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11
Registered:
11/26/05


Euclids postulates and nonEuclidean geometry
Posted:
Mar 7, 2006 5:42 PM


Hi,
I am trying to understand the motivation behind nonEuclidean geometry.
1. I do not understand why Euclid's fifth postulate is any different from the other postulates. For instance, it seems as intuitive to me to accept the fifth postulate as to accept the first one in one viewpoint. 2. If I assume that a Euclidean geometry refers to an infinite plane surface which closely matches our intuition at small scales, I find both the first and the fifth postulate to be equally believable (purely by intuition in both cases). 3. If I focus on the errors arising due to the approximation of a plane of intuitive scales actually being a part of the curved surface of the earth, then, of course, I begin to see the Euclidean rules failing, since we are on a different surface. I notice 'curved triangles' actually having 'curved sides' on the earth's surface. In this case, I completely redefine the angle between the 'curved sides' of the 'curved triangles' on the surface to be the angle betwen the tangents of the curves at that point. Now we are no longer talking about the angle between strictly straight lines  we are referring to three angles between tangents to 'bulging' curves, which, quit intuitively would add to more than 180 degrees. 4. If we set aside intuition for a moment and focus completely on the abstract platonic world of ideal forms, then every postulate and axiom can be questioned, and we can have very many amusing platonic worlds based on different rules. I could have one platonic world where Euclid's first postulate is wrong and the fifth is valid and see what interesting behaviors I can find in that world. I could create several such worlds and derive interesting properties in all of them. To that extent, a nonEuclidean geometry may be accepted as existing in one world among many worlds, where other worlds had other forms of nonEuclidean geometries formed by questioning each postulate and axiom of Euclid in different combinations. 5. If the justification for singling out the fifth postulate to focus on is merely driven by the utility of the resulting worlds, then I am also willing to accept a statement such as " there are several platonic worlds formed by negating each axiom and postulate of Euclid, but the only ones using practically useful results and map to our intuition are the Euclidean geometry (at small scales) and some hyperbolic geometries (at the large scale of the universe)". However, this seems more of a convenience argument than a scientific one.
I am very new to this entire field, so could someone help me get past these fundamental confusions?
Thanks



