Is Euclid's 5th Postulate Crucial for Defining Euclidean Geometry?

[radou]

Is Euclid's 5th postulate the basic thing which, if valid or not, makes a geometry Euclidean or non-Euclidean?

 

[HallsofIvy]

Fundamentally, yes. There is also the axiom (I confess I don't rember the number- it might be #2!) that asserts that there exist exactly one line between any two points. "Hyperbolic geometry", in particular, the geometry on the surface of a sphere, does not satisfy that but generally speaking, the distinction between Euclidean geometry and "elliptic geometry"- normally thought of as "non-Euclidean" geometry is the requirement that, through a given point, there exist exactly one line parallel to a given point (known as "Playfair's axiom). While elliptic geometry allows that there exist more than one axiom, hyperbolic geometry requires exist exactly one such line.

 

[tacman]

Euclid's 5th postulate, also known as the parallel postulate, states that if a line intersects two other lines and forms interior angles on the same side that sum to less than 180 degrees, then the two lines will eventually intersect on that side. This postulate has been a subject of much debate and has led to the development of non-Euclidean geometries.

The validity of Euclid's 5th postulate is indeed crucial in determining whether a geometry is Euclidean or non-Euclidean. If the postulate is true, then the geometry is considered to be Euclidean, and the resulting geometry is known as Euclidean geometry. However, if the postulate is not true, then the geometry is non-Euclidean, and the resulting geometry is known as non-Euclidean geometry.

Non-Euclidean geometries have been developed as alternatives to Euclidean geometry, and they do not assume the parallel postulate to be true. Instead, they propose different axioms and postulates that lead to different geometric systems. These non-Euclidean geometries have been used in various fields, such as physics and astronomy, to better explain certain phenomena that cannot be explained by Euclidean geometry.

In conclusion, Euclid's 5th postulate is indeed a fundamental factor in determining whether a geometry is Euclidean or non-Euclidean. Its validity or lack thereof has led to the development of different geometric systems and has expanded our understanding of the world around us.