# The universe is flat? (split from Time for a different view)

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I don’t want to go off track understanding time but I think I’m getting confused over the ‘universe is flat’ perspective. Does the universe being ‘flat’ just mean it is infinite in every direction?

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1 hour ago, MPMin said:

I don’t want to go off track understanding time but I think I’m getting confused over the ‘universe is flat’ perspective. Does the universe being ‘flat’ just mean it is infinite in every direction?

That’s off track and off topic. Interesting question. Maybe it deserves its own separate thread.

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5 hours ago, MPMin said:

Does the universe being ‘flat’ just mean it is infinite in every direction?

No. 'Flat' means that e.g. parallel lines stay parallel, no matter how far you compare their distance between them. Best example is two light beams. When these do not converge or diverge, the universe is flat.

It is difficult to imagine how a positive curved universe can be infinite because it is closed in itself, like the surface of a sphere. But with negative curvature and flat space that problem does not arise.

Just take care that the universe seems to be flat on average. Locally, due to mass and energy, the flat universe can be curved.

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11 minutes ago, Eise said:

No. 'Flat' means that e.g. parallel lines stay parallel, no matter how far you compare their distance between them. Best example is two light beams. When these do not converge or diverge, the universe is flat.

It is difficult to imagine how a positive curved universe can be infinite because it is closed in itself, like the surface of a sphere. But with negative curvature and flat space that problem does not arise.

Just take care that the universe seems to be flat on average. Locally, due to mass and energy, the flat universe can be curved.

Back to your clear and concise self I see. An excellent answer.  +1

I would, however, like to offer a small correction.

12 minutes ago, Eise said:

No. 'Flat' means that e.g. parallel lines stay parallel, no matter how far you compare their distance between them.

Is  "Parallel lines stay parallel" not a tautology ?

Further how can you say "locally flat"  and say "no matter how far you compare their distance between them"  ?

Local means nearby, not far away.

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This might help

1 hour ago, studiot said:

Is  "Parallel lines stay parallel" not a tautology ?

Further how can you say "locally flat"  and say "no matter how far you compare their distance between them"  ?

Local means nearby, not far away.

Parallel locally might be a situation like on earth, that longitudinal lines are perpendicular to the equator (as explained in the article) which is a condition for parallel lines in a flat geometry.

But the geometry of the earth's surface is curved, so these lines do not remain parallel.

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Posted (edited)
1 hour ago, swansont said:

Parallel locally might be a situation like on earth, that longitudinal lines are perpendicular to the equator (as explained in the article) which is a condition for parallel lines in a flat geometry.

But the geometry of the earth's surface is curved, so these lines do not remain parallel.

However you have completely missed my point.

'Parallel lines' that do not 'remain parallel'  are, by definition, not parallel.

Yes lines of longitude are a good example of 'parallel lines' that intersect somewhere.
(Yet lines of latitude do not intersect).

'Non intersection' has long been recognised as an inadequate definition of 'parallel', although it is often offered at primary/junior level geometry,

Equally the notion of maintaining a constant separation distance is untenable.

Furthermore we live in a 3 dimensional universe, where lines are either intersction or non intersecting.

Non intersecting lines are either parallel or skew.

Parallel lines are in a common plane, skew lines have no common plane.

Edited by studiot
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4 minutes ago, studiot said:

However you have completely missed my point.

'Parallel lines' that do not 'remain parallel'  are, by definition, not parallel.

Yes lines of longitude are a good example of 'parallel lines' that intersect somewhere.
(Yet lines of latitude do not intersect).

'Non intersection' has long been recognised as an inadequate definition of 'parallel', although it is often offered at primary/junior level geometry,

Equally the notion of maintaining a constant separation distance is untenable.

Furthermore we live in a 3 dimensional universe, where lines are either intersction or non intersecting.

Non intersecting lines are either parallel or skew.

Parallel lines are in a common plane, skew lines have no common plane.

I don't think I did miss the point.

