Non Euclidian Geometry

[SGS]

Ok... for an assignment I am to make a definition of non-Euclidian geometry. How does this sound?

Given that Euclidian geometry theory is based off of five postulates stated by Euclid, non Euclidian geometry is then any geometry that does not follow any one of these five postulates. The most essential difference is the nature of parallel lines in each of these cases. Given a line and a point not on the line in non Euclidian geometry, there can be infinitely many lines through the point that are parallel to the given line or there can be none at all. In Euclidian geometry given these same circumstances, there is exactly one line through the given point that is parallel to the given line.

SGS

[atrifix]

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Ok... for an assignment I am to make a definition of non-Euclidian geometry. How does this sound?

[/ QUOTE ]
This sounds like a terrible assignment.

Definition: a point is a Euclidean triangle. QED.

[lightw1thoutheat]

well euclid had five axioms.
1) two points can be joined by a straight line
2) any straight line segment can be extended infinitely to a straight line
3)Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.
4)all right angles are congruent.
5) If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.

if you disregard the last one, the parallel line one you get non euclidean geometries.



In Euclidean geometry, if we start with a line l and a point A, which is not on l, then we can only draw one line through A that is parallel to l. In hyperbolic geometry, by contrast, there are infinitely many lines through A parallel to l, and in elliptic geometry, parallel lines do not exist.

consider 2 lines that are perpendicular to a third line. in eiptical geometries, these lines curve towards eachother, ie they dont remain the same distance from each other. in hyperbolic geometries, these lines curve away from each other.

otherwise, these are consistent geometries with euclidean geometry.
(from wiki)

-light

[bunny]

There are other non-euclidean geometries. The one I studied didnt have the 3rd axiom you list either, for example.

[atrifix]

Just in case anyone didn't get my point:

Nearly any definition will do, all that's required is violating one of the axioms. So just define a geometry that violates one of the axioms.

Now, perhaps the professor also wants a consistent geometry, although who knows if that's the case or not. I don't know whether typical geometrical axioms can also prove their own consistency; that seems beyond the scope of the assignment. At any rate, consistent geometries can be defined by anything that contradicts one of Euclid's axioms. So: "(df) Two points can not be connected by a line segment" is a consistent non-Euclidean geometry.

If you want consistent non-Euclidean geometries from which you can derive interesting results (for example, one consistent with Einstein's general theory of relativity), then you have another question altogether.