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Author Topic: Geometry and parallel lines.  (Read 10009 times)

Offline yor_on

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Geometry and parallel lines.
« on: 30/06/2011 06:26:16 »
If I imagine I drive a car around the globe I will leave two parallel lines by its wheels. If I then folded that 'earth' out, you know, Godlike like? Making it a flat surface (two dimensional), would now those two parallel lines intersect, meaning that they cross each other?

And you are sure about that?
ahem?
==

And just for fun, take a look at this one, and tell me what you see :)


« Last Edit: 30/06/2011 06:37:08 by yor_on »


 

Offline Soul Surfer

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Geometry and parallel lines.
« Reply #1 on: 30/06/2011 08:52:24 »
Assuming that you were allowed to stretch the surface to flatten it out as with all other conformal transforms (cup to ring donut for example.  the lines would not cross and there is no transform that I can think off that would cause them to cross.  however in no case are the lines ever geometrically straight in a Euclidian sense.

The pattern you show in this comment is very interesting and illustrates well aspects of the way that our brains process information.  The patterns are concentric circles of slightly angled squares.  If you look at them from a distance or very closely when you cannot see clearly the outlines of the squares you can easily see them as circles however when you look at them at a normal viewing distance the angled squares are visible and the brain tries to want to link them together in chains so that looked at generally you tend do see an intersecting "spirograph" pattern while concentrating on a small part you see a segment of a circle of angled squares.  This gives a very ambiguous nature to the image.
 

Offline yor_on

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Geometry and parallel lines.
« Reply #2 on: 30/06/2011 09:47:27 »
Yeah, it caught my attention directly as I saw it. It's a very strange effect, feels like my brain is fighting itself when I look at it, even though I can see that they are concentric circles my eyes swear to me that 'they move'(as in 'flipping from side to side':)

And I'm glad you agreed Soul Surfer, cause that's my take on it too. But, there seems to be a different opinion on it geometrically. So I hope we will get the other side of it too, as I expect it to make sense when one (and that should be me:) understand 'what the he* they really mean by it' :)
 

Offline imatfaal

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Geometry and parallel lines.
« Reply #3 on: 30/06/2011 12:54:15 »
SSurfer - is the cup to ring donut really a conformal mapping?  My memory of these transformations was that angles and arbitrarilly small shapes were all maintained  - would this hold for teacup to donut?
 

Offline Don_1

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Geometry and parallel lines.
« Reply #4 on: 30/06/2011 16:57:03 »
Looking at that, those two, three........ Oooo I feel quite
 

Offline JP

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Geometry and parallel lines.
« Reply #5 on: 30/06/2011 18:47:22 »
As the comments indicate, there are a lot of ways to make the globe flat.  You have to tell us which one you want to use. 

More generally, parallel lines are very special constructions that rely on the geometry looking flat.  After all, what's a "straight line" on the curved surface of the globe?  http://en.wikipedia.org/wiki/Parallel_(geometry)#Extension_to_non-Euclidean_geometry

(Knowing that you like general relativity, yor_on, you'll see a lot of familiar terms.  The point is that motion in straight lines along a flat surface translates to motion along geodesics if you have a curved surface.)
 

Offline yor_on

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Geometry and parallel lines.
« Reply #6 on: 30/06/2011 22:48:32 »
Yes, I understand that there is a 'hidden truth' to the statement JP, but as I look at gravity from my way I fail to connect it to this geometric statement.

Those are my dissents

1. To say that parallel tracks intersect can't mean that they do so in 'the same point'

(because if they did there would be no such things as parallel lines. As it then always would be a axiom, that you from some other 'dimensionality' can prove not to hold)

A parallel line is one that to me will hold true in all dimensionality, meaning that even though you can find them 'crossing' they should not meet in any point under that 'crossing'. And this is my main beef with it. calling it 'crossing' is a result from applying more dimensions on something that never intersects, a little like seeing a picture of a bridge 'crossing' a road on a aerial photo, if that now make sense :). From looking at the 2D photo you might find it a cross, but in reality they still are separated from each other.

