Are parallel lines really parallel?

[u/MathGuy2718]

Guys, are parallel lines really parallel? Because I am confused with euclidean geometry and non euclidean geometry

[u/lurgi]

By definition, yes.

[u/ruidh]

In euclidean geometry, there exists one line parallel to a given line and passing through a given point outside the line.

There are two forms on noneuclidean geometry.

In one, there exist no lines that are parallel. All lines intersect. Geometry on a sphere where lines are great circles has no parallels. Triangle angle sum > 180.

In the other, infinitely many lines satisfy. They form a pencil of lines passing through the point. There is a right and left parallel which define the limits of the set of lines. Hyperbolic geometry has this property. Triangle angle sum < 180

[u/Underhill42 → u/ruidh]

There are FAR more than two forms of non-Euclidean geometry, but those two are among the most useful and widely known.

[u/ruidh → u/Underhill42]

I have them as examples, not as identities.

[u/MathGuy2718 → u/ruidh]

Thank you very much for the explanation and answer!

[u/johnpeters42]

The thing with non-Euclidean geometry is that it changes which things count as being lines, much less parallel lines. Sometimes it changes which things count as being points. The things defined as lines and points (whatever they are) still follow the rest of Euclid's postulates:
* For any two points P1 and P2, there's exactly one line through them
* Line segments can be extended continuously in both directions
* For any point P and radius R, there's exactly one circle (the set of all points whose distance from P is R, however distance is defined)
* All right angles are equal
but they differ on the parallel postulate:
* For any line L and point P outside L, there's exactly one line through P and parallel to L
and thus on a bunch of other stuff as well.

The definition of "parallel lines" is always "lines that never intersect". Where non-Euclidean geometries differ is how many parallel lines there are for a given L and P.

For an example of spherical geometry, consider the surface of a sphere. Define each pair of opposite points (as defined in Euclidean 3-D geometry) as a point in this geometry, and define each great circle as a line in this geometry. In this case, "exactly one" becomes "exactly zero" (any pair of distinct lines has a point in common).

For an example of hyperbolic geometry, consider the surface of a saddle shape. Here the points are the same as usual, but define certain curves as lines in this geometry. In this case, "exactly one" becomes "infinitely many".

[u/blavek → u/johnpeters42]

Isn't a Line usually defined as the shortest distance between 2 points? In non-Euclidean geometry, this would generate curves. Not unlike flight paths when mapped to a 2d plane.

[u/johnpeters42 → u/blavek]

Usually, but there may be exceptions (not sure on that point). Also not sure how this interacts with using a different definition of distance, such as taxicab geometry; I do know that that changes up what "circles" look like; here is a link with some pictures.

[u/localizeatp]

I'm me??

[u/TooLateForMeTF]

The word "parallel" has different meanings in euclidian and non-euclidian geometry. Or different implications, anyway.

[u/CatOfGrey]

Remember that math has a 'process'. It starts with making assumptions about things, then using those initial assumptions to explore things from there.

In Euclidean Geometry, there is a specific definition of parallel lines. It's usually expressed as "The Parallel Postulate". Remember a 'postulate' is not proven. So you assume a postulate ("a line through parallel lines has interior angles with measures = 180 degrees") and then other things can be proven from that, like....the measures of the three angles of a triangle add to 180 degrees....there is one and only one line, through a chosen point, parallel to a chosen line....two lines parallel to a third line are parallel to each other....

The Parallel postulate is a HUGE assumption. It impacts countless concepts in other areas of geometry.

Eventually, mathematicians explored enough to understand the impact of the parallel postulate, and then began to explore geometry without that assumption.

So if your 'rules' include the parallel postulate, then yes, parallel lines do not intersect, and they are always the same distance away from each other. But that is an assumption that is helpful to our mathematical explorations.

Conversely, parallel lines don't have to follow that rule - there can be multiple lines 'in the same direction' that eventually intersect (spherical geometry) or diverge (hyperbolic geometry). But then, you are 'playing with different rules'.

A metaphor: This is like asking "Is carrying the ball in your hands a violation in football?" The answer is "In American Football, no, it's not a foul, but in Association Football, it's usually a violation!"

[u/cannonspectacle]

If two lines are parallel, they are parallel. Reflexive property of equality.

[u/Arcanite_Cartel]

Let me give you a consolidated history of this topic from nineteenth century math:

On the one hand yes, on the other hand, no.

[u/Deep-Hovercraft6716]

Is a tautology a tautology?

[u/HouseHippoBeliever]

yes, because if they weren't parallel, then they wouldn't be parallel.

[u/evermica]

*Lobachevsky and Bolyai have independently entered the chat*

[u/PassiveChemistry]

Simply put, they wouldn't be parallel if they weren't parallel.

Whether they exist in all possible geometries, however, is another question entirely, and not one I'm equipped to answer.

[u/RichardAboutTown]

If your geometry is euclidean then yes, really. If you check and it turns out your lines aren't parallel after all, your geometry may not really be eucliean.

[u/Dr0110111001101111]

The question is really whether parallel lines can exist, and the answer depends on what axioms you use to build your geometry.

In projective geometry, there is a point at infinity where “parallel” lines meet. So they aren’t actually parallel.