number of angles than 180 degrees.

Short History

Non-Euclidean Geometry began to appear at the beginning of the 19th century, thanks to the contributions of some mathematicians, including Nikolai Lobacevsky and János Bolyya, who independently developed hyperbolic geometry. Meanwhile, elliptical geometry was developed by Bernhard Riemann. Their discovery paved the way for the development of relativity theory by Albert Einstein, which suggests that space-time can have different structures than what is understood in classical geometry.

NonEuclidean Geometry Application

Non-Euclidean geometry is not only attractive in theoretical terms; it also has various practical applications:

1. Relativity TheoryNon-Euclidean is very important in physics, especially in general relativity theory. In this theory, gravity is not understood as force, but as a curve in space-time. The elliptical geometry is used to explain phenomena such as light lenses around large objects, such as stars or galaxies.
2. 3D Computer and Modeling Graphics: In computer graphics development, non-Euclidean modeling allows for more complex and realistic environmental creation. It's useful in game design and simulation, where objects can interact in unexpected ways.
3. Architecture and Design: The principles of non-Euclidean geometry can be applied to architecture to create innovative and aesthetic structures. Arching and unconventional buildings are often inspired by the concept of non-Euclidean geometry.

Challenge in Understanding NonEuclidean Geometry

Although nonEuclidean geometry offers new insights, understanding the concept could be a challenge. Many people find it difficult to adapt to the idea that there is more than one type of "space" and that geometric properties can dramatically vary depending on the basic assumptions used. However, learning about non-Euclidean geometry can expand our understanding of the world around us and give new perspective in different areas of science.

Conclusion

Non-Euclidean geometry has revolutionized the way we understand space and shape. By opening the boundaries of classical geometry, he has contributed significantly to science and technology. Understanding nonEuclidean geometry is not just about studying shape and size, but also about exploring deeper and complex concepts of space.

source: Greenberg, M. J. (2008). W. H. Freeman.