The other way to look at it can't be right, because what it state is that what you have as a distance between those lines not only disappear, but also as I see it, more or less 'invert' itself? As we then will find a new distance on the other side. All of this under the assumption that what the definition state is that they not only cross but also meet in a plane. It just don't feel right to me.
==

Those? I need coffeee :)
 

Offline yor_on

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Geometry and parallel lines.
« Reply #7 on: 30/06/2011 22:55:25 »
And to assume that gravity 'intersect' with itself depends on your definition I think. Treat 'gravity' as vectors and you can state that they 'intersect', stating that is the 'whole gravity' that does so. Treat it as spilling a dye in (streaming) water you can see it spin etc but that 'gravity' will always have a direction in time, so at no point will it 'intersect' even though it may meet itself. This one is more subtle and the way I look at it.
==

Maybe it's dense? I definitely need some coffee here.
« Last Edit: 30/06/2011 22:58:00 by yor_on »
 

Offline JP

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Geometry and parallel lines.
« Reply #8 on: 01/07/2011 04:37:54 »
Ok, but what's your definition of parallel lines?
 

Offline CliffordK

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Geometry and parallel lines.
« Reply #9 on: 01/07/2011 07:53:20 »
I have definitive proof that the lines simply merge when flattened!!!!


 

Offline yor_on

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Geometry and parallel lines.
« Reply #10 on: 01/07/2011 12:00:57 »
Heh :)

Sweet one.

And, I don't know JP. It's been on my mind for some time now, if I assume that parallel lines can merge then the idea of them 'crossing' is just a small step to take. But the axiom is that they can't intersect, and it makes sense to me. If I assume a equal distance between those parallel lines then it seems to me that this distance should stay assuming that the lines are parallel. At least as long as it is in a 'plane', like drawing it on a paper. Introducing new dimensions is a very strange idea, both as we assume that they can be incredibly small, as in string theory, as well as stretch outside our 'room perception', as when using logic to build from the eight corners we see in SpaceTime to sixteen etc. That assumes then that we on one hand have rolled up dimensions and on the other hand dimensions that are 'larger' in some for us invisible way.

Let's take that globe and place in in a plane like Clifford did, sort of :) My assumption would then be that what this 2D inhabitant would see would be a 'singularity'. A circle that wouldn't be possible to penetrate. Replace those parallel lines by one, if two equally distanced lines at some point can merge into one, doesn't that also state that one single line at some point could disappear? So a infinite line wouldn't be possible, not that it is, but still? Can you explain under what circumstances two parallel lines will intersect?
==

One more thing, when looking at the axiom it does not involve time. Most (actually all) of the things we do is 'fluid', in that it/they changes with time. But the axiom is not involving time, at least not as I see it. Those parallel lines will at any instant in time be parallel, whereas in 'real' SpaceTime everything is a fluid system. One is what we have, and the logic that follows from 'fluidity', the other is just pure logic. Maybe that's it?

It's a geometric design, has it do with affine planes?
« Last Edit: 01/07/2011 12:26:54 by yor_on »
 

Offline CliffordK

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Geometry and parallel lines.
« Reply #11 on: 01/07/2011 12:32:37 »
I think the point of the original post was a very cute optical illusion...  which your eyes resolve into a flower-like pattern that drives your brain bonkers.

Most ordinary projections of a 3-D globe into 2-D space should maintain non crossing lines, although they may diverge significantly from being parallel.

However, one could imagine a projection which would actually create crossing lines.  For example, consider a transparent sphere with a pair of parallel lines drawn around the equator.  Looking at it from the side and you will see 2 perfect parallel lines.  Look at it from the top and you'll see a perfect circle.  Look at it at other angles, and you'll see lines that cross.

If you make the loop off-center, say around the 45th parallel, entirely in the Northern Hemisphere, then the top view would show 2 circles, with oblique views again giving you crossing lines.
 

Offline yor_on

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Geometry and parallel lines.
« Reply #12 on: 01/07/2011 12:40:27 »
Yes, that is true, but they do not cross. It's a optical illusion, they are still at a equal distance from each other, even if we assume that gravity would act that way they would still be parallel lines to me. Maybe if we define it such as that in a plane they never can cross we can allow ourselves to also define that they at some 'angle' in some space would seem to be crossed. It's like Lorentz contraction, either you see it as a illusion (twisted geometry) or as a reality.
==

And that is part of my beef with the idea actually, that they are stated to 'merge'. If the definition is that they from some space/dimensionality/angle will seem to 'merge' I have no problems with it, but that's not what I've seen.
« Last Edit: 01/07/2011 12:44:13 by yor_on »
 

