1. ## Non-Euclidean Space

Hello All,

David Hilbert's contribution to Non Euclidean geometry has been written in Hilbert's Foundations of Geometry. Can anybody please explain what Hilbert tried to mean by non-Euclidean space?

Thank you.

Anyone moved to answer can post the answer here or PM to the OP

2. ## Re: Non-Euclidean Space

Originally Posted by shounakbhatta
Hello All,

David Hilbert's contribution to Non Euclidean geometry has been written in Hilbert's Foundations of Geometry. Can anybody please explain what Hilbert tried to mean by non-Euclidean space?

Thank you.

Anyone moved to answer can post the answer here or PM to the OP
Have you tries googling non-euclidean geometry

CB

3. ## Re: Non-Euclidean Space

Originally Posted by CaptainBlack
Have you tries googling non-euclidean geometry

CB
Hello,

Yes, I am on it. Non-Euclidean Geometry refers to certain more dimensions rather than length, breadth and height, may be another dimension like time.

Non-Euclidean geometry generally starts with modifying the 5th.postulate which is called the parallel postulate.

4. ## Re: Non-Euclidean Space

Originally Posted by shounakbhatta
Hello,

Yes, I am on it. Non-Euclidean Geometry refers to certain more dimensions rather than length, breadth and height, may be another dimension like time.
No, that's not true. "Non-Euclidean Geometry" has nothing to do with dimensions (you may have found examples of non-Euclidean geometries in which other dimensions were added for other, perhaps physics, reasons).

Non-Euclidean geometry generally starts with modifying the 5th.postulate which is called the parallel postulate.
Strictly speaking, non-Euclidean geometry is any geometry in which at least one of Euclid's postulates is denied. However, it is most commonly used to mean geometries in which the fifth postulate is denied. The most commonly used rephrasing of Euclid's fifth postulate is "Playfair's axiom" that "through a given point, P, not on a given line, l, there exist one and only one line through P parallel to l".

There are two ways to deny that:
1) Assert that there are NO parallels to l through p which leads to the statement that no line has any parallels. In order to be consistent with other postulates, you must also drop postulate that two lines intersect in, at most, one point. That is called "elliptic geometry". Spherical geometry, where "lines" are great circles on a sphere, is a model for elliptic geometry.

2) Assert that the exist two or more parallels to l thorugh p. This is "hyperbolic geometry". The Poincare "half plane" and "disc" models are the simplest models for hyperbolic geometric although you can also use the "hypersphere" (a sort of sphere with "negative" radius) to mimic the sphere model for elliptic geometry.

5. ## Re: Non-Euclidean Space

Originally Posted by HallsofIvy
No, that's not true. "Non-Euclidean Geometry" has nothing to do with dimensions (you may have found examples of non-Euclidean geometries in which other dimensions were added for other, perhaps physics, reasons).

Strictly speaking, non-Euclidean geometry is any geometry in which at least one of Euclid's postulates is denied. However, it is most commonly used to mean geometries in which the fifth postulate is denied. The most commonly used rephrasing of Euclid's fifth postulate is "Playfair's axiom" that "through a given point, P, not on a given line, l, there exist one and only one line through P parallel to l".

There are two ways to deny that:
1) Assert that there are NO parallels to l through p which leads to the statement that no line has any parallels. In order to be consistent with other postulates, you must also drop postulate that two lines intersect in, at most, one point. That is called "elliptic geometry". Spherical geometry, where "lines" are great circles on a sphere, is a model for elliptic geometry.

2) Assert that the exist two or more parallels to l thorugh p. This is "hyperbolic geometry". The Poincare "half plane" and "disc" models are the simplest models for hyperbolic geometric although you can also use the "hypersphere" (a sort of sphere with "negative" radius) to mimic the sphere model for elliptic geometry.
Hello,

Thank you very much for providing such an easy way to understand Non Euclidean geometry. I was reading through PlayFair's postulate, which defines on a 2 dimensional plane give a line "L" an a point "A" which is not on that line (L) there's exactly one line through A that does not intersect.

Well, in hyperbolic geometry by contrast the are infinitely many lines (ultra parallels) and in elliptic geometry they do INTERSECT. But Intersecting does not mean it denies Euclid's 5th. postulate. Am I right? Please do correct me if I am wrong?

What I mean to say is that what you replies is exactly right. But my point is in Non-Euclidean geometry as the parallels do not intersect whereas in hyperbolic and elliptic geometry they curve away and intersect. Does that mean it is denying on of the postulates of Euclid?

6. ## Re: Non-Euclidean Space

Originally Posted by shounakbhatta
But my point is in Non-Euclidean geometry as the parallels do not intersect whereas in hyperbolic and elliptic geometry they curve away and intersect. Does that mean it is denying on of the postulates of Euclid?

