Riemannian Geometry: What is It?

[Ebolamonk3y]

I heard one of my friends talking about a math or a geometry invented by some famous Russian guy... It goes something like...

2 parallel lines come from the same point... And that's the base of everything else... Some crazy stuff! I been thinking about this... You can be standing on a road, and you have that vantage point effect where the horizon kind of fades off to the distance... But its still 2 parallel lines, the street/road... Yet to your view it looks like its coming from a point...

So is this geometry a mere shift of POV or something vastly different?

 

[master_coda]

I heard one of my friends talking about a math or a geometry invented by some famous Russian guy... It goes something like...

2 parallel lines come from the same point... And that's the base of everything else... Some crazy stuff! I been thinking about this... You can be standing on a road, and you have that vantage point effect where the horizon kind of fades off to the distance... But its still 2 parallel lines, the street/road... Yet to your view it looks like its coming from a point...

So is this geometry a mere shift of POV or something vastly different?

You seem to be referring to Non-Euclidean geometry. For example in Hyperbolic geometry you can have multiple parallel lines passing through the same point.

It's a somewhat dramatic shift in thinking. At the time is was developed, it was very dramatic; Euclidean geometry was generally accepted as "real" geometry. The drama is mostly just psycological...from an everyday point of view, reality appears to obey Euclidean geometry, so the idea of using a different type of geometry can be disturbing.

 

[Ebolamonk3y]

multiple parallel lines parallel to each other? That would be what I am talking about... A is parallel to B... A and B come from the same origin... Hehe, what would be better is if A and B ended at the same point as well! Squeezing space. :)

 

[recon]

I'm no expert on higher mathematics but anyway:
Just think of Longitudes. Don't they all intersect at the North Pole (and South Pole) even though they're parellel?

 

[matt grime]

lattitudes aren't geodesics, and longitudes aren't parallel. .

For the record, the simplest model of hyperbolic geometry is the unit disc in the plane, where the geodesics are circles that meet the edge of the disc at right angles. You can easily imagine there being an infinite number of geodesics passing through a given point and parallel to another given geodesic.

 

[HallsofIvy]

The "Russian Guy" was Lobachevskii. And the model Matt Grime is talking about is "Euler's disk model" (although, personally, I think Euler's "half plane model" is simpler).

 

[Ebolamonk3y]

Yeah! I think that's the guy! What is this math called?

 

[JonF]

Yeah! I think that's the guy! What is this math called?

it is called hard

 

[Ebolamonk3y]

LOL, ahahahah

 

[HallsofIvy]

The very first reply told you that it was "non-Euclidean geometry". There is a classic book called "non-Euclidean" geometry, written by Bonola, that has been reprinted by Dover.

 

[matt grime]

The "Russian Guy" was Lobachevskii. And the model Matt Grime is talking about is "Euler's disk model" (although, personally, I think Euler's "half plane model" is simpler).

Genuine query, and explanation of why I prefer the disk model:

I know the disk model generalizes to higher dimensions, is there a generalization for the half plane? I can think of two possibilities for 3-dim space, and I guess the one where geodesics are hemispheres and planes orthogonal to the x-y plane (where I take the model to be the triples (x,y,z) in R^3 with z>0) is the 'correct' one.

 

[HallsofIvy]

The disk and half-plane models are really the same thing. Imagine a small section of the disk, next to the bounding circle is "blown up" (expanded). If you make it big enough, the bounding circle is indistinguishable from a straight line and you have the half-plane model.

 

[franz32]

Interesting, huh?! =)