Offline imatfaal

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Geometry and parallel lines.
« Reply #13 on: 01/07/2011 12:41:15 »
Cliff - you are dead right; however, those are views not projections or transforms
 

Offline yor_on

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Geometry and parallel lines.
« Reply #14 on: 01/07/2011 12:47:56 »
And Imatfaal, any transformation/projection is a logical construct following some predefined idea over how to relate points to each other. But those parallel lines are to me a axiom making a immediate sense. Pick up one stone in each hand and then tell me how many you have.
==

Maybe it has to do with time after all? I'm thinking of a Lorentz contraction there.
But it hasn't, has it :)
Da*n
« Last Edit: 01/07/2011 12:59:24 by yor_on »
 

Offline imatfaal

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Geometry and parallel lines.
« Reply #15 on: 01/07/2011 13:14:56 »
Yoron - but what is an axiom if it is not a logical construct?
 

Offline CliffordK

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Geometry and parallel lines.
« Reply #16 on: 01/07/2011 13:19:30 »
There are some interesting non-parallel shapes you can get.

For example, if you take the typical "box projection" that seems to be popular with the weather maps.

And then drew your parallel lines on each side of the prime/anti meridian (0 & 180), you would get something that looks like this (ignoring my not so steady touch-pad hand.

 

Offline yor_on

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Geometry and parallel lines.
« Reply #17 on: 01/07/2011 13:39:53 »
To me axioms are where we start. It's from those we build our logic, and those parallel lines are a axiom to me. Then, on the other hand, I do see a Lorentz contraction as being nothing less than the travelers truth :) which then makes it as evident that from Earth the ship really has shrunk, as we can't just take the travelers version as real and ignore earths :) But there we have time as a parameter. Still, if you get it, tell me how the magic is done? :)
 

Offline yor_on

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Geometry and parallel lines.
« Reply #18 on: 01/07/2011 14:00:57 »
Look at it this way, when defining a triangle we get one result when drawing it on paper, and another if drawing it on a sphere. Just as a thought. And that do make sense. But ..
 

Offline JP

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Geometry and parallel lines.
« Reply #19 on: 01/07/2011 14:03:10 »
And, I don't know JP. It's been on my mind for some time now, if I assume that parallel lines can merge then the idea of them 'crossing' is just a small step to take. But the axiom is that they can't intersect, and it makes sense to me. If I assume a equal distance between those parallel lines then it seems to me that this distance should stay assuming that the lines are parallel. At least as long as it is in a 'plane', like drawing it on a paper.

The definitions and axioms you seem to be referring to have to do with flat space or planes.  Your question is about a sphere, though, for which these axioms don't hold.  What do you mean by parallel lines on a sphere?
 

Offline imatfaal

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Geometry and parallel lines.
« Reply #20 on: 01/07/2011 14:18:08 »
Yoron - axioms are assumptions/postulations which you will not prove but will form thje basis of your theories.  The axioms of euclidian space can be shown to be false or non-applicable in non-euclidian space - you need to spell out your axiom
 

Offline yor_on

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Geometry and parallel lines.
« Reply #21 on: 01/07/2011 14:59:30 »
No JP, I used the triangle on a sphere as a definition that I found understandable, but my question here is how two parallel lines can be seen to cross, and as I've seen it described 'merge'? As I see it there must be a space between those lines, and at no point will they share a same point (a slight pun, still, how I see it). And it's a axiom to me, as self evident as picking up those stones to me to count.

That is if I got the definition of how they are thought to 'cross' right? Because if they can be proved to share a same point somewhere 'on the line', ahem, they can't be called parallel, can they?
==

A triangle drawn on a sphere (using a ruler) won't give the same angles as when drawn on a plane, the sum of those three angles will be greater that 180. And that you already know :) right, and I agree. That one is simple to see. For those that want to 'test' the statement, get a flexible ruler that 'bends as you draw those lines, or if you have Java you can test it here, cyber-like of course :) (called 'positive curvature' as earths.)
« Last Edit: 01/07/2011 15:28:37 by yor_on »
 

Offline JP

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Geometry and parallel lines.
« Reply #22 on: 01/07/2011 15:42:25 »
No JP, I used the triangle on a sphere as a definition that I found understandable but my question here is how two parallel lines can be seen to cross, and as I've seen it described 'merge'? As I see it there must be a space between those lines, and at no point will they share a same point (a slight pun, still, how I see it). And it's a axiom to me, as self evident as picking up those stones to me to count.