This is an inherent problem of trying to address problems/questions with colloquial expressions and analogies, with the inevitable failure of rigor at some point along the way.

It's a cousin to the expression "All models are wrong. Some are useful." where one party is trying to convey the useful part and another party is focused on the "wrong" part.

So when someone offers up an example of parallel lines (or, more specifically, what one might assume are parallel lines, because they meet a criterion for parallel lines) and then you find out that they are not, the conclusion is you are not in a flat geometry.

Someone who is not clear on the concept might find that example helpful. Someone who does understand the concept pointing out that the lines are not actually parallel is probably not nearly as helpful.

This is reminiscent of when folks show up to ask a question about basic physics, and people jump in with some advanced physics that has no direct bearing on the discussion (e.g. pointing out how GR treats a problem when Newtonian gravity is sufficient to answer the question). Being right and being helpful are not necessarily the same thing.

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2 hours ago, studiot said:

Is  "Parallel lines stay parallel" not a tautology ?

More or less. But I think I should wait for MPMin's reaction. You see, if you introduce too technical details, there is the risk of a 'conceptual overload'. Either MPMin understands what I mean, or (s)he asks more detailed questions him (her?) self.

You see, sometimes my toes curve, not due to space curvature, but because I see that a curious beginner gets confronted with all kinds of technical details that he cannot see through. So this is a didactic principle of mine: try to connect as much as possible to the level of understanding that the questioner has. This risk is exactly shown here. You and Swansont get in a detailed discussion, that might just demotivate the questioner. I would always try to avoid that. And if the technical discussion is interesting enough, one could start a new thread.

In the end, MPMin wanted to know if the (none)-infinity of the universe has something to with its (none-)flatness. I think I did sufficiently reacted on that question.

1 minute ago, swansont said:

Being right and being helpful are not necessarily the same thing.

Exactly.

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1 hour ago, Eise said:

In the end, MPMin wanted to know if the (none)-infinity of the universe has something to with its (none-)flatness. I think I did sufficiently reacted on that question.

Did he ?

I wonder if the question was prompted by the cant that is so often taught to juniors near the beginning of geometry.

"Parallel lines meet at infinity"

MPMin did not actually ask his question in your terms; he asked if an infinite universe was an (inevitable) consequence of flatness.
I don't see anywhere that he mentioned parallel lines, but if he somehow equated flatness with parallelism then it is easy to see how this is a very good question, as iNow said in his post.

Nor did MPMin actually indicate if he thinks the universe is finite or infinite.

1 hour ago, Eise said:

You see, sometimes my toes curve, not due to space curvature, but because I see that a curious beginner gets confronted with all kinds of technical details that he cannot see through. So this is a didactic principle of mine: try to connect as much as possible to the level of understanding that the questioner has. This risk is exactly shown here. You and Swansont get in a detailed discussion, that might just demotivate the questioner. I would always try to avoid that. And if the technical discussion is interesting enough, one could start a new thread.

Yes i fully agree and commend this approach, but how much of a beginner is MPMin ?

I look back at that long thread where he was introducing Swartzchild geometry, black holes and other heady stuff, whilst (some) other members were throwing the vector calculus version of Maxwell at him and I offered a few basic comments along the lines you indicated.
For my pains MPMin and I exchanged the following

On 9/26/2019 at 5:05 PM, MPMin said:
On 9/26/2019 at 10:16 AM, studiot said:

Does this help ?

Yes it does

thank you

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Posted (edited)
17 hours ago, MPMin said:

I don’t want to go off track understanding time but I think I’m getting confused over the ‘universe is flat’ perspective. Does the universe being ‘flat’ just mean it is infinite in every direction?

Scientific probes such as WMAP have shown the universe to be flat with small error bars...flat as others have noted to mean that two beams of light emmitted parallel to each other, will remain parallel. A flat universe also denotes an infinite universe. But of course while any error bars remain, we can never really be sure that the universe is really, totally flat and as a consequence infinite. The flatness that our probes measure, may simply be part of a much larger curvature.

