Exploring Non-Flat Surfaces and Euclid's Fifth Postulate

[JamieLam]

Hi. I'm from singapore. I'm now interested in maths. I only studied till secondary school(high school) I wasn't interested in maths then. Now I am. I read maths book written by David Berlinski , John allen paulos and others to try to understand. I do not have mathematician friends so I couldn't ask them. I like to ask about Eculid fifth postulate. As you all know its something like there is a line L and a point P. One can draw only a line through P that is parellel to L. I wonder if the fifth postulate applies to surfaces that are not flat like say a sphere? I think its possible but the books I read suggest otherwise. (Maybe its due to my lack of knowledge in math) Can someone explain to me? Thanks.

 

[Ackbach]

I'm not sure I totally understand the question. Let me elaborate a little, and then maybe ask a few clarifying questions.

What you've stated is technically Playfair's Axiom: in a plane, given a line and a point not on it, at most one line parallel to the given line can be drawn through the point. It is equivalent to Euclid's Fifth Postulate.

You could, no doubt, prove a 3D analogue of Euclid's Fifth Postulate (here I'll state it in a Playfair style): given a plane and a point not on it, at most one plane parallel to the given plane can be drawn through the point.

But now (here I'm guessing at your meaning), could we say the following: given a sphere and a point not on it, at most one non-intersecting sphere can be drawn through the point? That is definitely false in Euclidean geometry, since I could (theoretically) draw lots of spheres of varying radii through the point not on the given sphere. The uniqueness depends on (among other things) the fact that a line or a plane extends infinitely in two (for the line) or many directions (for the plane). So any finite object in Euclidean space will definitely not substitute in the postulate.

But perhaps you meant something else?

 

[Evgeny.Makarov]

One can draw only a line through P that is parellel to L. I wonder if the fifth postulate applies to surfaces that are not flat like say a sphere?

I'm not sure I totally understand the question.

The OP is asking if there exists a single (suitable definied) straight line $l_2$ passing through a point not on a straigt line $l_1$ such that $l_2$ is parallel to $l_1$. Is it true on non-flat surfaces such as spheres?

 

[JamieLam]

The OP is asking if there exists a single (suitable definied) straight line $l_2$ passing through a point not on a straigt line $l_1$ such that $l_2$ is parallel to $l_1$. Is it true on non-flat surfaces such as spheres?

Right.

 

[Evgeny.Makarov]

One can draw only a line through P that is parellel to L.

"Only a line" $\mapsto$ "a single line" or "at most one line", depending on what you mean.

I wonder if the fifth postulate applies to surfaces that are not flat like say a sphere?

In general not. On a sphere, it is natural to define straight lines to be great circles. Then there are no parallel lines passing through a point not on the given line. (Isn't it strange that such a simple observation did not occur to critics of the non-Euclidean geometry? There are some discoveries in math that are extremely technically complex, and there are others that are simple but require fresh look at things.) There is also a tractricoid, which can be called a pseudosphere. On it, there is an infinite number of parallel lines passing through a point not on the given line.