In elliptic geometry there are no parallels.

In hyperbolic geometry there are a number of definitions of parallelism with more than one such parallel associated with a given line and point.

But here is the biggie - geodesics are playing the part of straight lines in Euclidean geometry in both of these types of geometry but curvature of a line is not simply defined and to speak of the parallels in a hyperbolic geometry curving is not possible without defining what you mean by curving within that geometry.

What you perceive as parallels curving is the curvature of the geodesics in a Euclidean model of hyperbolic geometry, but when dealing with abstract geometries we need to define concepts in a way intrinsic to the geometry and not use some feature of a model.

Also, the homogeneous geometries (geometries of constant curvature) do not exhaust the possibilities of non-Euclidean geometry.

CB

7. ## Re: Non-Euclidean Space

Originally Posted by CaptainBlack
In elliptic geometry there are no parallels.

In hyperbolic geometry there are a number of definitions of parallelism with more than one such parallel associated with a given line and point.

But here is the biggie - geodesics are playing the part of straight lines in Euclidean geometry in both of these types of geometry but curvature of a line is not simply defined and to speak of the parallels in a hyperbolic geometry curving is not possible without defining what you mean by curving within that geometry.

What you perceive as parallels curving is the curvature of the geodesics in a Euclidean model of hyperbolic geometry, but when dealing with abstract geometries we need to define concepts in a way intrinsic to the geometry and not use some feature of a model.

Also, the homogeneous geometries (geometries of constant curvature) do not exhaust the possibilities of non-Euclidean geometry.

CB
Hello,

Thank you again for your response. Sorry for replying late. To summarize, what I feel, following the Playfair's axiom, there is exactly one line through A that does not intersect the line L. Extending Playfair's axiom to hyperbolic and elliptical geometry it violated Euclid's 5th.postulate, hence comes the concept of Non-Euclidean geometry. Please correct me if I am wrong.

8. ## Re: Non-Euclidean Space

Originally Posted by shounakbhatta
Hello,

Thank you again for your response. Sorry for replying late. To summarize, what I feel, following the Playfair's axiom, there is exactly one line through A that does not intersect the line L. Extending Playfair's axiom to hyperbolic and elliptical geometry it violated Euclid's 5th.postulate, hence comes the concept of Non-Euclidean geometry. Please correct me if I am wrong.

As has already been posted here, any system which is based on the axioms of Euclidean geometry with one or more axioms deleted or changed is a non-Euclidean geometry. The non-Euclidean geometries seen most often by laymen and elementary students are those in which the parallel postulate, or whatever is being used in its place, is replaced by one or other of those that give rise to elliptic and hyperbolic geometries.

(Playfair's axiom is equivalent to the Parallel Postulate)

CB

9. ## Re: Non-Euclidean Space

Originally Posted by shounakbhatta
Hello,

Thank you again for your response. Sorry for replying late. To summarize, what I feel, following the Playfair's axiom, there is exactly one line through A that does not intersect the line L. Extending Playfair's axiom to hyperbolic and elliptical geometry it violated Euclid's 5th.postulate, hence comes the concept of Non-Euclidean geometry. Please correct me if I am wrong.

It's a bit difficult to understand exactly what you are saying. Playfair's axiom is equivalent to Euclid's 5th postulate. It is not a matter of "extending" Playfair's axiom but of denying it.

10. ## Re: Non-Euclidean Space

Originally Posted by HallsofIvy
It's a bit difficult to understand exactly what you are saying. Playfair's axiom is equivalent to Euclid's 5th postulate. It is not a matter of "extending" Playfair's axiom but of denying it.
Well, PLayfair's axiom which is equivalent to Euclid's 5th. postulate. Playfair's postulate which states that any given line L and a point A which does not lie on L, there is only one line L which does not intersect A.

As what you are telling Non-Euclidean geometry does not following Playfair's axiom as in the case of hyperbolic and elliptic geometry. That's what exactly you are trying to say, isn't it?

11. ## Re: Non-Euclidean Space

Originally Posted by shounakbhatta
Well, PLayfair's axiom which is equivalent to Euclid's 5th. postulate. Playfair's postulate which states that any given line L and a point A which does not lie on L, there is only one line L which does not intersect A.

As what you are telling Non-Euclidean geometry does not following Playfair's axiom as in the case of hyperbolic and elliptic geometry. That's what exactly you are trying to say, isn't it?
Please use google to research non-Euclidean geometry. Try reading what the main sites say. If you have any specific problems understanding what you read post a question on MHF but ask definite questions.

The discussion here is going around in circles, so I am closing this thread.

CB