Euclidean and Non Euclidean Space?

[sepulshan]

Hi

Can someone explain the difference between Euclidean and Non Euclidean Space and how does one classify a space as Euclidean or Non Euclidean?? I heard about Gauss coming up with Non Euclidean Spaces when he was doing surveying of a piece of land. I am wondering what the word 'FLAT' really means??Any examples or references would be appreciated..

Regards
Shankar

 

[Hurkyl]

You know Euclids axioms for geometry, right? Well, a space is Euclidean if and only if all of them hold. :tongue:

(well, Euclid missed a couple axioms, but you get the point)

 

[tacman]

Euclidean space refers to the traditional three-dimensional space that we are familiar with, where the basic principles of geometry and mathematics apply. This includes the concept of "flatness," where parallel lines never intersect and the sum of angles in a triangle is always 180 degrees. It is named after the ancient Greek mathematician Euclid, who laid out the fundamental principles of geometry in his book "Elements."

On the other hand, Non-Euclidean space refers to a type of geometry where the traditional principles of Euclidean space do not apply. In these spaces, parallel lines may intersect, and the sum of angles in a triangle can be greater or less than 180 degrees. These spaces were first explored by mathematicians such as Carl Friedrich Gauss, János Bolyai, and Nikolai Lobachevsky in the 19th century, who challenged the long-held belief that Euclidean geometry was the only possible way to understand space.

Non-Euclidean space can be further divided into two types: hyperbolic and elliptic. Hyperbolic geometry is characterized by the property that the sum of angles in a triangle is less than 180 degrees, while elliptic geometry has the opposite property, with the sum of angles being greater than 180 degrees.

The concept of "flatness" in Non-Euclidean space is also different from that in Euclidean space. In Non-Euclidean space, "flatness" refers to the shape of a space, rather than the absence of curvature. For example, a flat surface in hyperbolic space may have a saddle shape, while in elliptic space, it may have a spherical shape.

One way to classify a space as Euclidean or Non-Euclidean is by its curvature. In Euclidean space, the curvature is zero, while in Non-Euclidean space, it is either positive (elliptic) or negative (hyperbolic). Another way is to look at the parallel postulate, which states that through a point not on a given line, there is only one parallel line that can be drawn to the given line. In Euclidean space, this postulate holds true, but in Non-Euclidean space, it does not.

In terms of applications, Non-Euclidean geometry has been used in various fields, such as physics, computer graphics, and navigation. For example, Einstein's theory of general relativity, which describes the curvature of space-time, uses Non-Euclidean geometry