Ok, but it's a problem to use axioms that are "self evident," since our brains aren't really constructed to intuitively understand non-Euclidean geometry.

Here's one of the problems: what is a "straight line" on a sphere?  A sphere is a curved surface, so nothing on it is straight, but how do we get close to a straight line?

The answer is that we figure out the rules for constructing straight lines in 2D and then figure out how to construct those on a sphere.  One way of constructing a line would be to put your pencil on the paper and trace a path without picking the pencil up or having the path cross itself.  But this isn't necessarily straight.  I could draw a curve or a spiral if I wanted (Fig. A) 


The usual definition of parallel is that "lines" (whatever we define them as" don't ever touch except possibly at infinity.  I could draw another curve that never touches the first curve.  Are these parallel lines?  Well, they never touch, but they're not lines. (Fig. B).


To make the line straight, I can think of constructing it from a stretched rubber band on the plane.  No matter how much I try to bend the rubber band, it snaps back into a straight line.  The rubber band minimizes the distance between any two points along it's path in the plane, and is therefore known as a geodesic, which is a straight line on a 2D Euclidean plane.

Now I have a unique definition that captures what it means to be straight parallel lines:
1) The lines are constructed from the set of points that minimizes the distance between any two points on the line.
2) The lines don't intersect.

If you try to put a tight rubber band on a sphere, you'll find that it pops off as it tightens unless it's a great circle of the sphere.  And if you put two such rubber bands on the sphere, the have to intersect.  You cannot apply both 1) and 2) simultaneously to the surface of a sphere.  You either have "straightness" or "parallelness" but not both.  Geodesics always intersect and non-intersecting lines are never geodesics. 

From your description, I'm guessing you want to give up "straightness" and keep "parallelness."  But then you can end up with weird cases like Fig. B above where you have wiggly contours on the sphere that don't ever touch, but clearly aren't intuitively parallel.

In terms of this, what do you mean by parallel lines on a sphere?  Is it just non-intersecting?
 

Offline yor_on

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Geometry and parallel lines.
« Reply #23 on: 01/07/2011 17:19:00 »
You can also express it as gravity definition of a straight line is where you don't expend energy. But that doesn't state that it is a straight line from our geometric definitions, at least not mine? As for using rubber bands on spheres I can accept your statement but I fail to see why they allow parallel lines to intersect in some dimensional space, also, by definition you can't define intersecting lines as being parallel.

The new limitation for a straight line seems then to be that it should not expend energy? Which in a way is what I used for the longest time to define gravity's geodesics as, the straightest line in SpaceTime :) But, I can draw two straight lines, ignoring gravity, and even when gravity over longer distances will influence them I can still expect those parallel lines to have a space between them. I see no fault in that statement and this is what I see it to be about. So we must be discussing a 'dimension' more for them to meet in a point, as it seems to me?
=


And yes JP :)

You're perfectly correct, I can accept them to be crooked, and still conceptually define them as parallel, and 'straight', knowing about 'gravity'. It works any which way we define it as long as they don't intersect in a same point, meeting each other. That's where my imagination ends (for now) And I think you gave a good intuitive picture of it, except with the sphere where you lost me, because I cant imagine a parallel line meeting a 'sphere' that will separate to 'climb it' crosswise so to speak. But I expect that to be a limitation of translating the mathematics to a intuitive picture. And I can accept a parallel line to become severely twisted, even 'crossing itself' as long as there still is a space between our 'ideal lines' :)

« Last Edit: 01/07/2011 17:40:58 by yor_on »
 

Offline JP

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Geometry and parallel lines.
« Reply #24 on: 01/07/2011 17:40:03 »
There are no straight lines on a sphere.  Period. 

If you want something similar to a straight line, you need to come up with the definition of a straight line in a plane and then apply that rule to the sphere.  That process gives you geodesics.  All geodesics on a sphere must cross, so they cannot be parallel.  Period.

If you loosen up the restrictions on the paths you're considering so that they aren't geodesics, then you can define many paths that are parallel since they don't intersect.  But they're not really geometrically equivalent to straight parallel lines.  They're more like what I showed in Fig. B above.
 

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« Reply #24 on: 01/07/2011 17:40:03 »

 

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