The other topological alternatives are an open universe or a closed universe, or possibly even some more exotic topology like a donut for example.

The curvature of the universe is generally denoted by omega, Ω...when Ω equals exactly 1, then the universe is truly flat. That of course cannot really be known other then for our observable universe.

11 hours ago, Eise said:

Just take care that the universe seems to be flat on average. Locally, due to mass and energy, the flat universe can be curved.

Bingo and what I was trying to convey.

Which of course is why we can not determine whether the universe is finite or infinite.

Edited by beecee
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As we can't seem to agree on the concept of 'parallel lines', let's just say that a triangle will have angles that add up to 180o in a 'flat' universe, or alternatively, the Pythagoras theorem has equivalent unit multipliers for each of the three squared terms ( this is also useful for higher dimensional spaces ).

As to whether a 'flat' universe is infinite ...
Of course.
If it wasn't, it would need to have an 'edge'.
( and that is non-sensical )

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1 hour ago, MigL said:

As we can't seem to agree on the concept of 'parallel lines', let's just say that a triangle will have angles that add up to 180o ..

What would be the greatest number of degrees that a triangle could have in a (closed?) curved space?

Is it 360?

Is it always 180 if the triangle is small enough and 360 if the triangle is large enough?

And what is the smallest number of degrees that a triangle can have in a (open?) curved space?

Zero?

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For the surface of a sphere ( positive curvature ), 360o is correct; formad by two right angles at the equator, and a 180o at the pole.
You can see that any angle at the pole would make the total greater than 180o, as all equatorial angles are 90o.
I'm not sure about higher dimensional positively curved spaces.

If by ( open ) curved space, you mean a negatively curved, saddle shaped space, then the minimum total angles would approach zero.

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16 minutes ago, MigL said:

For the surface of a sphere ( positive curvature ), 360o is correct; formad by two right angles at the equator, and a 180o at the pole.
You can see that any angle at the pole would make the total greater than 180o, as all equatorial angles are 90o.
I'm not sure about higher dimensional positively curved spaces.

If by ( open ) curved space, you mean a negatively curved, saddle shaped space, then the minimum total angles would approach zero.

A coincidence/connection  that 360 degrees is the same as the number of degrees that a radius of a circle completes in a complete revolution ? (or just the length of the perimeter of a circle)

(that was why I guessed 360)

Yes ,"open space" seems to  be another description of  negatively curved spaces.

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Posted (edited)

It approaches 360o.
Thae sphere would have to be infinitely large for the pole angle to be 180o.

So, no connection, I would think.

Edited by MigL
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Sorry Geordief.
I was sleepy last night and not thinking clearly.

Of course it is relaed to the 360o of a circle.o
The maximal triangle would be two half circumferences, in orthogonal planes.

And disregard what I said about 'approaching 360o, I was 'looking at the wrong plane; it would be exactly 360o ( for the maximal triangle ).

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The following quite lengthy and informative Wiki article is excellent imo......

a small sample extract.....

"The exact shape is still a matter of debate in physical cosmology, but experimental data from various independent sources (WMAP, BOOMERanG, and Planck for example) confirm that the universe is flat with only a 0.4% margin of error.[4][5][6] On the other hand, any non-zero curvature is possible for a sufficiently large curved universe (analogously to how a small portion of a sphere can look flat)"

and an image....

and finally towards the end...

"One of the presently unanswered questions about the universe is whether it is infinite or finite in extent".

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3 hours ago, MigL said:

Sorry Geordief.
I was sleepy last night and not thinking clearly.

Of course it is relaed to the 360o of a circle.o
The maximal triangle would be two half circumferences, in orthogonal planes.

And disregard what I said about 'approaching 360o, I was 'looking at the wrong plane; it would be exactly 360o ( for the maximal triangle ).

OK

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On 5/18/2021 at 11:30 PM, Eise said:

More or less. But I think I should wait for MPMin's reaction. You see, if you introduce too technical details, there is the risk of a 'conceptual overload'. Either MPMin understands what I mean, or (s)he asks more detailed questions him (her?) self.

Thank you all for reacting to my questions.

My reaction is firstly, it was not my intention to start a new thread. The subject of ‘flat universe’ was raised in my previous thread and I asked for clarification. I do however appreciate this new thread to further expand my learning.

On the subject of the universe with reference to parallel lines, there seems to be some definitive wriggle room due to colloquialisms. My understanding of parallel lines is that two parallel lines do not intersect or diverge on a two dimensional plane. However, when considering parallel lines in a three dimensional context then the parallel lines require another qualifying feature to contextualise the lines for the purpose of the intended discussion. The word ‘straight’ must accompany the term parallel when describing the lines in the context of converging or diverging when travelling through the three dimensional universe. I’d argue that lines following the contour of a sphere may be parallel in some respects but they can not be considered as being straight lines and as such are out of context when considering ‘parallel straight’ lines travelling through the universe.

My understanding of ‘parallel straight lines’ (or ‘straight parallel lines’ grammar aside) is that they will never intersect or diverge unless acted upon by an external force or influence. Perhaps the whole is the universe flat debate is just a confusion of the terms with reference to geodesic lines because if the lines were truly flat and straight then there’s no reason for them to intersect or diverge unless they weren’t truly straight and parallel to begin with just like the lines of longitude and latitude are not truly straight and parallel. To my understanding, the key issue is the lines must be straight and parallel simultaneously.

I would therefore deduce that if two lines were truly straight and parallel and never intersected or diverged (unless acted upon) then the universe must be infinite.

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Posted (edited)
2 hours ago, MPMin said:

Thank you all for reacting to my questions.

My reaction is firstly, it was not my intention to start a new thread. The subject of ‘flat universe’ was raised in my previous thread and I asked for clarification. I do however appreciate this new thread to further expand my learning.

On the subject of the universe with reference to parallel lines, there seems to be some definitive wriggle room due to colloquialisms. My understanding of parallel lines is that two parallel lines do not intersect or diverge on a two dimensional plane. However, when considering parallel lines in a three dimensional context then the parallel lines require another qualifying feature to contextualise the lines for the purpose of the intended discussion. The word ‘straight’ must accompany the term parallel when describing the lines in the context of converging or diverging when travelling through the three dimensional universe. I’d argue that lines following the contour of a sphere may be parallel in some respects but they can not be considered as being straight lines and as such are out of context when considering ‘parallel straight’ lines travelling through the universe.

My understanding of ‘parallel straight lines’ (or ‘straight parallel lines’ grammar aside) is that they will never intersect or diverge unless acted upon by an external force or influence. Perhaps the whole is the universe flat debate is just a confusion of the terms with reference to geodesic lines because if the lines were truly flat and straight then there’s no reason for them to intersect or diverge unless they weren’t truly straight and parallel to begin with just like the lines of longitude and latitude are not truly straight and parallel. To my understanding, the key issue is the lines must be straight and parallel simultaneously.

I would therefore deduce that if two lines were truly straight and parallel and never intersected or diverged (unless acted upon) then the universe must be infinite.

A pretty good summary for one who did not introduce parallel lines. +1

As ever there is more to this subject than first meets the eye.

Take a sheet of paper.

Draw a series of parallel lines on a sheet of paper or use lined paper.

Now roll the lined paper into a cylinder so that all the lines run along axially the surface and do not form loops around it.

The lines are parallel but-

You can pick a plane containing any two lines, but never three or more lines.

In the other hand, if you rolled the paper into a cylinder the other way then all the loops would be parallel, although no two are in the same plane!

A surface with this property is called a developable surface or a ruled surface.

You are also right about terminology and this thread was one I had in mind when I started another one in the Philosophy section that has bearing on your question.

My sheet of paper does not extend to infinity.

Mathematically the lines drawn on it do.

But they are 'mathematically' superimposed on the paper.

There are difference between maths and physics about the meaning of some subjects such as space, 'the universe', that need to be borne in mind.

Edited by studiot

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